- Email: [email protected]

CK-12 Foundation March 18, 2010

CK-12 Foundation is a non-proﬁt organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, webbased collaborative model termed the “FlexBook,” CK-12 intends to pioneer the generation and distribution of high quality educational content that will serve both as core text as well as provide an adaptive environment for learning. Copyright ©2009 CK-12 Foundation This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Contents

1 Functions, Limits, and Continuity

7

1.1

Equations and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2

Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.3

Models and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

1.4

The Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

1.5

Finding Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

1.6

Evaluating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

1.7

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

1.8

Inﬁnite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

2 Derivatives

89

2.1

Tangent Lines and Rates of Change . . . . . . . . . . . . . . . . . . . . . . .

89

2.2

The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

2.3

Techniques of Diﬀerentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.4

Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 113

2.5

The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.6

Implicit Diﬀerentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.7

Linearization and Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . 128

3 Applications of Derivatives

137

3.1

Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.2

Extrema and the Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . 146

3

www.ck12.org

3.3

The First Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.4

The Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

3.5

Limits at Inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.6

Analyzing the Graph of a Function . . . . . . . . . . . . . . . . . . . . . . . 176

3.7

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

3.8

Approximation Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4 Integration

205

4.1

Indeﬁnite Integrals Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

4.2

The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4.3

The Area Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.4

Deﬁnite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

4.5

Evaluating Deﬁnite Integrals

4.6

The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . 230

4.7

Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.8

Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

. . . . . . . . . . . . . . . . . . . . . . . . . . 225

5 Applications of Deﬁnite Integrals

251

5.1

Area Between Two Curves

. . . . . . . . . . . . . . . . . . . . . . . . . . . 251

5.2

Volumes

5.3

The Length of a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 278

5.4

Area of a Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 282

5.5

Applications from Physics, Engineering, and Statistics . . . . . . . . . . . . 288

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6 Transcendental Functions

307

6.1

Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

6.2

Exponential and Logarithmic Functions

6.3

Diﬀerentiation and Integration of Logarithmic and Exponential Functions

6.4

Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . 330

6.5

Derivatives and Integrals Involving Inverse Trigonometric Functions . . . . . 340

www.ck12.org

4

. . . . . . . . . . . . . . . . . . . . 315 . 320

6.6

L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7 Integration Techniques

351

7.1

Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

7.2

Integration By Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

7.3

Integration by Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 368

7.4

Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

7.5

Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

7.6

Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

7.7

Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 399

8 Inﬁnite Series

407

8.1

Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.2

Inﬁnite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

8.3

Series Without Negative Terms . . . . . . . . . . . . . . . . . . . . . . . . . 433

8.4

Series With Odd or Even Negative Terms . . . . . . . . . . . . . . . . . . . 440

8.5

Ratio Test, Root Test, and Summary of Tests . . . . . . . . . . . . . . . . . 446

8.6

Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

8.7

Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

8.8

Calculations with Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

5

www.ck12.org

www.ck12.org

6

Chapter 1 Functions, Limits, and Continuity 1.1

Equations and Graphs

Learning Objectives A student will be able to:

• • • •

Find solutions of graphs of equations. Find key properties of graphs of equations including intercepts and symmetry. Find points of intersections of two equations. Interpret graphs as models.

Introduction In this lesson we will review what you have learned in previous classes about mathematical equations of relationships and corresponding graphical representations and how these enable us to address a range of mathematical applications. We will review key properties of mathematical relationships that will allow us to solve a variety of problems. We will examine examples of how equations and graphs can be used to model real-life situations. Let’s begin our discussion with some examples of algebraic equations: Example 1: y = x2 The equation has ordered pairs of numbers (x, y) as solutions. Recall that a particular pair of numbers is a solution if direct substitution of the x and y values into the original equation yields a true equation statement. In this example, several solutions can be seen in the following table:

7

www.ck12.org

y = x2 + 2x − 1 7 2 −1 −2 −1 2 7 14

x −4 −3 −2 −1 0 1 2 3

We can graphically represent the relationships in a rectangular coordinate system, taking the x as the horizontal axis and the y as the vertical axis. Once we plot the individual solutions, we can draw the curve through the points to get a sketch of the graph of the relationship:

We call this shape a parabola and every quadratic function, f (x) = ax2 + bx + c, a ̸= 0 has a parabola-shaped graph. Let’s recall how we analytically ﬁnd the key points on the parabola. The vertex will be the lowest point, (−1, −2). In general, the vertex is located at the point (b/2a, f (b/2a)). We then can identify points crossing the x and y axes. These are called the intercepts of the equation. The y−intercept is found by setting x = 0 in the equation, and then solving for y as follows: y = 02 + 2(0) − 1 = −1. The y−intercept is located at (0, −1). The x−intercept is found by setting y = 0 in the equation, and solving for x as follows: www.ck12.org

8

0 = x2 + 2x − 1 Using the √ quadratic formula, √ we ﬁnd that x = −1 ± (−1 − 2, 0) and (−1 + 2, 0).

√

2. The x−intercepts are located at

Finally, recall that we deﬁned the symmetry of a graph. We noted examples of vertical and horizontal line symmetry as well as symmetry about particular points. For the current example, we note that the graph has symmetry in the vertical line x = −1. The graph with all of its key characteristics is summarized below:

Let’s look at a couple of more examples. Example 2: Here are some other examples of equations with their corresponding graphs: y = 2x + 3

9

www.ck12.org

x2 + y 2 = 4

y = x3 − 9x

www.ck12.org

10

Example 3: We recall the ﬁrst equation as linear so that its graph is a straight line. Can you determine the intercepts? Solution: x−intercept at (−3/2, 0) and y−intercept at (0, 3). Example 4: We recall from pre-calculus that the second equation is that of a circle with center (0, 0) and radius r = 2. Can you show analytically that the radius is 2 Solution: Find the four intercepts, by setting x = 0 and solving for y, and then setting y = 0 and solving for x. Example 5: The third equation is an example of a polynomial relationship. Can you ﬁnd the intercepts analytically? Solution: We can ﬁnd the x−intercepts analytically by setting y = 0 and solving for x. So, we have

11

www.ck12.org

x3 − 9x = 0 x(x2 − 9) = 0 x(x − 3)(x + 3) = 0 x = 0, x = −3, x = 3. So the x−intercepts are located at (−3, 0), (0, 0), and (3, 0). Note that (0, 0) is also the y−intercept. The y−intercepts can be found by setting x = 0. So, we have

x3 − 9x = y (0)3 − 9(0) = y y = 0. Sometimes we wish to look at pairs of equations and examine where they have common solutions. Consider the linear and quadratic graphs of the previous examples. We can sketch them on the same axes:

We can see that the graphs intersect at two points. It turns out that we can solve the problem of ﬁnding the points of intersections analytically and also by using our graphing calculator. Let’s review each method. Analytical Solution www.ck12.org

12

Since the points of intersection are on each graph, we can use substitution, setting the general y−coordinates equal to each other, and solving for x.

2x + 3 = x2 + 2x − 1 0 = x2 − 4 x = 2, x = −2.

We substitute each value of x into one of the original equations and ﬁnd the points of intersections at (−2, −1) and (2, 7). Graphing Calculator Solution Once we have entered the relationships on the Y= menu, we press 2nd [CALC] and choose #5 Intersection from the menu. We then are prompted with a cursor by the calculator to indicate which two graphs we want to work with. Respond to the next prompt by pressing the left or right arrows to move the cursor near one of the points of intersection and press [ENTER]. Repeat these steps to ﬁnd the location of the second point. We can use equations and graphs to model real-life situations. Consider the following problem. Example 6: Linear Modeling The cost to ride the commuter train in Chicago is $2. Commuters have the option of buying a monthly coupon book costing $5 that allows them to ride the train for $1.5 on each trip. Is this a good deal for someone who commutes every day to and from work on the train? Solution: We can represent the cost of the two situations, using the linear equations and the graphs as follows:

C1 (x) = 2x C2 (x) = 1.5x + 5

13

www.ck12.org

As before, we can ﬁnd the point of intersection of the lines, or in this case, the break-even value in terms of days, by solving the equation:

C1 (x) = C2 (x) 2x = 1.5x + 5 x = 10. So, even though it costs more to begin with, after 10 days the cost of the coupon book pays oﬀ and from that point on, the cost is less than for those riders who did not purchase the coupon book. Example 7: Non-Linear Modeling The cost of disability beneﬁts in the Social Security program for the years 2000 - 2005 can be modeled as a quadratic function. The formula

Y = −0.5x2 + 2x + 4 indicates the number of people Y , in millions, receiving Disability Beneﬁts x years after 2000. In what year did the greatest number of people receive beneﬁts? How many people received beneﬁts in that year? Solution: www.ck12.org

14

We can represent the graph of the relationship using our graphing calculator.

The vertex is the maximum point on the graph and is located at (2, 6). Hence in year 2002 a total of 6 million people received beneﬁts.

Lesson Summary 1. Reviewed graphs of equations 2. Reviewed how to ﬁnd the intercepts of a graph of an equation and to ﬁnd symmetry in the graph 3. Reviewed how relationships can be used as models of real-life phenomena 4. Reviewed how to solve problems that involve graphs and relationships

Review Questions In each of problems 1 - 4, ﬁnd a pair of solutions of the equation, the intercepts of the graph, and determine if the graph has symmetry.

1. 2x − 3y = 5

15

www.ck12.org

2. 3x2 − y = 5

3. y = x3 − x

www.ck12.org

16

4. y = x3 + x2 − 6x

5. Once a car is driven oﬀ of the dealership lot, it loses a signiﬁcant amount of its resale value. The graph below shows the depreciated value of a BMW versus that of a Chevy after t years. Which of the following statements is the best conclusion about the data?

17

www.ck12.org

(a) You should buy a BMW because they are better cars. (b) BMWs appear to retain their value better than Chevys. (c) The value of each car will eventually be $0. 6. Which of the following graphs is a more realistic representation of the depreciation of cars.

7. A rectangular swimming pool has length that is 25 yards greater than its width. (a) Give the area enclosed by the pool as a function of its width. (b) Find the dimensions of the pool if it encloses an area of 264 square yards. 8. Suppose you purchased a car in 2004 for $18, 000. You have just found out that the current year 2008 value of your car is $8, 500. Assuming that the rate of depreciation of the car is constant, ﬁnd a formula that shows changing value of the car from 2004 to 2008. 9. For problem #8, in what year will the value of the vehicle be less than $1, 400? 10. For problem #8, explain why using a constant rate of change for depreciation may not be the best way to model depreciation.

Review Answers 1. (1, −1) and (4, 1) are two solutions. The intercepts are located at (0, 5/3) and (5/2, 0). We have a linear relationship between x and y, so its graph can be sketched as the line passing through any two solutions. 2 2. by solving for y, we have y = ( 3x√ − 5,) so two solutions are (−1, −2) and (1, −2). The x−intercepts are located at ±

www.ck12.org

5 ,0 3

and the y−intercept is located at (0, −5). The

18

3.

4.

5. 6. 7. 8. 9. 10.

1.2

graph is symmetric in the y−axis. Using your graphing calculator, enter the relationship on the Y= menu. Viewing a table of points, we see many solutions, say (2, 6) and (−2, −6), and the intercepts at (0, 0), (−1, 0) and (1, 0). By inspection we see that the graph is symmetric about the origin. Using your graphing calculator, enter the relationship on the Y= menu. Viewing a table of points, we see many solutions, say (2, 0), and (−1, 6), and the intercepts located at (0, 0), (−3, 0), and (2, 0). By inspection we see that the graph does not have any symmetry. b. c. because you would expect (1) a decline as soon as you bought the car, and (2) the value to be declining more gradually after the initial drop. (a) A(w) = w2 + 25w. (b) The pool has area 264 when w = 8, l = 33. The rate of change will be (−9500/4) = −2375. The formula will be y = −2375x + 18000. At the time x = 8, or equivalently in the year 2112, the car will be valued at $1375. A linear model may not be the best function to model depreciation because the graph of the function decreases as time increases; hence at some point the value will take on negative real number values, an impossible situation for the value of real goods and products.

Relations and Functions

Learning Objectives A student will be able to: • • • • • • •

Identify functions from various relationships. Review function notation. Determine domains and ranges of particular functions. Identify key properties of some basic functions. Sketch graphs of basic functions. Sketch variations of basic functions using transformations. Compose functions.

Introduction In our last lesson we examined a variety of mathematical equations that expressed mathematical relationships. In this lesson we will focus on a particular class of relationships called

19

www.ck12.org

functions, and examine their key properties. We will then review how to sketch graphs of some basic functions that we will revisit later in this class. Finally, we will examine a way to combine functions that will be important as we develop the key concepts of calculus. Let’s begin our discussion by reviewing four types of equations we examined in our last lesson. Example 1: y = x2

y = 2x + 3

www.ck12.org

20

x2 + y 2 = 4

y = x3 − 9x

21

www.ck12.org

Of these, the circle has a quality that the other graphs do not share. Do you know what it is? Solution: The circle’s graph includes points x−value has two points associated with √ where a particular √ it; for example, the points (1, 3) and (1, − 3) are both solutions to the equation x2 +y 2 = 4. For each of the other relationships, a particular x−value has exactly one y−value associated with it.) The relationships that satisfy the condition that for each x−value there is a unique y−value are called functions. Note that we could have determined whether the relationship satisﬁed this condition by a graphical test, the vertical line test. Recall the relationships of the circle, which is not a function. Let’s compare it with the parabola, which is a function.

If we draw vertical lines through the graphs as indicated, we see that the condition of a particular x−value having exactly one y−value associated with it is equivalent to having at www.ck12.org

22

most one point of intersection with any vertical line. The lines on the circle intersect the graph in more than one point, while the lines drawn on the parabola intersect the graph in exactly one point. So this vertical line test is a quick and easy way to check whether or not a graph describes a function. We want to examine properties of functions such as function notation, their domain and range (the sets of x and y values that deﬁne the function), graph sketching techniques, how we can combine functions to get new functions, and also survey some of the basic functions that we will deal with throughout the rest of this book. Let’s start with the notation we use to describe functions. Consider the example of the linear function y = 2x + 3. We could also describe the function using the symbol f (x) and read as ”f of x” to indicate the y−value of the function for a particular x−value. In particular, for this function we would write f (x) = 2x + 3 and indicate the value of the function at a particular value, say x = 4 as f (4) and ﬁnd its value as follows: f (4) = 2(4) + 3 = 11. This statement corresponds to the solution (4, 11) as a point on the graph of the function. It is read, ”f of x is 11.” We can now begin to discuss the properties of functions, starting with the domain and the range of a function. The domain refers to the set of x−values that are inputs in the function, while the range refers to the set of y−values that the function takes on. Recall our examples of functions: Linear Function g(x) = 2x + 3 Quadratic Function f (x) = x2 Polynomial Function p(x) = x3 − 9x We ﬁrst note that we could insert any real number for an x−value and a well-deﬁned y−value would come out. Hence each function has the set of all real numbers as a domain and we indicate this in interval form as D : (−∞, ∞). Likewise we see that our graphs could extend up in a positive direction and down in a negative direction without end in either direction. Hence we see that the set of y−values, or the range, is the set of all real numbers R : (−∞, ∞). Example 2: Determine the domain and range of the function.

f (x) = 1/(x2 − 4). Solution: We note that the condition for each y−value is a fraction that includes an x term in the denominator. In deciding what set of x−values we can use, we need to exclude those values that make the denominator equal to 0. Why? (Answer: division by 0 is not deﬁned

23

www.ck12.org

for real numbers.) Hence the set of all permissible x−values, is all real numbers except for the numbers (2, −2), which yield division by zero. So on our graph we will not see any points that correspond to these x−values. It is more diﬀicult to ﬁnd the range, so let’s ﬁnd it by using the graphing calculator to produce the graph.

From the graph, we see that every y ̸= 0 value in (−∞, ∞) (or ”All real numbers”) is represented; hence the range of the function is {−∞, 0} ∪ {0, ∞}. This is because a fraction with a non-zero numerator never equals zero. Eight Basic Functions We now present some basic functions that we will work with throughout the course. We will provide a list of eight basic functions with their graphs and domains and ranges. We will then show some techniques that you can use to graph variations of these functions. Linear f (x) = x Domain = All reals Range = All reals

www.ck12.org

24

Square (Quadratic) f (x) = x2 Domain = All reals Range = {y ≥ 0}

Cube (Polynomial) f (x) = x3 Domain = All reals Range = All reals

25

www.ck12.org

Square Root √ f (x) = x Domain = {x ≥ 0} Range = {y ≥ 0}

www.ck12.org

26

Absolute Value f (x) = |x| Domain = All reals Range = {y ≥ 0}

Rational f (x) = 1/x Domain = {x ̸= 0} Range = {y ̸= 0}

27

www.ck12.org

Sine f (x) = sin x Domain = All reals Range = {−1 ≤ y ≤ 1}

www.ck12.org

28

Cosine f (x) = cos x Domain = All reals Range = {−1 ≤ y ≤ 1}

Graphing by Transformations Once we have the basic functions and each graph in our memory, we can easily sketch variations of these. In general, if we have f (x), and c is some constant value, then the graph of f (x − c) is just the graph of f (x) shifted c units to the right. Similarly, the graph of f (x + c) is just the graph of f (x) shifted c units to the left. Example 3: f (x) = x2

29

www.ck12.org

f (x) = (x − 2)2

In addition, we can shift graphs up and down. In general, if we have f (x), and c is some constant value, then the graph of f (x) + c is just the graph of f (x) shifted c units up on the y−axis. Similarly, the graph of f (x) − c is just the graph of f (x) shifted c units down on the y−axis. www.ck12.org

30

Example 4: √ f (x) = x

f (x) =

√

x+3

We can also ﬂip graphs in the x−axis by multiplying by a negative coeﬀicient.

31

www.ck12.org

f (x) = x2

f (x) = −x2

www.ck12.org

32

Finally, we can combine these transformations into a single example as follows. Example 5: f (x) = −(x − 2)2 + 3. The graph will be generated by taking f (x) = x2 , ﬂipping in the y−axis, and moving it two units to the right and up three units.

33

www.ck12.org

Function Composition The last topic for this lesson involves a way to combine functions called function composition. Composition of functions enables us to consider the eﬀects of one function followed by another. Our last example of graphing by transformations provides a nice illustration. We can think of the ﬁnal graph as the eﬀect of taking the following steps:

x → −(x − 2)2 → −(x − 2)2 + 3 We can think of it as the application of two functions. First, g(x) takes x to −(x − 2)2 and then we apply a second function, f (x) to those y−values, with the second function adding +3 to each output. We would write the functions as f (g(x)) = −(x − 2)2 + 3 where g(x) = −(x − 2)2 and f (x) = x + 3. We call this operation the composing of f with g and use notation f ◦ g. Note that in this example, f ◦ g ̸= g ◦ f. Verify this fact by computing g ◦ f right now. (Note: this fact can be veriﬁed algebraically, by showing that the expressions f ◦ g and g ◦ f diﬀer, or by showing that the diﬀerent function decompositions are not equal for a speciﬁc value.)

Lesson Summary 1. Learned to identify functions from various relationships. 2. Reviewed the use of function notation. www.ck12.org

34

3. 4. 5. 6. 7.

Determined domains and ranges of particular functions. Identiﬁed key properties of basic functions. Sketched graphs of basic functions. Sketched variations of basic functions using transformations. Learned to compose functions.

Review Questions In problems 1 - 2, determine if the relationship is a function. If it is a function, give the domain and range of the function. 1.

2.

In problems 3 - 5, determine the domain and range of the function and sketch the graph if no graph is provided.

35

www.ck12.org

2

3. f (x) = x3x √ 2 −1 4. y = −x + 3

5. f (x) = |2x − 3| − 2

In problems 6 - 8, sketch the graph using transformations of the graphs of basic functions. 6. f (x) = −(x + 2)2 + 5 www.ck12.org

36

1 +3 7. f (x) = − x−2 √ 8. y = − −x − 2 + 3 9. Find the composites, f ◦ g and g ◦ f for the following functions.

f (x) = −3x + 2, g(x) =

√

x

10. Find the composites, f ◦ g and g ◦ f for the following functions. f (x) = x2 , g(x) =

√

x

Review Answers 1. The relationship is a function. Domain is All Real Numbers and range = {−2 ≤ y ≤ 2}. 2. The relationship is not a function. 3. The domain is {x = ̸ −1, 1}. Use the graph and view the table of solutions to determine the range. Using your graphing calculator, enter the relationship on the Y= menu. Viewing the table shows that range {y > 3} ∪ {y ≤ 0}.

4. Domain = {x ≤ −3}, range = {y ≥ 0}. 5. This is the basic absolute value function shifted 3/2 units to the right and down two units. Domain is All Reals and range is {y ≥ −2}. 6. Reﬂect and shift the general quadratic function as indicated here:

37

www.ck12.org

7. Reﬂect and shift the general rational function as indicated here:

8. Reﬂect and shift the general radical function as indicated here:

www.ck12.org

38

√ √ 9. f ◦ g = −3 x + 2, g ◦ f = −3x + 2. 10. f ◦ g = x, g ◦ f = x; any functions where f ◦ g = g ◦ f = x; are called inverses; in this problem f and g are inverses of one another. Note that the domain for f ◦ g is restricted to only positive numbers and zero.

1.3

Models and Data

Learning Objectives A student will be able to: • • • •

Fit Fit Fit Fit

data data data data

to to to to

linear models. quadratic models. trigonometric models. exponential growth and decay models.

Introduction In our last lesson we examined functions and learned how to classify and sketch functions. In this lesson we will use some classic functions to model data. The lesson will be a set of examples of each of the models. For each, we will make extensive use of the graphing calculator.

39

www.ck12.org

Let’s do a quick review of how to model data on the graphing calculator. Enter Data in Lists Press [STAT] and then [EDIT] to access the lists, L1 - L6. View a Scatter Plot Press 2nd [STAT PLOT] and choose accordingly. Then press [WINDOW] to set the limits of the axes. Compute the Regression Equation Press [STAT] then choose [CALC] to access the regression equation menu. Choose the appropriate regression equation (Linear, Quad, Cubic, Exponential, Sine). Graph the Regression Equation Over Your Scatter Plot Go to Y=> [MENU] and clear equations. Press [VARS], then enter 5 and EQ and press [ENTER] (This series of entries will copy the regression equation to your Y = screen.) Press [GRAPH] to view the regression equation over your scatter plot Plotting and Regression in Excel You can also do regression in an Excel spreadsheet. To start, copy and paste the table of data into Excel. With the two columns highlighted, including the column headings, click on the Chart icon and select XY scatter. Accept the defaults until a graph appears. Select the graph, then click Chart, then Add Trendline. From the choices of trendlines choose Linear. Now let’s begin our survey of the various modeling situations. Linear Models For these kinds of situations, the data will be modeled by the classic linear equation y = mx + b. Our task will be to ﬁnd appropriate values of m and b for given data. Example 1: It is said that the height of a person is equal to his or her wingspan (the measurement from ﬁngertip to ﬁngertip when your arms are stretched horizontally). If this is true, we should be able to take a table of measurements, graph the measurements in an x−y coordinate system, and verify this relationship. What kind of graph would you expect to see? (Answer: You would expect to see the points on the line y = x.) Suppose you measure the height and wingspans of nine of your classmates and gather the following data. Use your graphing calculator to see if the following measurements ﬁt this linear model (the line y = x).

www.ck12.org

40

Table 1.1: Height (inches)

Wingspan (inches)

67 64 56 60 62 71 72 68 65

65 63 57 61 63 70 69 67 65

We observe that only one of the measurements has the condition that they are equal. Why aren’t more of the measurements equal to each other? (Answer: The data do not always conform to exact speciﬁcations of the model. For example, measurements tend to be loosely documented so there may be an error arising in the way that measurements were taken.) We enter the data in our calculator in L1 and L2. We then view a scatter plot. (Caution: note that the data ranges exceed the viewing window range of [−10, 10]. Change the window ranges accordingly to include all of the data, say [40, 80].) Here is the scatter plot:

Now let us compute the regression equation. Since we expect the data to be linear, we will choose the linear regression option from the menu. We get the equation y = .76x + 14. In general we will always wish to graph the regression equation over our data to see the goodness of ﬁt. Doing so yields the following graph, which was drawn with Excel:

41

www.ck12.org

Since our calculator will also allow for a variety of non-linear functions to be used as models, we can therefore examine quite a few real life situations. We will ﬁrst consider an example of quadratic modeling. Quadratic Models Example 2: The following table lists the number of Food Stamp recipients (in millions) for each year after 1990. http://www.fns.usda.gov/pd/fssummar.htm. Table 1.2: years after 1990

Participants

1 2 3 4 5 6 7 8 9 10

22.6 25.4 27.0 27.5 26.6 2.55 22.5 19.8 18.2 17.2

We enter the data in our calculator in L3 and L4 (that enables us to save the last example’s data). We then will view a scatter plot. Change the window ranges accordingly to include all of the data. Use [−2, 10] for x and [−2, 30] for y. www.ck12.org

42

Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests a quadratic model for the data, we will choose Quadratic Regression from the menu. We get the equation:

y = −0.30x2 + 2.38x + 21.67. Let’s graph the equation over our data. We see the following graph:

Trigonometric Models The following example shows how a trigonometric function can be used to model data. Example 3: With the skyrocketing cost of gasoline, more people have looked to mass transit as an option for getting around. The following table uses data from the American Public Transportation Association to show the number of mass transit trips (in billions) between 1992 and 2000. Table 1.3: year

Trips (billions)

1992 1993 1994 1995 1996 1997 1998 1999 2000

8.5 8.2 7.93 7.8 7.87 8.23 8.6 9.08 9.4

43

www.ck12.org

We enter the data in our calculator in L5 and L6. We then will view a scatter plot. Change the window ranges accordingly to include all of the data. Use [−2, 10] for both x and y ranges. Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests a sine model for the data, we will choose Sine Regression from the menu. We get the equation:

y = .9327 ∗ sin(.4681x + 2.8734) + 8.7358. Let us graph the equation over our data. We see the following graph:

This example suggests that the sine over time t is a function that is used in a variety of modeling situations. Caution: Although the ﬁt to the data appears quite good, do we really expect the number of trips to continue to go up and down in the future? Probably not. Here is what the graph looks like when projected an additional ten years:

Exponential Models Our last class of models involves exponential functions. Exponential models can be used to model growth and decay situations. Consider the following data about the declining number of farms for the years 1980 - 2005. Example 4: The number of dairy farms has been declining over the past 20+ years. The following table charts the decline:

www.ck12.org

44

Table 1.4: Year

Farms (thousands)

1980 1985 1990 1995 2000 2005

334 269 193 140 105 67

We enter the data in our calculator in L5 (again entering the years as 1, 2, 3...) and L6. We then will view a scatter plot. Change the window ranges accordingly to include all of the data. For the large y−values, choose the range [−50, 350] with a scale of 25. Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests an exponential model for the data, we will choose Exponential Regression from the menu. We get the equation: y = 490.6317 ∗ .7266x Let’s graph the equation over our data. We see the following graph:

In the homework we will practice using our calculator extensively to model data.

Lesson Summary 1. 2. 3. 4.

Fit Fit Fit Fit

data data data data

to to to to

linear models. quadratic models. trigonometric models. exponential growth and decay models.

45

www.ck12.org

Review Questions 1. Consider the following table of measurements of circular objects: (a) (b) (c) (d)

Make a scatter plot of the data. Based on your plot, which type of regression will you use? Find the line of best ﬁt. Comment on the values of m and b in the equation. Table 1.5:

Object

Diameter (cm)

Circumference (cm)

Glass Flashlight Aztec calendar Tylenol bottle Popcorn can Salt shaker Coﬀee canister Cat food bucket Dinner plate Ritz cracker

8.3 5.2 20.2 3.4 13 6.3 11.3 33.5 27.3 4.9

26.5 16.7 61.6 11.6 41.4 20.1 35.8 106.5 85.6 15.5

2. Manatees are large, gentle sea creatures that live along the Florida coast. Many manatees are killed or injured by power boats. Here are data on powerboat registrations (in thousands) and the number of manatees killed by boats in Florida from 1987 - 1997. (a) Make a scatter plot of the data. (b) Use linear regression to ﬁnd the line of best ﬁt. (c) Suppose in the year 2000, powerboat registrations increase to 700, 000. Predict how many manatees will be killed. Assume a linear model and ﬁnd the line of best ﬁt. Table 1.6: Year

Boats

Manatees killed

1987 1988 1989 1990 1991 1992

447 460 480 497 512 513

13 21 24 16 24 21

www.ck12.org

46

Table 1.6: (continued) Year

Boats

Manatees killed

1993 1994 1995 1996 1997

526 557 585 614 645

15 33 34 34 39

3. A passage in Gulliver’s Travels states that the measurement of “Twice around the wrist is once around the neck.” The table below contains the wrist and neck measurements of 10 people. (a) Make a scatter plot of the data. (b) Find the line of best ﬁt and comment on the accuracy of the quote from the book. (c) Predict the distance around the neck of Gulliver if the distance around his wrist is found to be 52 cm. Table 1.7: Wrist (cm)

Neck (cm)

17.9 16 16.5 15.9 17 17.3 16.8 17.3 17.7 16.9

39.5 32.5 34.7 32 33.3 32.6 33 31.6 35 34

4. The following table gives women’s average percentage of men’s salaries for the same jobs for each 5-year period from 1960 - 2005. (a) (b) (c) (d)

Make a scatter plot of the data. Based on your sketch, should you use a linear or quadratic model for the data? Find a model for the data. Can you explain why the data seems to dip at ﬁrst and then grow?

47

www.ck12.org

Table 1.8: Year

Percentage

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

42 36 30 37 41 42 48 55 58 60

5. Based on the model for the previous problem, when will women make as much as men? Is your answer a realistic prediction? 6. The average price of a gallon of gas for selected years from 1975 - 2008 is given in the following table: (a) Make a scatter plot of the data. (b) Based on your sketch, should you use a linear, quadratic, or cubic model for the data? (c) Find a model for the data. (d) If gas continues to rise at this rate, predict the price of gas in the year 2012. Table 1.9: Year

Cost

1975 1976 1981 1985 1995 2005 2008

1 1.75 2 2.57 2.45 2.75 3.45

7. For the previous problem, use a linear model to analyze the situation. Does the linear method provide a better estimate for the predicted cost for the year 2011? Why or why not? 8. Suppose that you place $1, 000 in a bank account where it grows exponentially at a rate of 12% continuously over the course of ﬁve years. The table below shows the amount www.ck12.org

48

of money you have at the end of each year. (a) Find the exponential model. (b) In what year will you triple your original amount? Table 1.10: Year

Amount

0 1 2 3 4 5 6

1000 1127.50 1271.24 1433.33 1616.07 1822.11 2054.43 9. Suppose that in the previous problem, you started with $3, 000 but maintained the same interest rate. (a) Give a formula for the exponential model. (Hint: note the coeﬀicient and exponent in the previous answer!) (b) How long will it take for the initial amount, $3, 000, to triple? Explain your answer.

10. The following table gives the average daily temperature for Indianapolis, Indiana for each month of the year: (a) Construct a scatter plot of the data. (b) Find the sine model for the data. Table 1.11: Month

Avg Temp (F)

Jan Feb March April May June July Aug Sept Oct

22 26.3 37.8 51 61.7 75.3 78.5 84.3 68.5 53.2

49

www.ck12.org

Table 1.11: (continued) Month

Avg Temp (F)

Nov Dec

38.7 26.6

Review Answers 1. (a) (b) (c) (d) 2.

3.

4.

5. 6.

7.

8. 9.

. Linear. y = 3.1334x + .3296. m is an estimate of π, and b should be zero but due to error in measurement it is not. (a) . (b) . (c) y = .120546x − 39.0465; about 46 manatees will be killed in the year 2000. Note: there were actually 81 manatees killed in the year 2000. (a) . (b) . (c) y = 2.0131x − 0.2634; 104.42 cm. (a) . (b) Quadratic (c) y = .4848x2 − 2.4545x + 39.7333. (d) It might be because the ﬁrst wave of women into the workforce tended to take whatever jobs they could ﬁnd without regard for salary. The data suggest that women will reach 100% in 2009; this is unrealistic based on current reports that women still lag far behind men in equal salaries for equal work. (a) . (b) Cubic. (c) y = .0277x3 − 0.3497x2 + 1.6203x − 0.3157. (d) $12.15. Linear y = 0.35x + 0.88; Predicted cost in 2012 is $4.73; it is hard to say which model works best but it seems that the use of a cubic model may overestimate the cost in the short term. (a) . (b) A = 1000 ∗ 2.7182.12t ; the amount will triple early in Year 9. (a) A = 3000 ∗ 2.7182.12t . (b) The amount will triple early in Year 9 as in the last problem because the exponential equations 3000 = 1000 ∗ 2.7182.12t and 9000 = 1000 ∗ 2.7182.12t both reduce to the same equation 3 = 2.7182.12t and hence have the same solution.

www.ck12.org

50

10. (a) . (b) y = 30.07 ∗ sin(.5196x − 2.1503) + 51.46.

1.4

The Calculus

Learning Objectives A student will be able to:

• Use linear approximations to study the limit process. • Compute approximations for the slope of tangent lines to a graph. • Introduce applications of diﬀerential calculus.

Introduction In this lesson we will begin our discussion of the key concepts of calculus. They involve a couple of basic situations that we will come back to time and again throughout the book. For each of these, we will make use of some basic ideas about how we can use straight lines to help approximate functions. Let’s start with an example of a simple function to illustrate each of the situations. Consider the quadratic function f (x) = x2 . We recall that its graph is a parabola. Let’s look at the point (1, 1) on the graph.

Suppose we magnify our picture and zoom in on the point (1, 1). The picture might look like this:

51

www.ck12.org

We note that the curve now looks very much like a straight line. If we were to overlay this view with a straight line that intersects the curve at (1, 1), our picture would look like this:

We can make the following observations. First, this line would appear to provide a good estimate of the value of f (x) for x−values very close to x = 1. Second, the approximations appear to be getting closer and closer to the actual vale of the function as we take points on the line closer and closer to the point (1, 1). This line is called the tangent line to f (x) at (1, 1). This is one of the basic situations that we will explore in calculus. Tangent Line to a Graph Continuing our discussion of the tangent line to f (x) at (1, 1), we next wish to ﬁnd the equation of the tangent line. We know that it passes though (1, 1), but we do not yet have enough information to generate its equation. What other information do we need? (Answer: The slope of the line.) Yes, we need to ﬁnd the slope of the line. We would be able to ﬁnd the slope if we knew a second point on the line. So let’s choose a point P on the line, very close to (1, 1). We www.ck12.org

52

can approximate the coordinates of P using the function f (x) = x2 ; hence P (x, x2 ). Recall that for points very close to (1, 1), the points on the line are close approximate points of the function. Using this approximation, we can compute the slope of the tangent as follows: m = (x2 − 1)/(x − 1) = x + 1 (Note: We choose points very close to (1, 1) but not the point itself, so x ̸= 1). In particular, for x = 1.25 we have P (1.25, 1.5625) and m = x+1 = 2.25. Hence the equation of the tangent line, in point slope form is y − 1 = 2.25(x − 1). We can keep getting closer to the actual value of the slope by taking P closer to (1, 1), or x closer and closer to x = 1, as in the following table:

P (x, y) (1.2, 1.44) (1.15, 1.3225) (1.1, 1.21) (1.05, 1.1025) (1.005, 1.010025) (1.0001, 1.00020001)

m 2.2 2.15 2.1 2.05 2.005 2.0001

As we get closer to (1, 1), we get closer to the actual slope of the tangent line, the value 2. We call the slope of the tangent line at the point (1, 1) the derivative of the function f (x) at the point (1, 1). Let’s make a couple of observations about this process. First, we can interpret the process graphically as ﬁnding secant lines from (1, 1) to other points on the graph. From the diagram we see a sequence of these secant lines and can observe how they begin to approximate the tangent line to √ the graph√at (1, 1). The diagram shows a pair of secant lines, joining (1, 1) with points ( 2, 2) and ( 3, 3).

53

www.ck12.org

Second, in examining the sequence of slopes of these secants, we are systematically observing approximate slopes of the function as point P gets closer to (1, 1). Finally, producing the table of slope values above was an inductive process in which we generated some data and then looked to deduce from our data the value to which the generated results tended. In this example, the slope values appear to approach the value 2. This process of ﬁnding how function values behave as we systematically get closer and closer to particular x−values is the process of ﬁnding limits. In the next lesson we will formally deﬁne this process and develop some eﬀicient ways for computing limits of functions. Applications of Diﬀerential Calculus Maximizing and Minimizing Functions Recall from Lesson 1.3 our example of modeling the number of Food Stamp recipients. The model was found to be y = −0.5x2 + 4x + 19 with graph as follows: (Use viewing window ranges of [−2, 14] on x and [−2, 30] on y)

www.ck12.org

54

We note that the function appears to attain a maximum value about an x−value somewhere around x = 4. Using the process from the previous example, what can we say about the tangent line to the graph for that x value that yields the maximum y value (the point at the top of the parabola)? (Answer: the tangent line will be horizontal, thus having a slope of 0.)

Hence we can use calculus to model situations where we wish to maximize or minimize a particular function. This process will be particularly important for looking at situations from business and industry where polynomial functions provide accurate models. Velocity of a Falling Object We can use diﬀerential calculus to investigate the velocity of a falling object. Galileo found that the distance traveled by a falling object was proportional to the square of the time it has been falling:

s(t) = 4.9t2 . The average velocity of a falling object from t = a to t = b is given by (s(b) − s(a))/(b − a). HW Problem #10 will give you an opportunity to explore this relationship. In our discussion, we saw how the study of tangent lines to functions yields rich information about functions. We now consider the second situation that arises in Calculus, the central problem of ﬁnding the area under the curve of a function f (x). Area Under a Curve First let’s describe what we mean when we refer to the area under a curve. Let’s reconsider our basic quadratic function f (x) = x2 . Suppose we are interested in ﬁnding the area under the curve from x = 0 to x = 1.

55

www.ck12.org

We see the cross-hatched region that lies between the graph and the x−axis. That is the area we wish to compute. As with approximating the slope of the tangent line to a function, we will use familiar linear methods to approximate the area. Then we will repeat the iterative process of ﬁnding better and better approximations. Can you think of any ways that you would be able to approximate the area? (Answer: One ideas is that we could compute the area of the square that has a corner at (1, 1) to be A = 1 and then take half to ﬁnd an area A = 1/2. This is one estimate of the area and it is actually a pretty good ﬁrst approximation.)

We will use a variation of this covering of the region with quadrilaterals to get better approximations. We will do so by dividing the x−interval from x = 0 to x = 1 into equal sub-intervals. Let’s start by using four such subintervals as indicated:

www.ck12.org

56

We now will construct four rectangles that will serve as the basis for our approximation of the area. The subintervals will serve as the width of the rectangles. We will take the length of each rectangle to be the maximum value of the function in the subinterval. Hence we get the following ﬁgure:

If we call the rectangles R1–R4, from left to right, then we have the areas

57

www.ck12.org

R1 = R2 = R3 = R4 = and R1 + R2 + R3 + R4 =

30 64

=

1 4 1 4 1 4 1 4

( ) 1 1 ∗f = , 4 64 ( ) 1 1 ∗f = , 2 16 ( ) 3 9 ∗f = , 4 64 1 ∗ f (1) = , 4

15 . 32

Note that this approximation is very close to our initial approximation of 1/2. However, since we took the maximum value of the function for a side of each rectangle, this process tends to overestimate the true value. We could have used the minimum value of the function in each sub-interval. Or we could have used the value of the function at the midpoint of each sub-interval. Can you see how we are going to improve our approximation using successive iterations like we did to approximate the slope of the tangent line? (Answer: we will sub-divide the interval from x = 0 to x = 1 into more and more sub-intervals, thus creating successively smaller and smaller rectangles to reﬁne our estimates.) Example 1: The following table shows the areas of the rectangles and their sum for rectangles having width w = 1/8. Table 1.12: Rectangle Ri

Area of Ri

R1 R2 R3 R4 R5 R6 R7 R8

1 512 4 512 9 512 16 512 25 512 36 512 49 512 64 512

∑ . This value is approximately equal to .3803. Hence, the approximation is A= Ri = 195 512 1432 now quite a bit less than .5. For sixteen rectangles, the value is 4096 which is approximately equal to .34. Can you guess what the true area will approach? (Answer: using our www.ck12.org

58

successive approximations, the area will approach the value 1/3.) We call this process of ﬁnding the area under a curve integration of f (x) over the interval [0, 1]. Applications of Integral Calculus We have not yet developed any computational machinery for computing derivatives and integrals so we will just state one popular application of integral calculus that relates the derivative and integrals of a function. Example 2: There are quite a few applications of calculus in business. One of these is the cost function C(x) of producing x items of a product. It can be shown that the derivative of the cost function that gives the slope of the tangent line is another function that that gives the cost to produce an additional unit of the product. This is called the marginal cost and is a very important piece of information for management to have. Conversely, if one knows the marginal cost as a function of x, then ﬁnding the area under the curve of the function will give back the cost function C(x).

Lesson Summary 1. 2. 3. 4.

We We We We

used linear approximations to study the limit process. computed approximations for the slope of tangent lines to a graph. analyzed applications of diﬀerential calculus. analyzed applications of integral calculus.

Review Questions 1. For the function f (x) = x2 approximate the slope of the tangent line to the graph at the point (3, 9). (a) Use the following set of x−values to generate the sequence of secant line slopes: x = 2.9, 2.95, 2.975, 2.995, 2.999. (b) What value does the sequence of slopes approach? 2. Consider the function f (x) = x2 . (a) For what values of x would you expect the slope of the tangent line to be negative? (b) For what value of x would you expect the tangent line to have slope m = 0? (c) Give an example of a function that has two diﬀerent horizontal tangent lines? 3. Consider the function p(x) = x3 − x. Generate the graph of p(x) using your calculator. (a) Approximate the slope of the tangent line to the graph at the point (2, 6). Use the following set of x−values to generate the sequence of secant line slopes. x =

59

www.ck12.org

2.1, 2.05, 2.005, 2.001, 2.0001. (b) For what values of x do the tangent lines appear to have slope of 0? (Hint: Use the calculate function in your calculator to approximate the x−values.) (c) For what values of x do the tangent lines appear to have positive slope? (d) For what values of x do the tangent lines appear to have negative slope? 4. The cost of producing x Hi − F i stereo receivers by Yamaha each week is modeled by the following function: C(x) = 850 + 200x − .3x2 . (a) Generate the graph of C(x) using your calculator. (Hint: Change your viewing window to reﬂect the high y values.) (b) For what number of units will the function be maximized? (c) Estimate the slope of the tangent line at x = 200, 300, 400. (d) Where is marginal cost positive? 5. Find the area under the curve of f (x) = x2 from x = 1 to x = 3. Use a rectangle method that uses the minimum value of the function within sub-intervals. Produce the approximation for each case of the subinterval cases. (a) (b) (c) (d)

four sub-intervals. eight sub-intervals. Repeat part a. using a Mid-Point Value of the function within each sub-interval. Which of the answers in a. - c. provide the best estimate of the actual area?

6. Consider the function p(x) = −x3 + 4x. (a) Find the area under the curve from x = 0 to x = 1. (b) Can you ﬁnd the area under the curve from x = −1 to x = 0. Why or why not? What is problematic for this computation? √ 7. Find the area under the curve of f (x) = x from x = 1 to x = 4. Use the Max Value rectangle method with six sub-intervals to compute the area. 8. The Eiﬀel Tower is 320 meters high. Suppose that you drop a ball oﬀ the top of the tower. The distance that it falls is a function of time and is given by s(t) = 4.9t2 . Find the velocity of the ball after 4 seconds. (Hint: the average velocity for a time interval is average velocity = change in distance/change in time. Investigate the average velocity for t intervals close to t = 4 such as 3.9 ≤ t ≤ 4 and closer and see if a pattern is evident.)

Review Answers 1. (a) . (b) m = 6 www.ck12.org

60

2. (a) For x < 0 (b) At x = 0 the tangent line is horizontal and thus has slope of 0. (c) Many diﬀerent examples; for instance, a polynomial function such as p(x) = x3 − 4x. 3. (a) Slope tends toward m = 11. (b) x = ±57. (c) x < −.57, x > .57. (d) −.57 < x < .57. 4. (a) . (b) x = 500. (c) . (d) x < 500. 5. (a) . (b) . (c) . (d) . 6. (a) . (b) The graph drops below the x−axis into the third quadrant. Hence we are not ﬁnding the area below the curve but actually the area between the curve and the x-axis. But note that the curve is symmetric about the origin. Hence the region from x = −1 to x = 0 will have the same area as the region from x = 0 to x = 1. 7. . m 8. 39.2 sec .

1.5

Finding Limits

Learning Objectives A student will be able to: • • • •

Find the limit of a function numerically. Find the limit of a function using a graph. Identify cases when limits do not exist. Use the formal deﬁnition of a limit to solve limit problems.

Introduction In this lesson we will continue our discussion of the limiting process we introduced in Lesson 1.4. We will examine numerical and graphical techniques to ﬁnd limits where they exist and also to examine examples where limits do not exist. We will conclude the lesson with a more precise deﬁnition of limits.

61

www.ck12.org

Let’s start with the notation that we will use to denote limits. We indicate the limit of a function as the x values approach a particular value of x, say a, as

lim f (x).

x→a

So, in the example from Lesson 1.3 concerning the function f (x) = x2 , we took points that got closer to the point on the graph (1, 1) and observed the sequence of slope values of the corresponding secant lines. Using our limit notation here, we would write

x2 − 1 . x→1 x − 1 lim

Recall also that we found that the slope values tended to the value x = 2; hence using our notation we can write

x2 − 1 = 2. x→1 x − 1 lim

Finding Limits Numerically In our example in Lesson 1.3 we used this approach to ﬁnd that limx→1 this technique to a more complicated function. Consider the rational function f (x) =

x+3 . x2 +x−6

lim

x→−3 x2

x2 −1 x−1

= 2. Let’s apply

Let’s ﬁnd the following limit:

x+3 . +x−6

Unlike our simple quadratic function, f (x) = x2 , it is tedious to compute the points manually. So let’s use the [TABLE] function of our calculator. Enter the equation in your calculator and examine the table of points of the function. Do you notice anything unusual about the points? (Answer: There are error readings indicated for x = −3, 2 because the function is not deﬁned at these values.) Even though the function is not deﬁned at x = −3, we can still use the calculator to read the y−values for x values very close to x = −3. Press 2ND [TBLSET] and set Tblstart to −3.2 and △ to 0.1 (see screen on left below). The resulting table appears in the middle below. www.ck12.org

62

Can you guess the value of limx→−3 x2x+3 ? If you guessed −.20 = −(1/5) you would be +x−6 correct. Before we ﬁnalize our answer, let’s get even closer to x = −3 and determine its function value using the [CALC VALUE] tool. Press 2ND [TBLSET] and change Indpnt from Auto to Ask. Now when you go to the table, enter x = −2.99999. and press [ENTER] and you will see the screen on the right above. Press [ENTER] and see that the function value is x = −0.2, which is the closest the calculator can display in the four decimal places allotted in the table. So our guess is correct and limx→−3 x2x+3 = − 51 . +x−6 Finding Limits Graphically Let’s continue with the same problem but now let’s focus on using the graph of the function to determine its limit.

lim

x→−3 x2

x+3 +x−6

We enter the function in the Y = menu and sketch the graph. Since we are interested in the value of the function for x close to x = −3, we will look to [ZOOM] in on the graph at that point.

Our graph above is set to the normal viewing window [−10, 10]. Hence the values of the function appear to be very close to 0. But in our numerical example, we found that the function values approached −.20 = −(1/5). To see this graphically, we can use the [ZOOM] and [TRACE] function of our calculator. Begin by choosing [ZOOM] function and choose [BOX]. Using the directional arrows to move the cursor, make a box around the x value −3.

63

www.ck12.org

(See the screen on the left below Press [ENTER] and [TRACE] and you will see the screen in the middle below.) In [TRACE] mode, type the number −2.99999 and press [ENTER]. You will see a screen like the one on the right below.

The graphing calculator will allow us to calculate limits graphically, provided that we have the function rule for the function so that we can enter its equation into the calculator. What if we have only a graph given to us and we are asked to ﬁnd certain limits? It turns out that we will need to have pretty accurate graphs that include suﬀicient detail about the location of data points. Consider the following example. Example 1: Find limx→3 f (x) for the function pictured here. Assume units of value 1 for each unit on the axes.

By inspection, we see that as we approach the value x = 3 from the left, we do so along what appears to be a portion of the horizontal line y = 2. We see that as we approach the value x = 3 from the right, we do so along a line segment having positive slope. In either case, the y values of f (x) approaches y = 2. Nonexistent Limits We sometimes have functions where limx→a f (x) does not exist. We have already seen an example of a function where our a value was not in the domain of the function. In particular, the function was not deﬁned for x = −3, 2, but we could still ﬁnd the limit as x → −3.

lim

x→−3 x2

x+3 1 =− +x−6 5

What do you think the limit will be as we let x → 2? www.ck12.org

64

lim

x→2 x2

x+3 +x−6

Our inspection of the graph suggests that the function around x = 2 does not appear to approach a particular value. For x > 2, the points all lie in the ﬁrst quadrant and appear to grow very quickly to large positive numbers as we get close to x = 2. Alternatively, for x < 2 we see that the points all lie in the fourth quadrant and decrease to large negative numbers. If we inspect actual values very close to x = 2 we can see that the values of the function do not approach a particular value.

x 1.999 1.9999 2 2.001 2.0001

y − 1000 − 10000 ERROR 1000 10000

65

www.ck12.org

For this example, we say that limx→2

x+3 x2 +x−6

does not exist.

Formal Deﬁnition of a Limit We conclude this lesson with a formal deﬁnition of a limit. Deﬁnition: We say that the limit of a function f (x) at a is L, written as limx→a f (x) = L , if for every open interval D of L, there exists an open interval N of a, that does not include a, such that f (x) is in D for every x in N. This deﬁnition is somewhat intuitive to us given the examples we have covered. Geometrically, the deﬁnition means that for any lines y = b1 , y = b2 below and above the line y = L, there exist vertical lines x = a1 , x = a2 to the left and right of x = a so that the graph of f (x) between x = a1 and x = a2 lies between the lines y = b1 and y = b2 . The key phrase in the above statement is “for every open interval D”, which means that even if D is very, very small (that is, f (x) is very, very close to L), it still is possible to ﬁnd interval N where f (x) is deﬁned for all values except possibly x = a.

Example 2: Use the deﬁnition of a limit to prove that

lim (2x + 1) = 7.

x→3

We need to show that for each open interval of 7, we can ﬁnd an open neighborhood of 3, that does not include 3, so that all x in the open neighborhood map into the open interval of 7. Equivalently, we must show that for every interval of 7, say (7 − ε, 7 + ε), we can ﬁnd an interval of 3, say (3−δ, 3+δ), such that (7−ε < 2x+1 < 7+ε) whenever (3−δ < x < 3+δ). www.ck12.org

66

The ﬁrst inequality is equivalent to 6 − ε < 2x < 6 + ε and solving for x, we have

3− Hence if we take δ =

ε 2

ε ε

, we will have 3 − δ < x < 3 + δ ⇒ 7 − ε < 2x + 1 < 7 + ε.

Fortunately, we do not have to do this to evaluate limits. In Lesson 1.6 we will learn several rules that will make the task manageable. Multimedia Links For another look at the deﬁnition of a limit, the series of videos at Tutorials for the Calculus Phobe has a nice intuitive introduction to this fundamental concept (despite the whimsical name). If you want to experiment with limits yourself, follow the sequence of activities using a graphing applet at Informal Limits. Directions for using the graphing applets at this very useful site are also available at Applet Intro.

Lesson Summary 1. 2. 3. 4.

We We We We

learned to ﬁnd the limit of a function numerically. learned to ﬁnd the limit of a function using a graph. identiﬁed cases when limits do not exist. used the formal deﬁnition of a limit to solve limit problems.

Review Questions 1. Use a table of values to ﬁnd limx→−2

x2 −4 . x+2

(a) Use x−values of x = −1.9, −1.99, −1.999, −2.1, −2.099, −2.0099. (b) What value does the sequence of values approach? 2. Use a table of values to ﬁnd limx→ 1 2

2x−1 2x2 +3x−2

.

(a) Use x−values of x = .49, .495, .49999, .51, .5099, .500001. (b) What value does the sequence of values approach? 3. Consider the function p(x) = 3x3 −3x. Generate the graph of p(x) using your calculator. Find each of the following limits if they exist. Use tables with appropriate x values to determine the limits. (a) (b) (c) (d)

limx→4 (3x3 − 3x) limx→−4 (3x3 − 3x) limx→0 (3x3 − 3x) Find the values of the function corresponding to x = 4, −4, 0. How do these function values compare to the limits you found in #a-c? Explain your answer.

67

www.ck12.org

4. Examine the graph of f (x) below to approximate each of the following limits if they exist.

(a) (b) (c) (d)

limx→3 f (x) limx→2 f (x) limx→1 f (x) limx→4 f (x)

5. Examine the graph of f (x) below to approximate each of the following limits if they exist.

(a) (b) (c) (d)

limx→2 f (x) limx→0 f (x) limx→4 f (x) limx→50 f (x)

www.ck12.org

68

In problems #6-8, determine if the indicated limit exists. Provide a numerical argument to justify your answer. 6. limx→2 (x2 + 3) 7. limx→−1 xx+1 √ 2 −1 8. limx→2 −2x + 5 In problems #9-10, determine if the indicated limit exists. Provide a graphical argument to justify your answer. (Hint: Make use of the [ZOOM] and [TABLE] functions of your calculator to view functions values close to the indicated x value. 9. limx→4 (x2 + 3x) 10. limx→−1 |x+1| x+1

Review Answers 1. (a) . 2 −4 (b) limx→−2 xx+2 = −4 2. (a) . = 52 (b) limx→ 1 2x22x−1 +3x−2 2 3. (a) limx→4 (3x3 − 3x) = 180 (b) limx→−4 (3x3 − 3x) = 180 (c) limx→0 (3x3 − 3x) = 0 (d) They are the same values because the function is deﬁned for each of these x−values. 4. (a) limx→3 f (x) = 1.5 (b) limx→2 f (x) = 0 (c) limx→1 f (x) = 2 (d) limx→4 f (x) does not exist. 5. (a) limx→2 f (x) = 0 (b) limx→0 f (x) does not exist. (c) limx→4 f (x) is some number close to 1 and less than 1, but not equal to 1. (d) limx→50 f (x) is some number close to 1 and less than 1, but not equal to 1. 6. The limit does exist. This can be veriﬁed by using the [TRACE] or [TABLE] function of your calculator, applied to x values very close to x = 2. 7. The limit does exist. This can be veriﬁed by using the [TRACE] or [TABLE] function of your calculator, applied to x values very close to x = −1. 8. The limit does exist. This can be veriﬁed by using the [TRACE] or [TABLE] function of your calculator, applied to x values very close to x = 2. 9. The limit does exist. This can be veriﬁed with either the [TRACE] or [TABLE] function of your calculator. 10. The limit does not exist; [ZOOM] in on the graph around x = −1 and see that the y−values approach a diﬀerent value when approached from the right and from the left.

69

www.ck12.org

1.6

Evaluating Limits

Learning Objectives A student will be able to: • • • • •

Find the limit of basic functions. Use properties of limits to ﬁnd limits of polynomial, rational and radical functions. Find limits of composite functions. Find limits of trigonometric functions. Use the Squeeze Theorem to ﬁnd limits.

Introduction In this lesson we will continue our discussion of limits and focus on ways to evaluate limits. We will observe the limits of a few basic functions and then introduce a set of laws for working with limits. We will conclude the lesson with a theorem that will allow us to use an indirect method to ﬁnd the limit of a function. Direct Substitution and Basic Limits Let’s begin with some observations about limits of basic functions. Consider the following limit problems:

lim 5,

x→2

lim x.

x→4

These are examples of limits of basic constant and linear functions, f (x) = c and f (x) = mx + b. We note that each of these functions are deﬁned for all real numbers. If we apply our techniques for ﬁnding the limits we see that

lim 5 = 5,

x→2

lim x = 4,

x→4

and observe that for each the limit equals the value of the function at the x−value of interest: www.ck12.org

70

lim 5 = f (5) = 5,

x→2

lim x = f (4) = 4.

x→4

Hence limx→a f (x) = f (a). This will also be true for some of our other basic functions, in particular all polynomial and radical functions, provided √ that the function is deﬁned at x = a. For example, limx→3 x3 = f (3) = 27 and limx→4 x = f (4) = 2. The properties of functions that make these facts true will be discussed in Lesson 1.7. For now, we wish to use this idea for evaluating limits of basic functions. However, in order to evaluate limits of more complex function we will need some properties of limits, just as we needed laws for dealing with complex problems involving exponents. A simple example illustrates the need we have for such laws. Example 1:

√ Evaluate limx→2 (x3 + 2x). The problem here is that while we know √ that the limit of each 2 individual function of the sum exists, limx→2 x = 8 and limx→2 2x = 2, our basic limits above do not tell us what happens when we ﬁnd the limit of a sum of functions. We will state a set of properties for dealing with such sophisticated functions. Properties of Limits Suppose that limx→a f (x) and limx→a g(x) both exist. Then 1. 2. 3. 4. 5.

limx→a [cf (x)] = c limx→a f (x) where c is a real number, limx→a [f (x)]n = [limx→a f (x)]n where n is a real number, limx→a [f (x) ± g(x)] = limx→a f (x) ± limx→a g(x), limx→a [f [ (x)]· g(x)] = limx→a f (x) · limx→a g(x), (x) x→a f (x) limx→a fg(x) = lim provided that limx→a g(x) ̸= 0. limx→a g(x)

With these properties we can evaluate a wide range of polynomial and radical functions. Recalling our example above, we see that

lim (x3 +

x→2

√

√ 2x) = lim (x3 ) + lim ( 2x) = 8 + 2 = 10. x→2

x→2

Find the following limit if it exists:

lim (2x2 −

x→−4

71

√

−x). www.ck12.org

Since the limit of each function within the parentheses exists, we can apply our properties and ﬁnd

lim (2x2 −

x→−4

√

−x) = lim 2x2 − lim x→−4

√

x→−4

−x.

√ Observe that the second√limit, limx→−4 −x, is an application of Law #2 with n = 12 . So √ 2 2 we have limx→−4 (2x − −x) = limx→−4 2x − limx→−4 −x = 32 − 2 = 30. In most cases of sophisticated functions, we simplify the task by applying the Properties as indicated. We want to examine a few exceptions to these rules that will require additional analysis. Strategies for Evaluating Limits of Rational Functions Let’s recall our example x2 − 1 . x→1 x − 1 lim

We saw that the function did not have to be deﬁned at a particular value for the limit to exist. In this example, the function was not deﬁned for x = 1. However we were able to evaluate the limit numerically by checking functional values around x = 1 and found 2 −1 limx→1 xx−1 = 2. Note that if we tried to evaluate by direct substitution, we would get the quantity 0/0, which we refer to as an indeterminate form. In particular, Property #5 for ﬁnding limits does not apply since limx→1 (x − 1) = 0. Hence in order to evaluate the limit without using numerical or graphical techniques we make the following observation. The numerator of the function can be factored, with one factor common to the denominator, and the fraction simpliﬁed as follows: x2 − 1 (x + 1)(x − 1) = = x + 1. x−1 x−1 In making this simpliﬁcation, we are indicating that the original function can be viewed as a linear function for x values close to but not equal to 1, that is, x2 −1 x−1

= x + 1 for x ̸= 1. In terms of our limits, we can say x2 − 1 = lim (x + 1) = 1 + 1 = 2. x→1 x − 1 x→1 lim

www.ck12.org

72

Example 2: Find limx→0

x2 +5x . x

This is another case where direct substitution to evaluate the limit gives the indeterminate form 0/0. Reducing the fraction as before gives: x2 + 5x = lim (x + 5) = 5. x→0 x→0 x lim

Example 3: √ lim

x→9

x−3 . x−9

In order to evaluate the limit, we need to recall that the diﬀerence of squares of real numbers can be factored as x2 − y 2 = (x + y)(x − y). We then rewrite and simplify the original function as follows: √

√ x−3 x−3 1 √ = √ . = √ x−9 ( x + 3)( x − 3) ( x + 3)

Hence limx→9

√

x−3 x−9

= limx→9

√1 x+3

= 16 .

You will solve similar examples in the homework where some clever applications of factoring to reduce fractions will enable you to evaluate the limit. Limits of Composite Functions While we can use the Properties to ﬁnd limits of composite functions, composite functions will present some diﬀiculties that we will fully discuss in the next Lesson. We can illustrate with the following examples, one where the limit exists and the other where the limit does not exist. Example 4: Consider f (x) =

1 , x+1

g(x) = x2 . Find limx→−1 (f ◦ g)(x).

We see that (f ◦ g)(x) = x21+1 and note that property #5 does hold. Hence by direct substitution we have limx→−1 (f ◦ g)(x) = (−1)12 +1 = 21 . Example 5: 1 Consider f (x) = x+1 , g(x) = −1. Then we have that f (g(x)) is undeﬁned and we get the indeterminate form 1/0. Hence limx→−1 (f ◦ g)(x) does not exist.

73

www.ck12.org

Limits of Trigonometric Functions In evaluating limits of trigonometric functions we will look to rely more on numerical and graphical techniques due to the unique behavior of these functions. Let’s look at a couple of examples. Example 6: Find limx→0 sin(x). We can ﬁnd this limit by observing the graph of the sine function and using the [CALC VALUE] function of our calculator to show that limx→0 sin x = 0. While we could have found the limit by direct substitution, in general, when dealing with trigonometric functions, we will rely less on formal properties of limits for ﬁnding limits of trigonometric functions and more on our graphing and numerical techniques. The following theorem provides us a way to evaluate limits of complex trigonometric expressions. Squeeze Theorem Suppose that f (x) ≤ g(x) ≤ h(x) for x near a, and limx→a f (x) = limx→a h(x) = L. Then limx→a g(x) = L. In other words, if we can ﬁnd bounds for a function that have the same limit, then the limit of the function that they bound must have the same limit. Example 7: Find limx→0 x2 cos(10πx).

From the graph we note that: 1. The function is bounded by the graphs of x2 and −x2 2. limx→0 x2 = limx→0 (−x2 ) = 0. Hence the Squeeze Theorem applies and we conclude that limx→0 x2 cos(10πx) = 0. Multimedia Link For a brief, intuitive introduction to the Squeeze Theorem using everyday examples, see Khan Academy Squeeze Theorem (7:36) . This video includes a graphical presentation of the Squeeze Theorem. This video serves as an introduction to another (much longer) video, Khan academy proof of limit x -> 0 sin(x)/x (18:04). www.ck12.org

74

Figure 1.1: Intuition (but not a proof) of the Squeeze Theorem. http://www.youtube.com/ watch?v=igJdDN-DPgA&feature=youtube_gdata

Lesson Summary 1. 2. 3. 4.

We We We We

learned to ﬁnd the limit of basic functions. learned to ﬁnd the limit of polynomial, rational and radical functions. learned how to ﬁnd limits of composite and trigonometric functions. used the Squeeze Theorem to ﬁnd special limits.

Review Questions Find each of the following limits if they exist. 1. limx→2 (x2 − 3x + 4) 2 −16 2. limx→4 xx−4 √

x−2 3. limx→4 x−4 4. limx→−1 x−2 x+1 5. limx→−1 10x−2 3x+1 √

6. limx→1 x+3−2 x−1 x2 −25 7. limx→5 x3 −125 1 8. Consider f (x) = x+1 , g(x) = x2 . We found limx→−1 (f ◦ g)(x) = 21 Find limx→−1 (g ◦ f )(x). 9. Consider function f (x) such that 5x − 11 ≤ f (x) ≤ x2 − 4x + 9 for x ≥ 0. Use the Squeeze Theorem to ﬁnd limx→5 f (x). 10. Use the Squeeze Theorem to show that limx→0 x4 sin( x1 ) = 0

Review Answers 1. limx→2 (x2 − 3x + 4) = 2.

75

www.ck12.org

x2 −16 =8 √x−4 x−2 limx→4 x−4 = 41 does not limx→−1 x−2 x+1 10x−2 limx→−1 3x+1 = 6 √ limx→1 x+3−2 = 14 x−1 2 −25 2 limx→5 xx3 −125 = 15

2. limx→4 3. 4. 5.

exist.

6. 7. 8. limx→−1 (g ◦ f )(x) does not exist since g(f (x)) is undeﬁned. 9. limx→5 f (x) = 14 since limx→5 (5x − 11) = limx→5 (x2 − 4x + 9) = 14 10. Note that x4 ≥ sin( x1 ) ≥ −x4 , and since limx→0 x4 = limx→0 (−x4 ) = 0, then by the Squeeze Theorem must have limx→0 x4 sin( x1 ) = 0.

1.7

Continuity

Learning Objectives A student will be able to:

• • • • •

Learn to examine continuity of functions. Find one-sided limits. Understand properties of continuous functions. Solve problems using the Min-Max theorem. Solve problems using the Intermediate Value Theorem.

Introduction In this lesson we will discuss the property of continuity of functions and examine some very important implications. Let’s start with an example of a rational function and observe its graph. Consider the following function:

f (x) = (x + 1)/(x2 − 1). We know from our study of domains that in order for the function to be deﬁned, we must use x ̸= −1, 1. Yet when we generate the graph of the function (using the standard viewing window), we get the following picture that appears to be deﬁned at x = −1: www.ck12.org

76

The seeming contradiction is due to the fact that our original function had a common factor in the numerator and denominator, x + 1, that cancelled out and gave us a picture that appears to be the graph of f (x) = 1/(x − 1). But what we actually have is the original function, f (x) = (x + 1)/(x2 − 1), that we know is not deﬁned at x = −1. At x = −1, we have a hole in the graph, or a discontinuity of the function at x = −1. That is, the function is deﬁned for all other x−values close to x = −1. Loosely speaking, if we were to hand-draw the graph, we would need to take our pencil oﬀ the page when we got to this hole, leaving a gap in the graph as indicated:

77

www.ck12.org

Now we will formalize the property of continuity of a function and provide a test for determining when we have continuous functions. Continuity of a Function Deﬁnition: The function f (x) is continuous at x = a if the following conditions all hold: 1. a is in the domain of f (x); 2. limx→a f (x) exists; 3. limx→a f (x) = f (a) Note that it is possible to have functions where two of these conditions are satisﬁed but the third is not. Consider the piecewise function

www.ck12.org

78

{ x, if x ̸= 1 f (x) = 3, if x = 1 In this example we have limx→1 f (x) exists, x = 1 is in the domain of f (x), but limx→1 f (x) ̸= f (1). One-Sided Limits and Closed Intervals Let’s recall our basic square root function, f (x) =

√

x.

√ √ Since the domain of f (x) = x is x ≥ 0, we see that that limx→0 x does not exist. Speciﬁcally, we cannot ﬁnd open intervals around x = 0 that satisfy the limit deﬁnition. However we do note that as we approach x = 0 from the right-hand side, we see the successive values tending towards x = 0. This example provides some rationale for how we can deﬁne one-sided limits. Deﬁnition:

79

www.ck12.org

We say that the right-hand limit of a function f (x) at a is b, written as limx→a+ f (x) = b, if for every open interval N of b, there exists an open interval (a, a + δ) contained in the domain of f (x), such that f (x) is√ in N for every x in (a, a + δ). For the example above, we write limx→0+ x = 0. Similarly, we say that the left-hand limit of f (x) at a is b, written as limx→a− f (x) = b, if for every open interval N of b there exists an open interval (a − δ, a) contained in the domain of f (x), such that f (x) is in N for every x in (a − δ, a). Example 1: Find limx→0+

x . |x|

The graph has a discontinuity at x = 0 as indicated:

We see that limx→0+

x |x|

= 1 and also that limx→0−

x |x|

= −1.

Properties of Continuous Functions Let’s recall our example of the limit of composite functions: f (x) = 1/(x + 1), g(x) = −1. We saw that f (g(x)) is undeﬁned and has the indeterminate form of 1/0. Hence limx→−1 (f ◦ g)(x) does not exist. In general, we will require that f be continuous at x = g(a) and x = g(a) must be in the domain of (f ◦ g) in order for limx→a (f ◦ g)(x) to exist. We will state the following theorem and delay its proof until Chapter 3 when we have learned more about real numbers. Min-Max Theorem: If a function f (x) is continuous in a closed interval I, then f (x) has both a maximum value and a minimum value in I. www.ck12.org

80

Example 2: Consider f (x) = x3 + 1 and interval I = [−2, 2]. The function has a minimum value at value at x = −2, f (−2) = −7, and a maximum value at x = 2, where f (2) = 9

We will conclude this lesson with a theorem that will enable us to solve many practical problems such as ﬁnding zeros of functions and roots of equations. Intermediate Value Theorem If a function is continuous on a closed interval [a, b], then the function assumes every value between f (a) and f (b). The proof is left as an exercise with some hints provided. (Homework #10). We can use the Intermediate Value Theorem to analyze and approximate zeros of functions. Example 3: Use the Intermediate Value Function to show that there is at least one zero of the function in the indicated interval.

81

www.ck12.org

f (x) = 3x4 − 3x3 − 2x + 1, (1, 2) We recall that the graph of this function is shaped somewhat like a parabola; viewing the graph in the standard window, we get the following graph:

Of course we could zoom in on the graph to see that the lowest point on the graph lies within the fourth quadrant, but let’s use the [CALC VALUE] function of the calculator to verify that there is a zero in the interval (1, 2). In order to apply the Intermediate Value Theorem, we need to ﬁnd a pair of x−values that have function values with diﬀerent signs. Let’s try some in the table below.

x 1.1 1.2 1.3 www.ck12.org

f (x) −.80 −.36 .37

82

We see that the sign of the function values changes from negative to positive somewhere between 1.2 and 1.3. Hence, by the Intermediate Value theorem, there is some value c in the interval (1.2, 1.3) such that f (c) = 0.

Lesson Summary 1. 2. 3. 4. 5.

We We We We We

learned to examine continuity of functions. learned to ﬁnd one-sided limits. observed properties of continuous functions. solved problems using the Min-Max theorem. solved problems using the Intermediate Value Theorem.

Review Questions 1. Generate the graph of f (x) = (|x + 1|)/(x + 1) using your calculator and discuss the continuity of the function. 2. Generate the graph of f (x) = (3x − 6)/(x2 − 4) using your calculator and discuss the continuity of the function. Compute the limits in #3 - 6. 3. limx→0+ √ 4. limx→2− 5. limx→1+ 6. limx→2+

√

x √ 1+ x−1 x3 −8 |x−2|(x−2) 2x|x−1| x−1 |x+2|+x+2 |x+2|−x−2

In problems 7 and 8, explain how you know that the function has a root in the given interval. (Hint: Use the Intermediate Value Function to show that there is at least one zero of the function in the indicated interval.): 3 2 7. f (x) = x − x + 1, in the interval (−2, −3) √ + 2x √ 3 8. f (x) = x − x − 1, in the interval (9, 10) 9. State whether the indicated x−values correspond to maximum or minimum values of the function depicted below.

83

www.ck12.org

10. Prove the Intermediate Value Theorem: If a function is continuous on a closed interval [a, b], then the function assumes every value between f (a) and f (b).

Review Answers 1. While graph of the function appears to be continuous everywhere, a check of the table values indicates that the function is not continuous at x = −1. 2. While the function appears to be continuous for all x = −2, a check of the table values indicates that √ the function is not continuous at x = 2. 3. limx→0+ √ √x =2 4. limx→2− 5. limx→1+

1+ x−1 x3 −8 |x−2|(x−2) 2x|x−1| = (x−1) |x+2|+x+2 |x+2|−x−2

does not exist 2

6. limx→2+ =0 7. f (−2.5) = .375, f (−2.9) = −3.669. By the Intermediate Value Theorem, there is an x−value c with f (c) = 0. 8. f (9.1) = −.071, f (9.99) = .006. By the Intermediate Value Theorem, there is an x−value c with f (c) = 0. 9. x = a is a relative maximum, x = b is an absolute minimum, x = c is an absolute maximum and x = d is not a maximum nor a minimum. 10. Here is an outline of the proof: we need to show that for every number d between f (a) and f (b), there exists a number such that f (c) = d. (a) Assume that f (a) < f (c) < f (b). (b) Let S be the set of x ϵ [a, b] > for which f (x) < d. Note that a ϵ S, b ϵ S. so b is an upper bound for set S. Hence by the completeness property of the reals, S has an upper bound, c. (c) There are then three possibilities to explore: f (c) < d, f (c) = d, or f (c) > d. Explore these and show why f (c) = d.

1.8

Inﬁnite Limits

Learning Objectives A student will be able to: • • • •

Find inﬁnite limits of functions. Analyze properties of inﬁnite limits. Identify asymptotes of functions. Analyze end behavior of functions.

www.ck12.org

84

Introduction In this lesson we will discuss inﬁnite limits. In our discussion the notion of inﬁnity is discussed in two contexts. First, we can discuss inﬁnite limits in terms of the value a function as we increase x without bound. In this case we speak of the limit of f (x) as x approaches ∞ and write limx→∞ f (x). We could similarly refer to the limit of f (x) as x approaches -∞ and write limx→−∞ f (x). The second context in which we speak of inﬁnite limits involves situations where the function values increase without bound. For example, in the case of a rational function such as f (x) = (x + 1)/(x2 + 1), a function we discussed in previous lessons:

At x = 1, we have the situation where the graph grows without bound in both a positive and a negative direction. We say that we have a vertical asymptote at x = 1, and this is indicated by the dotted line in the graph above. In this example we note that limx→1 f (x) does not exist. But we could compute both onesided limits as follows. limx→1− f (x) = −∞ and limx→1− f (x) = +∞.

85

www.ck12.org

More formally, we deﬁne these as follows: Deﬁnition: The right-hand limit of the function f (x) at x = a is inﬁnite, and we write limx→a+ f (x) = ∞, if for every positive number k, there exists an open interval (a, a + δ) contained in the domain of f (x), such that f (x) is in (k, ∞) for every x in (a, a + δ). The deﬁnition for negative inﬁnite limits is similar. Suppose we look at the function f (x) = (x + 1)/(x2 − 1) and determine the inﬁnite limits limx→∞ f (x) and limx→−∞ f (x). We observe that as x increases in the positive direction, the function values tend to get smaller. The same is true if we decrease x in the negative direction. Some of these extreme values are indicated in the following table.

x 100 200 −100 −200

f (x) .0101 .0053 −.0099 −.005

We observe that the values are getting closer to f (x) = 0. Hence limx→∞ f (x) = 0 and limx→−∞ f (x) = 0. Since our original function was roughly of the form f (x) = x1 , this enables us to determine limits for all other functions of the form f (x) = x1p with p > 0. Speciﬁcally, we are able to conclude that limx→∞ x1p = 0. This shows how we can ﬁnd inﬁnite limits of functions by examining the end behavior of the function f (x) = x1p , p > 0. The following example shows how we can use this fact in evaluating limits of rational functions. Example 1: Find limx→∞

2x3 −x2 +x−1 . x6 −x5 +3x4 −2x+1

Solution: Note that we have the indeterminate form, so Limit Property #5 does not hold. However, if we ﬁrst divide both numerator and denominator by the quantity x6 , we will then have a function of the form

f (x) = g(x) www.ck12.org

2x3 x6 x6 x6

−

− x5 x6

x2 x6

+

+

x x6

−

1 x6

3x4 x6

−

2x x6

+

− x14 + x15 − x16 = . 1 − x1 + x32 − x25 + x16 2 x3

1 x6

86

We observe that the limits limx→∞ f (x) and limx→∞ g(x) both exist. In particular, limx→∞ f (x) = 2x3 −x2 +x−1 0 and limx→∞ g(x) = 1. Hence Property #5 now applies and we have limx→∞ x6 −x 5 +3x4 −2x+1 = 0 = 0. 1

Lesson Summary 1. 2. 3. 4.

We We We We

learned to ﬁnd inﬁnite limits of functions. analyzed properties of inﬁnite limits. identiﬁed asymptotes of functions. analyzed end behavior of functions.

Review Questions In problems 1 - 7, ﬁnd the limits if they exist. (x+2)2 (x−2)2 −1 (x+2)2 limx→∞ (x−2) 2 −1 (x+2)2 limx→1+ (x−2)2 −1 limx→∞ 2x−1 x+1 5 4 +1 limx→−∞ x +3x x3 −1 4 −2x2 +3x+1 limx→∞ 3x 2x4 −2x2 +x−3 2x2 −x+3 limx→∞ x5 −2x 3 +2x−3

1. limx→3+ 2. 3. 4. 5. 6. 7.

In problems 8 - 10, analyze the given function and identify all asymptotes and the end behavior of the graph. 2

(x+4) 8. f (x) = (x−4) 2 −1 9. f (x) = −3x3 − x2 + 2x + 2 2 −8 10. f (x) = 2xx+2

Review Answers (x+2)2 = +∞ (x−2)2 −1 (x+2)2 limx→∞ (x−2)2 −1 = 1 (x+2)2 limx→1+ (x−2) 2 −1 = −∞ =2 limx→∞ 2x−1 x+1 x5 +3x4 +1 limx→−∞ x3 −1 = −∞

1. limx→3+ 2. 3. 4. 5.

87

www.ck12.org

6. 7. 8. 9. 10.

−2x +3x+1 = 32 limx→∞ 3x 2x4 −2x2 +x−3 2x2 −x+3 limx→∞ x5 −2x 3 +2x−3 = 0 Zero at x = −4; vertical asymptotes at x = 3, 5; f (x) → 1 as x → ±∞. Zero at x = 1; no vertical asymptotes; f (x) → −∞ as x → ∞; f (x) → ∞ as x → −∞. Zero at x = 2; no vertical asymptotes but there is a discontinuity at x = −2; f (x) → −∞ as x → −∞; f (x) → ∞ as x → ∞.

www.ck12.org

4

2

88

Chapter 2 Derivatives 2.1

Tangent Lines and Rates of Change

Learning Objectives A student will be able to: • Demonstrate an understanding of the slope of the tangent line to the graph. • Demonstrate an understanding of the instantaneous rate of change. A car speeding down the street, the inﬂation of currency, the number of bacteria in a culture, and the AC voltage of an electric signal are all examples of quantities that change with time. In this section, we will study the rate of change of a quantity and how is it related to the tangent lines on a curve.

The Tangent Line If two points P (x0 , y0 ) and Q(x1 , y1 ) are two diﬀerent points of the curve y = f (x) (Figure 1), then the slope of the secant line connecting the two points is given by

msec =

y1 − y0 f (x1 ) − f (x0 ) = x1 − x0 x1 − x 0

(1)

Now if we let the the point x1 approach x0 , Q will approach P along the graph f . Thus the slope of the secant line will gradually approach the slope of the tangent line as x1 approaches x0 . Therefore (1) becomes

89

www.ck12.org

mtan = lim

x1 →x0

f (x1 ) − f (x0 ) . x1 − x0

(2)

If we let h = x1 − x0 , then x1 = x0 + h and h → 0 becomes equivalent to x1 → x0 , so (2) becomes f (x0 + h) − f (x0 ) . h→0 h

mtan = lim

If the point P (x0 , y0 ) is on the curve f , then the tangent line at P has a slope that is given by f (x0 + h) − f (x0 ) h→0 h

mtan = lim provided that the limit exists.

Multimedia Link The following applet illustrates how the slope of a secant line can become the slope of the tangent to a curve at a point as h → 0. Follow the directions on the page to explore how changing the distance between the two points makes the slope of the secant approach the slope of the tangent Slope at a Point Applet. Note that the function and the point of tangency can also be edited in this simulator. Recall from algebra that the point-slope form for the tangent line is given by

y − y0 = mtan (x − x0 ). Example 1: Find the slope of the tangent line to the curve f (x) = x3 passing through point P (2, 8). Solution: Since P (x0 , y0 ) = (2, 8), using the slope of the tangent equation, f (x0 + h) − f (x0 ) h→0 h

mtan = lim we get www.ck12.org

90

f (2 + h) − f (2) h→0 h (h3 + 6h2 + 12h + 8) − 8 = lim h→0 h 3 2 h + 6h + 12h = lim h→0 h 2 = lim (h + 6h + 12)

mtan = lim

h→0

= 12. Thus the slope of the tangent line is 12. Using the point-slope formula above,

y − 8 = 12(x − 2) or

y = 12x − 16 Next we are interested in ﬁnding a formula for the slope of the tangent line at any point on the curve f . Such a formula would be the same formula that we are using except we replace the constant x0 by the variable x. This yields f (x + h) − f (x) . h→0 h

mtan = lim We denote this formula by

f (x + h) − f (x) , h→0 h

f ′ (x) = lim

where f (x) is read ”f prime of x.” The next example illustrate its usefulness. Example 2: If f (x) = x2 − 3, ﬁnd f (x) and use the result to ﬁnd the slope of the tangent line at x = 2 and x = −1. Solution:

91

www.ck12.org

Since

f ′ (x) = lim

h→0

f (x + h) − f (x) , h

then

[(x + h)2 − 3] − [x2 − 3] h→0 h 2 x + 2xh + h2 − 3 − x2 + 3 = lim h→0 h 2xh + h2 = lim h→0 h = lim (2x + h)

f ′ (x) = lim

h→0

= 2x To ﬁnd the slope, we simply substitute x = 2 into the result f (x),

f ′ (x) = 2x f ′ (2) = 2(2) =4 and

f ′ (x) = 2x f ′ (−1) = 2(−1) = −2 Thus slopes of the tangent lines at x = 2 and x = −1 are 4 and −2, respectively. Example 3: Find the slope of the tangent line to the curve y = Solution: Using the slope of the tangent formula www.ck12.org

92

1 x

that passes through the point (1, 1).

f ′ (x) = lim

h→0

f (x + h) − f (x) h

and substituting y = x1 , ( ′

y = lim

1 x+h

−

1 x

h

h→0

= lim

)

x−x−h x(x+h)

h x−x−h = lim h→0 hx(x + h) −h = lim h→0 hx(x + h) −1 = lim h→0 x(x + h) −1 = lim h→0 x(x + h) −1 = 2. x h→0

Substituting x = 1, −1 1 = −1.

y′ =

Thus the slope of the tangent line at x = 1 for the curve y = x1 is m = −1. To ﬁnd the equation of the tangent line, we simply use the point-slope formula, y − y0 = m(x − x0 ), where (x0 , y0 ) = (1, 1). y − 1 = −1(x − 1) = −x + 1 + 1 = −x + 2, which is the equation of the tangent line.

93

www.ck12.org

Average Rates of Change The primary concept of calculus involves calculating the rate of change of a quantity with respect to another. For example, speed is deﬁned as the rate of change of the distance travelled with respect to time. If a person travels 120 miles in four hours, his speed is 120/4 = 30 mi/hr. This speed is called the average speed or the average rate of change of distance with respect to time. Of course the person who travels 120 miles at a rate of 30 mi/hr for four hours does not do so continuously. He must have slowed down or sped up during the four-hour period. But it does suﬀice to say that he traveled for four hours at an average rate of 30 miles per hour. However, if the driver strikes a tree, it would not be his average speed that determines his survival but his speed at the instant of the collision. Similarly, when a bullet strikes a target, it is not the average speed that is signiﬁcant but its instantaneous speed at the moment it strikes. So here we have two distinct kinds of speeds, average speed and instantaneous speed. The average speed of an object is deﬁned as the object’s displacement △x divided by the time interval △t during which the displacement occurs:

v=

△x x1 − x0 = . △t t1 − t0

Notice that the points (t0 , x0 ) and (t1 , x1 ) lie on the position-versus-time curve, as Figure 1 shows. This expression is also the expression for the slope of a secant line connecting the two points. Thus we conclude that the average velocity of an object between time t0 and t1 is represented geometrically by the slope of the secant line connecting the two points (t0 , x0 ) and (t1 , x1 ). If we choose t1 close to t0 , then the average velocity will closely approximate the instantaneous velocity at time t0 .

www.ck12.org

94

Geometrically, the average rate of change is represented by the slope of a secant line and the instantaneous rate of change is represented by the slope of the tangent line (Figures 2 and 3). Average Rate of Change (such as the average velocity) The average rate of change of x = f (t) over the time interval [t0 , t1 ] is the slope msec of the secant line to the points (t0 , f (t0 )) and (t1 , f (t1 )) on the graph (Figure 2).

msec =

f (t1 ) − f (t0 ) t1 − t0

Instantaneous Rate of Change The instantaneous rate of change of x = f (t) at the time t0 is the slope mtan of the tangent line at the time t0 on the graph.

95

www.ck12.org

mtan = f ′ (t0 ) = lim

t1 →t0

f (t1 ) − f (t0 ) t1 − t0

Example 4: Suppose that y = x2 − 3. 1. Find the average rate of change of y with respect to x over the interval [0, 2]. 2. Find the instantaneous rate of change of y with respect to x at the point x = −1. Solution: 1. Applying the formula for Average Rate of Change with f (x) = x2 − 3 and x0 = 0 and x1 = 2 yields f (x1 ) − f (x0 ) x1 − x 0 f (2) − f (0) = 2−0 1 − (−3) = 1 =4

msec =

This means that the average rate of change of y is 4 units per unit increase in x over the interval [0, 2]. www.ck12.org

96

2. From the example above, we found that f (x) = 2x, so mtan = f ′ (x0 ) = f ′ (−1) = 2(−1) = −2 This means that the instantaneous rate of change is negative. That is, y is decreasing at x = −1. It is decreasing at a rate of 1 units per unit increase in x.

Review Questions 1. Given the function y = 1/2 x2 and the values of x0 = 3 and x1 = 4, ﬁnd (a) (b) (c) (d) (e)

The average rate of change of y with respect to x over the interval [x0 , x1 ]. The instantaneous rate of change of y with respect to x at x0 . The slope of the tangent line at x1 . The slope of the secant line between points x0 and x1 . Make a sketch of y = 1/2 x2 and show the secant and tangent lines at their respective points.

2. Repeat problem #1 for f (x) = 1/x and the values x0 = 2 and x1 = 3. 3. Find the slope of the graph f (x) = x2 + 1 at a general point x. What is the slope of the tangent line at x√ 0 = 6? 4. Suppose that y = 1/ x. (a) Find the average rate of change of y with respect to x over the interval [1, 3]. (b) Find the instantaneous rate of change of y with respect to x at point x = 1. 5. A rocket is propelled upward and reaches a height of h(t) = 4.9t2 in t seconds. (a) (b) (c) (d)

How high does it reach in 35 seconds? What is the average velocity of the rocket during the ﬁrst 35 seconds? What is the average velocity of the rocket during the ﬁrst 200 meters? What is the instantaneous velocity of the rocket at the end of the 35 seconds?

6. A particle moves in the positive direction along a straight line so that after t nanoseconds, its traversed distance is given by χ(t) = 9.9t3 nanometers. (a) What is the average velocity of the particle during the ﬁrst 2 nanoseconds? (b) What is the instantaneous velocity of the particle at t = 2 nanoseconds?

Review Answers 1. (a)

7 2

97

www.ck12.org

(b) 3 (c) 4 (d) 72 2. (a) −1 6 (b) −1 4 (c) −1 9 (d) −1 6 3. 2x, 12. (√ ) 4. (a) 63 − 21 (b) 5. (a) (b) (c) (d) 6. (a) (b)

2.2

− 12 6002.5 m 171.5 m/sec 31.3 m/sec 343 m/sec 39.6 m/sec 118.8 m/sec

The Derivative

Learning Objectives A student will be able to: • Demonstrate an understanding of the derivative of a function as a slope of the tangent line. • Demonstrate an understanding of the derivative as an instantaneous rate of change. • Understand the relationship between continuity and diﬀerentiability. The function f ′ (x) that we deﬁned in the previous section is so important that it has its own name. The Derivative The function f ′ is deﬁned by the new function f (x + h) − f (x) h→0 h

f ′ (x) = lim

where f is called the derivative of f with respect to x. The domain of f consists of all the values of x for which the limit exists. www.ck12.org

98

Based on the discussion in previous section, the derivative f ′ represents the slope of the tangent line at point x. Another way of interpreting it is to say that the function y = f (x) has a derivative f ′ whose value at x is the instantaneous rate of change of y with respect to point x. Example 1: Find the derivative of f (x) =

x . x+1

Solution: We begin with the deﬁnition of the derivative, f (x + h) − f (x) 1 = lim [f (x + h) − f (x)] , h→0 h→0 h h

f ′ (x) = lim where

x x+1 x+h f (x + h) = x+h+1 f (x) =

Substituting into the derivative formula, [ ] 1 x+h x f (x) = lim − h→0 h x + h + 1 x+1 [ ] 1 (x + h)(x + 1) − x(x + h + 1) = lim h→0 h (x + h + 1)(x + 1) [ 2 ] 1 x + x + hx + h − x2 − xh − x = lim h→0 h (x + h + 1)(x + 1) [ ] 1 h = lim h→0 h (x + h + 1)(x + 1) 1 = lim h→0 (x + h + 1)(x + 1) 1 = . (x + 1)2 ′

Example 2: Find the derivative of f (x) =

√

x and the equation of the tangent line at x0 = 1.

99

www.ck12.org

Solution: Using the deﬁnition of the derivative, f (x + h) − f (x) h→0 h √ √ x+h− x = lim h→0 h √ √ √ √ x+h− x x+h+ x √ = lim √ h→0 h x+h+ x 1 x+h−x = lim √ √ h→0 h x+h+ x 1 = lim √ √ h→0 x+h+ x 1 = √ . 2 x

f ′ (x) = lim

Thus the slope of the tangent line at x0 = 1 is 1 1 f ′ (1) = √ = . 2 2 1 For x0 = 1, we can ﬁnd y0 by simply substituting into f (x).

f (x0 ) ≡ y0 √ f (1) = 1 = 1 y0 = 1. Thus the equation of the tangent line is

y − y0 = m(x − x0 ) 1 y − 1 = (x − 1) 2 1 1 y = x+ . 2 2 Multimedia Link The following applet allows you to explore the relationship between a function and its derivative on a graph. Notice that as you move x along the curve, the slope www.ck12.org

100

of the tangent line to f (x) is the height of the derivative function, f ′ (x) Derivative Applet. This applet is customizable–after doing the steps outlined on the page, feel free to change the function deﬁnition and explore the derivative of many functions.

Notation Calculus, just like all branches of mathematics, is rich with notation. There are many ways to denote the derivative of a function y = f (x) in addition to the most popular one, f ′ (x). They are:

f ′ (x)

dy dx

y′

df dx

df (x) dx

In addition, when substituting the point x0 into the derivative we denote the substitution by one of the following notations:

f ′ (x0 )

df x − x0 dx

dy | x − x0 dx

df (x0 ) dx

Existence and Diﬀerentiability of a Function If, at the point (x0 , f (x0 )), the limit of the slope of the secant line does not exist, then the derivative of the function f (x) at this point does not exist either. That is, if

msec = lim

x→x0

f (x) − f (x0 ) = Does not exist x − x0

then the derivative f ′ (x) also fails to exist as x → x0 . The following examples show four cases where the derivative fails to exist. 1. At a corner. For example f (x) = |x|, where the derivative on both sides of x = 0 diﬀer (Figure 4). 2. At a cusp. For example f (x) = x2/3 , where the slopes of the secant lines approach +∞ on the right and −∞ on the left (Figure 5). 3. A vertical tangent. For example f (x) = x1/3 , where the slopes of the secant lines approach +∞ on the right and −∞ on the left (Figure 6).

101

www.ck12.org

4. A jump discontinuity. For example, the step function (Figure 7)

{ −2, f (x) = 2,

x<0 x≥0

where the limit from the left is −2 and the limit from the right is 2.

www.ck12.org

102

Many functions in mathematics do not have corners, cusps, vertical tangents, or jump discontinuities. We call them diﬀerentiable functions. From what we have learned already about diﬀerentiability, it will not be diﬀicult to show that continuity is an important condition for diﬀerentiability. The following theorem is one of the most important theorems in calculus: Diﬀerentiability and Continuity If f is diﬀerentiable at x0 , then f is also continuous at x0 .

103

www.ck12.org

The logically equivalent statement is quite useful: If f is not continuous at x0 , then f is not diﬀerentiable at x0 . (The converse is not necessarily true.) We have already seen that the converse is not true in some cases. The function can have a cusp, a corner, or a vertical tangent and still be continuous, but it is not diﬀerentiable. Multimedia Link The following simulator traces the instantaneous slope of a curve and graphs a qualitative form of derivative function on an axis below the curve Surﬁng the Derivative.

Review Questions In problems 16, use the deﬁnition of the derivative to ﬁnd f ′ (x) and then ﬁnd the equation of the tangent line at x = x0 . f (x) = √ 6x2 ; x0 = 3 f (x) = x + 2; x0 = 8 f (x) = 3x3 − 2; x0 = −1 1 f (x) = x+2 ; x0 = −1 2 f (x) = ax − b, (where a and b are constants); x0 = b f (x) = x1/3 ; x0 = 1. Find dy/dx|x=1 given that y = 5x2 − 2. √ Show that f (x) = 3 x is continuous at x = 0 but it is not diﬀerentiable at x = 0. Sketch the graph. 9. Show that { x2 + 1 x ≥ 1 f (x) = 2x x>1 1. 2. 3. 4. 5. 6. 7. 8.

is continuous and diﬀerentiable at x = 1. Sketch the graph of f . 10. Suppose that f is a diﬀerentiable function and has the property that f (x + y) = f (x) + f (y) + 3xy and f (h) = 4. h→0 h lim

Find f (0) and f ′ (x). www.ck12.org

104

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

f ′ (x) = 12x, y = 36x −( 54 ) 1 f ′ (x) = 2√x+2 , y = √16 12 x + 6 f ′ (x) = 9x2 , y = 9x + 4 −1 f ′ (x) = (x+2) 2 , y = −x ′ f (x) = 2ax, y = 2abx − b(ab + 1) f ′ (x) = 3x12/3 , y = 31 x + 23 10 Hint: Take the limit from both sides. Hint: Take the limit from both sides. f (0) = 0, f ′ (x) = 4 + 3x

2.3

Techniques of Diﬀerentiation

Learning Objectives A student will be able to: • Use various techniques of diﬀerentiations to ﬁnd the derivatives of various functions. • Compute derivatives of higher orders. Up to now, we have been calculating derivatives by using the deﬁnition. In this section, we will develop formulas and theorems that will calculate derivatives in more eﬀicient and quick ways. It is highly recommended that you become very familiar with all of these techniques.

The Derivative of a Constant If f (x) = c where c is a constant, then f ′ (x) = 0. In other words, the derivative or slope of any constant function is zero. Proof:

f (x + h) − f (x) c−c = lim =0 h→0 h→0 h h

f ′ (x) = lim Example 1:

If f (x) = 16 for all x, then f ′ (x) = 0 for all x. We can also write d/dx(16) = 0.

105

www.ck12.org

The Power Rule If n is a positive integer, then for all real values of x

d n [x ] = nxn−1 dx Multimedia Link The proof of the power rule is omitted in this text, but it is available at http://en.wikipedia.org/wiki/Calculus_with_polynomials and also in video form at Khan Academy Proof of the Power Rule . Note that this proof depends on using the

Figure 2.1: Proof that d/dx(xn̂) = n*x(̂n-1) http://www.youtube.com/watch?v= dZnc3PtNaN4&feature=youtube_gdata binomial theorem from precalculus. Example 2: If f (x) = x3 , then

f ′ (x) = 3x3−1 = 3x2 and

d [x] = 1 · x1−1 = x0 = 1, dx d √ d 1/2 1 1 1 1 [ x] = [x ] = x1/2−1 = x−1/2 = 1/2 = √ , dx dx 2 2 2x 2 x [ ] d −3 −3 d 1 = [x ] = −3x−3−1 = −3x−4 = 4 , 3 dx x dx x www.ck12.org

106

The Power Rule and a Constant If c is a constant and f is diﬀerentiable at all x, then

d d [cf (x)] = c [f (x)]. dx dx In simpler notation,

(cf )′ = c(f )′ · cf ′ . In other words, the derivative of a constant times a function is equal to the constant times the derivative of the function. Example 3:

d d [4x3 ] = 4 [x3 ] = 4[3x2 ] = 12x2 . dx dx Example 4: [ ] d −2 d d 8 = [−2x−4 ] = −2 [x−4 ] = −2[−4x−4−1 ] = −2[−4x−5 ] = 8x−5 = 5 . 4 dx x dx dx x Derivatives of Sums and Diﬀerences If f and g are two diﬀerentiable functions at x, then

d d d [f (x) + g(x)] = [f (x)] + [g(x)] dx dx dx and

d d d [f (x) − g(x)] = [f (x)] − [g(x)]. dx dx dx In simpler notation,

107

www.ck12.org

(f ± g)′ = f ′ ± g ′ . Example 5:

d d d [3x2 + 2x] = [3x2 ] + [2x] dx dx dx d 2 d = 3 [x ] + 2 [x] dx dx = 3[2x] + 2[1] = 6x + 2. Example 6:

d 3 d 3 d [x − 5x2 ] = [x ] − 5 [x2 ] dx dx dx 2 = 3x − 5[2x] = 3x2 − 10x. The Product Rule If f and g are diﬀerentiable at x, then

d d d [f (x) · g(x)] = f (x) g(x) + g(x) f (x). dx dx dx In a simpler notation,

(f · g)′ = f · g ′ + g · f ′ . The derivative of the product of two functions is equal to the ﬁrst times the derivative of the second plus the second times the derivative of the ﬁrst. Keep in mind that

(f · g)′ ̸= f ′ · g ′ . www.ck12.org

108

Example 7: Find

dy dx

for y = (3x4 + 2)(7x3 − 1).

Solution: There are two methods to solve this problem. One is to multiply the product and then use the derivative of the sum rule. The second is to directly use the product rule. Either rule will produce the same answer. We begin with the sum rule.

y = (3x4 + 2)(7x3 − 1) = 21x7 − 3x4 + 14x3 − 2. Taking the derivative of the sum yields dy = 147x6 − 12x3 + 42x2 + 0 dx = 147x6 − 12x3 + 42x2 . Now we use the product rule, dy = (3x4 + 2) · (7x3 − 1)′ + (3x4 + 2)′ · (7x3 − 1) dx = (3x4 + 2)(21x2 ) + (12x3 )(7x3 − 1) = (63x6 + 42x2 ) + (84x6 − 12x3 ) = 147x6 − 12x3 + 42x2 , which is the same answer. The Quotient Rule If f and g are diﬀerentiable functions at x and g(x) ̸= 0, then [ ] d d g(x) dx [f (x)] − f (x) dx [g(x)] d f (x) = . 2 dx g(x) [g(x)] In simpler notation, ( )′ f g · f ′ − f · g′ = . g g2

109

www.ck12.org

The derivative of a quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared. Keep in mind that the order of operations is important (because of the minus sign in the numerator) and ( )′ f f′ ̸= ′ . g g Example 8: Find dy/dx for

y=

x2 − 5 x3 + 2

Solution: [ ] dy d x2 − 5 = dx dx x3 + 2 (x3 + 2) · (x2 − 5)′ − (x2 − 5) · (x3 + 2)′ = (x3 + 2)2 (x3 + 2)(2x) − (x2 − 5)(3x2 ) = (x3 + 2)2 2x4 + 4x − 3x4 + 15x2 = (x3 + 2)2 4 −x + 15x2 + 4x = (x3 + 2)2 x(−x3 + 15x + 4) = . (x3 + 2)2 Example 9: At which point(s) does the graph of y =

x x2 +9

have a horizontal tangent line?

Solution: Since the slope of a horizontal line is zero, and since the derivative of a function signiﬁes the slope of the tangent line, then taking the derivative and equating it to zero will enable us to ﬁnd the points at which the slope of the tangent line equals to zero, i.e., the locations of the horizontal tangents. www.ck12.org

110

x , x2 + 9 (x2 + 9)(1) − x(2x) y′ = = 0. (x2 + 9)2 y=

Multiplying by the denominator and solving for x,

x2 + 9 − 2x2 = 0 x2 = 9 x = ±3. Therefore the tangent line is horizontal at x = −3, +3. Higher Derivatives If the derivative f of the function f is diﬀerentiable, then the derivative of f , denoted by f ′′ , is called the second derivative of f . We can continue the process of diﬀerentiating derivatives and obtain third, fourth, ﬁfth and higher derivatives of f . They are denoted by f ′ , f ′′ , f ′′′ , f (4) , f (5) , . . . Example 10: Find the ﬁfth derivative of f (x) = 2x4 − 3x3 + 5x2 − x − 1. Solution:

f ′ (x) = 8x3 − 9x2 + 5x − x f ′′ (x) = 24x2 − 18x + 5 f ′′′ (x) = 48x − 18 f (4) (x) = 48 f (5) (x) = 0 Example 11: Show that y = x3 + 3x + 2 satisﬁes the diﬀerential equation y ′′′ + xy ′′ − 2y ′ = 0. Solution: We need to obtain the ﬁrst, second, and third derivatives and substitute them into the diﬀerential equation.

111

www.ck12.org

y y′ y ′′ y ′′′

= x3 + 3x + 2 = 3x2 + 3 = 6x = 6.

Substituting,

y ′′′ + xy ′′ − 2y ′ = 6 + x(6x) − 2(3x2 + 3) = 6 + 6x2 − 6x2 − 6 =0 which satisﬁes the equation.

Review Questions Use the results of this section to ﬁnd the derivatives dy/dx. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

y = 5x7 y = 12 (x3 − 2x2 + 1) √ √ y = 2x3 − √12 x2 + 2x + 2 y = a2 − b2 + x2 − a − b + x (where a, b are constants) y = x−3 + x17 y = (x3 − 3x2 + x)(2x3 + 7x4 ) y = (√x1 + x12 )(3x4 − 7) y = x + √1x 3 y = √x+3 y = 4x+1 x2 −9 Newton’s Law of Universal Gravitation states that the gravitational force between two masses (say, the earth and the moon), m and M , is equal to their product divided by the square of the distance r between them. Mathematically, mM , r2 ( ) 2 where G is the Universal Gravitational Constant 1.602 × 10−11 Nkgm2 . If the distance r between the two masses is changing, ﬁnd a formula for the instantaneous rate of change of F with respect to the separation distance r. F =G

www.ck12.org

112

12. Find

[ ] d ψψ0 + ψ 3 dψ 3 − ψ0

where ψ0 is a constant. d3 y 13. Find dx , where y = 3 x=1

2 x3

.

Review Answers (some answers simplify further than the given responses) 1. 2. 3. 4. 5. 6. 7. 8. 9.

y′ y′ y′ y′ y′ y′ y′ y′ y′

= 35x6 = 32 x2 − 2x √ √ = 3 2x2 − 2x + 2 = 2x + 1 = −3 − x78 x4 = (x3 − 3x2 + x)(6x2 + 28x3 ) + (3x2 − 6x + 1)(2x3 + 7x4 ) = ( x1 + x12 )(12x3 ) + (− x12 − x23 )(3x4 − 7) = 2√1 x − 2x13/2 = 2√x(√3x+3)2

10. y ′ = −4x(x2−2x−36 −9)2 mM 11. dF = −2G dr r3 ψ0 +3ψ 2 12. 3−ψ0 13. −120 2

2.4

Derivatives of Trigonometric Functions

Learning Objectives A student will be able to: • Compute the derivatives of various trigonometric functions. Recall from Chapter 1 that if the angle h is measured in radians, limh→0

sin h h

= 1 and limh→0

1−cos h h

= 0.

We now want to ﬁnd an expression for the derivative of the six trigonometric functions sin x, cos x, tan x, sec x, csc x, and cot x. We ﬁrst consider the problem of diﬀerentiating sin x, using the deﬁnition of the derivative.

113

www.ck12.org

d sin(x + h) − sin x [sin x] = lim h→0 dx h Since

sin(α + β) = sin α cos β + cos α sin β. The derivative becomes

d sin x cos h + cos x sin h − sin x [sin x] = lim h→0 dx h ) [ ( ( )] cos h − 1 sin h = lim sin x + cos x h→0 h h ( ) ( ) 1 − cos h sin h = −sin x · lim + cos x · lim h→0 h→0 h h = −sin x · (0) + cos x · (1) = cos x. Therefore,

d [sin x] = cos x. dx It will be left as an exercise to prove that

d [cos x] = −sin x. dx The derivatives of the remaining trigonometric functions are shown in the table below. Derivatives of Trigonometric Functions www.ck12.org

114

d [sin x] = cos x dx d [cos x] = −sin x dx d [tan x] = sec2 x dx d [sec x] = sec x tan x dx d [csc x] = −csc x cot x dx d [sin x] = −csc2 x dx Keep in mind that for all the derivative formulas for the trigonometric functions, the argument x is measured in radians. Example 1: Show that

d [tan x] dx

= sec2 x.

Solution: It is possible to prove this relation by the deﬁnition of the derivative. However, we use a simpler method. Since

tan x =

sin x , cos x

then [ ] d d sin x [tan x] = dx dx cos x Using the quotient rule, (cos x)(cos x) − (sin x)(−sin x) = cos2 x 2 2 cos x + sin x = cos2 x 1 = cos2 x = sec2 x

115

www.ck12.org

Example 2: Find f (x) if f (x) = x2 cos x + sin x. Solution: Using the product rule and the formulas above, we obtain

f ′ (x) = x2 (−sin x) + 2x cos x + cos x = −x2 sin x + 2x cos x + cos x. Example 3: Find dy/dx if y =

cos x 1−tan x

. What is the slope of the tangent line at x = π/3?

Solution: Using the quotient rule and the formulas above, we obtain

dy (1 − tan x)(−sin x) − (cos x)(−sec2 x) = dx (1 − tan x)2 −sin x + tan x sin x + cos x sec2 x = (1 − tan x)2 To calculate the slope of the tangent line, we simply substitute x = π/3: dy dx

x= π3

=

−sin( π3 ) + tan( π3 ) sin( π3 ) + cos( π3 ) sec2 ( π3 ) . (1 − tan( π3 ))2

We ﬁnally get the slope to be approximately dy dx

= 4.9. x= π3

Example 4: If y = sec x, ﬁnd y ′′ (π/3). Solution: www.ck12.org

116

y ′ = sec x tan x y ′′ = sec x(sec2 x) + (sec x tan x) tan x = sec3 x + sec x tan2 x. Substituting for x = π/3, (π )

(π ) (π ) + sec tan2 3 √ 3 3 3 2 = (2) + (2)( 3) = 8 + (2)(3) = 14.

y ′′ = sec3

Thus y ′′ (π/3) = 14.

Review Questions Find the derivative y ′ of the following functions: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

y y y y y

= x sin x + 2 = x2 cos x − x tan x − 1 = sin2 x x−1 = sin sin x+1 x+sin x = cos cos x−sin x √

y = tanxx + 2 y = csc x sin x + x x y = sec csc x If y = csc x, ﬁnd y ′′ (π/6). Use the deﬁnition of the derivative to prove that

d [cos x] dx

= − sin x.

Review Answers 1. 2. 3. 4.

y′ y′ y′ y′

= x cos x + sin x = 2x cos x − x2 sin x − tan x − x sec2 x = 2 cos x sin x 2 cos x = (sin x+1)2 ( tan x+1 )2 ′ 5. y = 1 + 1−tan √ x 2 √ − 6. y ′ = 2cot x csc x x

117

www.ck12.org

7. 8. 9. 10.

2.5

y′ = 1 y ′ = sec2 x y(π/6) = 14 .

The Chain Rule

Learning Objectives A student will be able to: • Know the chain rule and its proof. • Apply the chain rule to the calculation of the derivative of a variety of composite functions. We want to derive a rule for the derivative of a composite function of the form f ◦ g in terms of the derivatives of f and g. This rule allows us to diﬀerentiate complicated functions in terms of known derivatives of simpler functions. The Chain Rule If g is a diﬀerentiable function at x and f is diﬀerentiable at g(x), then the composition function f ◦ g = f (g(x)) is diﬀerentiable at x. The derivative of the composite function is:

(f ◦ g)′ (x) = f ′ (g(x))g ′ (x). Another way of expressing, if u = u(x) and f = f (u), then

d du [f (u)] = f ′ (u) . dx dx And a ﬁnal way of expressing the chain rule is the easiest form to remember: If y is a function of u and u is a function of x, then

dy du dy = . . dx du dx Example 1: www.ck12.org

118

Diﬀerentiate f (x) = (2x3 − 4x2 + 5)2 . Solution: Using the chain rule, let u = 2x3 − 4x2 + 5. Then d d 2 [(2x2 − 4x2 + 5)2 ] = [u ] dx dx du = 2u dx = 2(2x3 − 4x2 + 5)(6x2 − 8x). The example above is one of the most common types of composite functions. It is a power function of the type

y = [u(x)]n . The rule for diﬀerentiating such functions is called the General Power Rule. It is a special case of the Chain Rule. The General Power Rule if

y = [u(x)]n then dy = n[u(x)]n−1 u′ (x). dx In simpler form, if

y = un then

y = nun−1 · u′ .

119

www.ck12.org

Example 2: What is the slope of the tangent line to the function y = point x = 3?

√

x2 − 3x + 2 that passes through

Solution: We can write y = (x2 − 3x + 2)1/2 . This example illustrates the point that n can be any real number including fractions. Using the General Power Rule, 1 dy 1 = (x2 − 3x + 2) 2 −1 (2x − 3) dx 2 1 2 = (x − 3x + 2)−1/2 (2x − 3) 2 (2x − 3) = √ 2 x2 − 3x + 2

To ﬁnd the slope of the tangent line, we simply substitute x = 3 into the derivative: dy dx

√ 3 3 2 = √ = . = √ 4 2 2 2 32 − 3(3) + 2 2(3) − 3

x=3

Example 3: Find dy/dx for y = sin3 x. Solution: The function can be written as y = [sin x]3 . Thus dy = 3[sin x]2 [cos x] dx = 3 sin2 x cos x Example 4: Find dy/dx for y = 5 cos(3x2 − 1). Solution: Let u = 3x2 − 1. By the chain rule, d du [f (u)] = f ′ (u) dx dx www.ck12.org

120

where f (u) = 5 cos u. Thus

dy = 5(− sin u).(6x) dx = −5 sin u.(6x) = −30x sin(3x2 − 1) Example 5: Find dy/dx for y = [cos(πx2 )]3 . Solution: This example applies the chain rule twice because there are several functions embedded within each other. Let u be the inner function and w be the innermost function.

y = (u(w))3 u(x) = cos x w(x) = πx2 . Using the chain rule,

d du [f (u)] = f ′ (u) dx dx d 3 d [u ] = [cos3 (πx2 )] dx dx d = [cos(πx2 )]3 dx = 3[cos(πx2 )]2 [− sin(πx2 )](2πx) = −6πx[cos(πx2 )]2 sin(πx2 ). Notice that we used the General Power Rule and, in the last step, we took the derivative of the argument.

Review Questions Find f ′ (x).

121

www.ck12.org

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

f (x) = (2x2 − 3x)39 ( )−3 f (x) = x3 − x52 1 f (x) = √3x2 −6x+2 f (x) = sin3 x f (x) = sinx f (x) = sin3 x3 5 f (x) = tan(4x ) √ f (x) = 4x − sin2 2x sin x f (x) = cos(3x−2) f (x) = (5x + 8)3 (x3 + 7x)13 ( x−3 )3 f (x) = 2x−5

Review Answers 1. f ′ (x) = 39(2x2 − 3x)38 (4x − 3) ( 2 )4 ( 5 ) 3x −10 ′ 2. f (x) = −3 x5x−5 x3 3. f ′ (x) = √ 4. 5. 6. 7. 8.

−3(x−1) (3x2 −6x+1) 2

3

f ′ (x) = 3 sin x cos x f ′ (x) = 3x2 cos x3 f ′ (x) = 9x2 cos x3 sin2 x3 f ′ (x) = 20x4 sec2 (4x5 ) √ 2xcosx) f ′ (x) = 2(1−sin

4x−sin2 2x cos(3x−2) cos x+3 sin(3x−2) sin x cos2 (3x−2) 3 3 12 2

9. f ′ (x) = 10. f ′ (x) = 13(5x + 8) (x + 7x) (3x + 7) + 15(x3 + 7x)13 (5x + 8)2 11. .

2.6

Implicit Diﬀerentiation

Learning Objectives A student will be able to: • Find the derivative of variety of functions by using the technique of implicit diﬀerentiation. Consider the equation www.ck12.org

122

2xy = 1. We want to obtain the derivative dy/dx. One way to do it is to ﬁrst solve for y,

y=

1 , 2x

and then project the derivative on both sides, [ ] dy d 1 = dx dx 2x −1 = 2. 2x There is another way of ﬁnding dy/dx. We can directly diﬀerentiate both sides: d d [2xy] = [1]. dx dx Using the Product Rule on the left-hand side,

y

d d [2x] + 2x [y] = 0 dx dx dy y[2] + 2x = 0. dx

Solving for dy/dx, dy −2y −y = = . dx 2x x But since y =

1 2x

, substitution gives −1 dy = dx x(2x) −1 = 2. 2x

123

www.ck12.org

which agrees with the previous calculations. This second method is called the implicit diﬀerentiation method. You may wonder and say that the ﬁrst method is easier and faster and there is no reason for the second method. That’s probably true, but consider this function:

3y 2 − cos y = x3 . How would you solve for y? That would be a diﬀicult task. So the method of implicit diﬀerentiation sometimes is very useful, especially when it is inconvenient or impossible to solve for y in terms of x. Explicitly deﬁned functions may be written with a direct relationship between two variables with clear independent and dependent variables. Implicitly deﬁned functions or relations connect the variables in a way that makes it impossible to separate the variables into a simple input output relationship. More notes on explicit and implicit functions can be found at http://en.wikipedia.org/wiki/Implicit_function. Example 1: Find dy/dx if 3y 2 − cos y = x3 . Solution: Diﬀerentiating both sides with respect to x and then solving for dy/dx,

d d 3 [3y 2 − cos y] = [x ] dx dx d d 3 [y 2 ] − [cos y] = 3x2 dx dx dy dy 3(2y ) − (− sin y) = 3x2 dx dx dy dy 6y + sin y = 3x2 dx dx dy [6y + sin y] = 3x2 . dx Solving for dy/dx, we ﬁnally obtain

dy 3x2 = . dx 6y + sin y Implicit diﬀerentiation can be used to calculate the slope of the tangent line as the example below shows. www.ck12.org

124

Example 2: Find the equation of the tangent line that passes through point (1, 2) to the graph of 8y 3 + x2 y − x = 3. Solution: First we need to use implicit diﬀerentiation to ﬁnd dy/dx and then substitute the point (1, 2) into the derivative to ﬁnd slope. Then we will use the equation of the line (either the slope-intercept form or the point-intercept form) to ﬁnd the equation of the tangent line. Using implicit diﬀerentiation,

d d [8y 3 + x2 y − x] = [3] dx dx dy dy 24y 2 + [(x2 )(1) + y(2x)] − 1 = 0 dx dx dy dy 24y 2 + x2 + 2xy − 1 = 0 dx dx dy = 1 − 2xy [24y 2 + x2 ] dx dy 1 − 2xy = . dx 24y 2 + x2 Now, substituting point (1, 2) into the derivative to ﬁnd the slope,

1 − 2(1)(2) dy = dx 24(2)2 + (1)2 −3 = . 97 So the slope of the tangent line is −3/97, which is a very small value. (What does this tell us about the orientation of the tangent line?) Next we need to ﬁnd the equation of the tangent line. The slope-intercept form is

y = mx + b, where m = −3/97 and b is the y−intercept. To ﬁnd it, simply substitute point (1, 2) into the line equation and solve for b to ﬁnd the y−intercept.

125

www.ck12.org

(

−3 2= 97 197 b= . 97

) (1) + b

Thus the equation of the tangent line is

y=

−3 197 x+ . 97 97

Remark: we could have used the point-slope form y − y1 = m(x − x1 ) and obtained the same equation. Example 3: Use implicit diﬀerentiation to ﬁnd d2 y/dx2 if 5x2 − 4y 2 = 9. Also ﬁnd does the second derivative represent? Solution:

d d [5x2 − 4y 2 ] = [9] dx dx dy 10x − 8y = 0. dx Solving for dy/dx,

dy 5x = . dx 4y Diﬀerentiating both sides implicitly again (and using the quotient rule), (4y)(5) − (5x)(4dy/dx) d2 y = 2 dx (4y)2 20y 20x dy = − 2 16y 16y 2 dx 5 5x dy = − . 4y 4y dx www.ck12.org

126

d2 y dx2

. What (x, y)=(2, 3)

But since dy/dx = 5x/4y, we substitute it into the second derivative: d2 y 5 5x 5x = − . 2 dx 4y 4y 4y 2 5 25x2 dy = − . dx2 4y 16y 2 This is the second derivative of y. d2 y The next step is to ﬁnd: dx 2

(x, y)=(2, 3)

d y dx2 2

(2, 3)

5 25(2)2 = − 4(2) 16(3)2 =−

5 . 72

Since the ﬁrst derivative of a function represents the rate of change of the function y = f (x) with respect to x, the second derivative represents the rate of change of the rate of change of the function. For example, in kinematics (the study of motion), the speed of an object (y ′ ) signiﬁes the change of position with respect to time but acceleration (y ′′ ) signiﬁes the rate of change of the speed with respect to time.

Review Questions Find dy/dx by implicit diﬀerentiation. 1. 2. 3. 4. 5. 6.

x2 + y 2 = 500 x2 y + 3xy − 2 = 1 1 + y1 = 12 x √ √ √ x− y = 3 sin(25xy 2 ) = x tan3 (x2 − y 2 ) = tan(π/4)

In problems #7 and 8, use implicit diﬀerentiation to ﬁnd the slope of the tangent line to the given curve at the speciﬁed point. 7. 8. 9. 10.

x2 y − y 2 x = −1 at (1, 1) sin(xy) = y at (π, 1) Find y ′′ by implicit diﬀerentiation for x3 y 3 = 5. Use implicit diﬀerentiation to show that the tangent line to the curve y 2 = kx at (x0 , y0 ) is given by y0 y = 21 k(x + x0 ), where k is a constant.

127

www.ck12.org

Review Answers 1. y ′ = 2. y ′ = 3. 4. 5. 6. 7. 8. 9. 10.

2.7

y′ y′ y′ y′ 1

= = = =

−1 1+π ′

y = .

−x y −y(2x+3) x(x+3) −y 2 x2 √ y x 1−25y 2 cos(25xy 2 ) 50xy cos(25xy 2 ) x y

2y x2

Linearization and Newton’s Method

Learning Objectives A student will be able to: • Approximate a function by the method of linearization. • Know Newton’s Method for approximating roots of a function.

Linearization: The Tangent Line Approximation If f is a diﬀerentiable function at x0 , then the tangent line, y = mx+b, to the curve y = f (x) at x0 is a good approximation to the curve y = f (x) for values of x near x0 (Figure 8a). If you “zoom in” on the two graphs, y = f (x) and the tangent line, at the point of tangency, (x0 , f (x0 )), or if you look at a table of values near the point of tangency, you will notice that the values are very close (Figure 8b). Since the tangent line passes through point (x0 , f (x0 )) and the slope is f ′ (x0 ), we can write the equation of the tangent line, in point-slope form, as

y − y0 = m(x − x0 ) y − f (x0 ) = f ′ (x0 )(x − x0 ) Solving for y, www.ck12.org

128

y = f (x0 ) + f ′ (x0 )(x − x0 )

So for values of x close to x0 , the values of y of this tangent line will closely approximate f (x). This gives the approximation

129

www.ck12.org

f (x) = f (x0 ) + f ′ (x0 )(x − x0 ).

The Tangent Line Approximation (Linearization) If f is a diﬀerentiable function at x = x0 , then the approximation function

L(x) = f (x) ≈ f (x0 ) + f ′ (x0 )(x − x0 )

is a linearization of f near x0 . Example 1: Find the linearization of f (x) =

√

x + 3 at point x = 1.

Solution: Taking the derivative of f (x),

1 f ′ (x) = (x + 3)−1/2 , 2

we have f (1) =

√

4 = 2, f ′ (1) = 1/4, and

f (x) ≈ f (x0 ) + f ′ (x0 )(x − x0 ) 1 ≈ 2 + (x − 1) 4 1 7 ≈ x+ . 4 4 √ This tells us that near the point x = 1, the function f (x) = x + 3 approximates the line y = (x/4) + 7/4. As we move away from x = 1, we lose accuracy (Figure 9). www.ck12.org

130

Example 2: Find the linearization of y = sin x at x = π/3. Solution: Since f (π/3) = sin(π/3) =

√

3/2, and f ′ (x) = cos x, f ′ (π/3) = cos(π/3) = 1/2, we have

√

3 1( π) + x− 2 3 √2 3 x π + − ≈ 2 2 6 x ≈ + 0.343. 2

f (x) ≈

131

www.ck12.org

Newton’s Method When faced with a mathematical problem that cannot be solved with simple algebraic means, such as ﬁnding the roots of the polynomial x3 − 2x + 3 = 0, calculus sometimes provides a way of ﬁnding the approximate solutions. √ Let’s say we are interested in computing 5 without using a calculator or a table. To do so, think about this problem in a diﬀerent way. Assume that we are interested in solving the quadratic equation

f (x) = x2 − 5 = 0 √ which leads to the roots x = ± 5. The idea here is to ﬁnd the linearization of the above function, which is a straight-line equation, and then solve the linear equation for x. Since √

4<

√

5<

√

9

or

2<

√

5 < 3,

We choose the linear approximation of f (x) to be near x0 = 2. Since f (x) = x2 − 5, f ′ (x) = 2x and thus f (2) = −1 and f ′ (2) = 4. Using the linear approximation formula,

f (x) ≈ f (x0 ) + f ′ (x0 )(x − x0 ) ≈ −1 + (4)(x − 2) ≈ −1 + 4x − 8 ≈ 4x − 9. Notice that this equation is much easier to solve than f (x) = x2 − 5. Setting f (x) = 0 and solving for x, we obtain, www.ck12.org

132

4x − 9 = 0 9 x= 4 = 2.25. If you use a calculator, you will get x = 2.236 . . . As you can see, this is a fairly good approximation. To be sure, calculate the percent diﬀerence [% diff] between the actual value and the approximate value,

% diﬀ =

2|A − X| 100%, |A + X|

where A is the accepted value and X is the calculated value. 2|2.236 − 2.25| 100% |2.236 + 2.25| = 0.62%,

% diﬀ =

which is less than 1%. We can actually make our approximation even better by repeating what we have just done √ not for x = 2, but for x1 = 2.25 = 94 , a number that is even closer to the actual value of 5. Using the linear approximation again,

f (x) ≈ f (x1 ) + f ′ (x1 )(x − x1 ) ( ) 1 9 9 ≈ + x− 16 2 4 9 161 ≈ x− . 2 16 Solving for x by setting f (x) = 0, we obtain

x = x2 = 2.236111, which is even a better approximation √ than x1 = 9/4. We could continue this process generating a better approximation to 5. This is the basic idea of Newton’s Method.

133

www.ck12.org

Here is a summary of Newton’s method. Newton’s Method

1. Guess the ﬁrst approximation to a solution of the equation f (x) = 0. A graph would be very helpful in ﬁnding the ﬁrst approximation (see Figure below). 2. Use the ﬁrst approximation to ﬁnd the second, the second to ﬁnd the third and so on by using the recursion relation

xn+1 = xn −

f (xn ) . f ′ (xn )

Example 3: Use Newton’s method to ﬁnd the roots of the polynomial f (x) = x3 + x − 1. Solution:

f (x) = x3 + x − 1 f ′ (x) = 3x2 + 1. Using the recursion relation, www.ck12.org

134

f (xn ) f ′ (xn ) x3 + xn − 1 = xn − n 2 . 3xn + 1

xn+1 = xn −

To help us ﬁnd the ﬁrst approximation, we make a graph of f (x). As Figure 11 suggests, set x1 = 0.6. Then using the recursion relation, we can generate x2 , x3 , . . ..

x3n + xn − 1 3x2n + 1 (0.6)3 + (0.6) − 1 x2 = 0.6 − 3(0.6)2 + 1 = 0.6884615.

xn+1 = xn −

Using the recursion relation again to ﬁnd x3 , we get

x3 = 0.6836403. We conclude that the solution to the equation x3 + x − 1 = 0 is about 0.6836403.

135

www.ck12.org

Review Questions 1. Find the linearization of f (x) =

x2 + 1 x

at a = 1. 2. Find the linearization of f (x) = tan x at a = π. 3. Use the linearization method to show that when x ≪ 1 (much less than 1), then (1 + x)n ≈ 1 + nx. 4. Use the result of problem #3, (1 + x)n ≈ 1 + nx , to ﬁnd the approximation for the following: 4 (a) f (x) = (1 √ − x) (b) f (x) = 1 − x 5 (c) f (x) = √1+x √( )2 3 (d) f (x) = 3 1 − (x−1)

(e) Without using a calculator, approximate (1.003)99 . 5. Use Newton’s Method to ﬁnd the roots of x3 + 3 = √ 0. 6. Use Newton’s Method to ﬁnd the roots of −x + 3 −1 + x = 0.

Review Answers f (x) ≈ 2 f (x) ≈ x − π Hint: Let x = 0. (a) 1 − 4x (b) 1 − 12 x (c) 5 + 5x (d) −1 + 2x (e) 1.297 5. x ≈ −1.442 6. x ≈ 1.146 and x ≈ 7.854 1. 2. 3. 4.

www.ck12.org

136

Chapter 3 Applications of Derivatives 3.1

Related Rates

Learning Objectives A student will be able to:

• Solve problems that involve related rates.

Introduction In this lesson we will discuss how to solve problems that involve related rates. Related rate problems involve equations where there is some relationship between two or more derivatives. We solved examples of such equations when we studied implicit diﬀerentiation in Lesson 2.6. In this lesson we will discuss some real-life applications of these equations and illustrate the strategies one uses for solving such problems. Let’s start our discussion with some familiar geometric relationships. Example 1: Pythagorean Theorem

x2 + y 2 = z 2

137

www.ck12.org

We could easily attach some real-life situation to this geometric ﬁgure. Say for instance that x and y represent the paths of two people starting at point p and walking North and West, respectively, for two hours. The quantity z represents the distance between them at any time t. Let’s now see some relationships between the various rates of change that we get by implicitly diﬀerentiating the original equation x2 + y 2 = z 2 with respect to time t.

x2 + y 2 = z 2 dx dy dz 2x + 2y = 2z . dt dt dt Simplifying, we have Equation 1. x dx + y dy = z dz dt dt dt So we have relationships between the derivatives, and since the derivatives are rates, this is an example of related rates. Let’s say that person x is walking at 5 mph and that person y is walking at 3 mph. The rate at which the distance between the two walkers is changing at any time is dependent on the rates at which the two people are walking. Can you think of any problems you could pose based on this information? One problem that we could pose is at what rate is the distance between x and y increasing after one hour. That is, ﬁnd dz/dt. Solution: Assume that they have walked for one hour. So x = 5 mi and y = 3.√Using the Pythagorean Theorem, we ﬁnd the distance between them after one hour is z = 34 = 5.83 miles. www.ck12.org

138

If we substitute these values into Equation 1 along with the individual rates we get √

dz dt √ dz 34 = 34 dt 34 dz √ = . dt 34

5(5) + 3(3) =

34

Hence after one hour the distance between the two people is increasing at a rate of √34 ≈ 5.83 mph. 34

dz dt

=

Our second example lists various formulas that are found in geometry. As with the Pythagorean Theorem, we know of other formulas that relate various quantities associated with geometric shapes. These present opportunities to pose and solve some interesting problems Example 2: Perimeter and Area of a Rectangle We are familiar with the formulas for Perimeter and Area:

P = 2 ∗ l + 2 ∗ w, A = l ∗ w. Suppose we know that at an instant of time, the length is changing at the rate of 8 ft/hour and the perimeter is changing at a rate of 24 ft/hour. At what rate is the width changing at that instant?

139

www.ck12.org

Solution: If we diﬀerentiate the original equation, we have Equation 2:

dp dt

=2∗

dl dt

+2∗

dw . dt

Substituting our known information into Equation II, we have

24 = (2 ∗ 8) + 2 ∗ 8=2∗ 4=

dw dt

dw dt

dw . dt

The width is changing at a rate of 4 ft/hour. Okay, rather than providing a related rates problem involving the area of a rectangle, we will leave it to you to make up and solve such a problem as part of the homework (HW #1). Let’s look at one more geometric measurement formula. Example 3: Volume of a Right Circular Cone 1 V = πr2 h 3

We have a water tank shaped as an inverted right circular cone. Suppose that water ﬂows into the tank at the rate of 5 ft3 /min. At what rate is the water level rising when the height of the water in the tank is 6 feet? www.ck12.org

140

Solution: We ﬁrst note that this problem presents some challenges that the other examples did not. When we diﬀerentiate the original equation, V = (1/3)πr2 h, we get

dV 1 dr 1 dh = π(h)(2r) + πr2 . dt 3 dt 3 dt The diﬀiculty here is that we have no information about the radius when the water level is at 6 feet. So we need to relate the radius a quantity that we do know something about. Starting with the original equation, let’s ﬁnd a relationship between h and r. Let r1 be the radius of the surface of the water as it ﬂows out of the tank.

Note that the two triangles are similar and thus corresponding parts are proportional. In particular,

8 r1 = h 20 8h 2h r1 = = . 20 5

141

www.ck12.org

Now we can solve the problem by substituting r1 = (2h/5) into the original equation:

1 V = π 3 Hence

dV dt

=

12π 2 dh h dt , 15

(

2h 5

)2 h=

4π 3 h. 15

and by substitution, 12π dh (36) 15 dt dh 75 ft = ≈ .06 . dt 432π min 5=

Multimedia Link Related rates make for great animations. In the following applet you can explore a problem about a melting snowball where the radius is decreasing at a constant rate. Calculus Applets Snowball Problem. Experiment with changing the time to see how the volume does not change at a constant rate in this problem. If you’d like to see a video of another example of a related rate problem worked out, see Khan Academy Rates of Change (Part 2) (5:38) .

Figure 3.1: Another (simpler) example of using the chain rule to determine rates-of-change http://www.youtube.com/watch?v=xmgk8_l3lig&feature=youtube_gdata

Lesson Summary 1. We learned to solve problems that involved related rates.

Review Questions 1. (a) Make up a related rates problem about the area of a rectangle. www.ck12.org

142

2.

3.

4.

5.

6. 7.

8.

9.

10.

(b) Illustrate the solution to your problem. Suppose that a particle is moving along the curve 4x2 + 16y 2 = 32. When it reaches the point (2, 1), the x−coordinate is increasing at a rate of 3 ft/sec. At what rate is the y−coordinate changing at that instant? A regulation softball diamond is a square with each side of length 60 ft. Suppose a player is running from ﬁrst base to second base at a speed of 18 ft/sec. At what rate is the distance between the runner and home plate changing when the runner is 2/3 of the way from ﬁrst to second base? At a recent Hot Air Balloon festival, a hot air balloon was released. Upon reaching a height of 300 ft, it was rising at a rate of 20 ft/sec. Mr. Smith was 100 ft away from the launch site watching the balloon. At what rate was the distance between Mr. Smith and the balloon changing at that instant? Two trains left the St. Louis train station in the late morning. The ﬁrst train was traveling East at a constant speed of 65 mph. The second train traveled South at a constant speed of 75 mph. At 3 PM, the ﬁrst train had traveled a distance of 120 miles while the second train had traveled a distance of 130 miles. How fast was the distance between the two trains changing at that time? Suppose that a 17 ft ladder is sliding down a wall at a rate of −6 ft/sec. At what rate is the bottom of the ladder moving when the top is 8 ft from the ground? Suppose that the length of a rectangle is increasing at the rate of 6 ft/min and the width is increasing at a rate of 2 ft/min. At what rate is the area of the rectangle changing when its length is 25 ft and its width is 15 ft? Suppose that the quantity demand of new 40′′ plasma TVs is related to its unit price by the formula p + x2 = 1200, where p is measured in dollars and x is measured in units of one thousand. How is the quantity demand changing when x = 20, p = 1500, and the price per TV is decreasing at a rate of $10? The volume of a cube with side s is changing. At a certain instant, the sides of the cube are 6 inches and increasing at the rate of 1/4 in/min. How fast is the volume of the cube increasing at that time? (a) Suppose that the area of a circle is increasing at a rate of 24 in2 / min. How fast is the radius increasing when the area is 36π in2 ? (b) How fast is the circumference changing at that instant?

Review Answers 1. Answers will vary. ft 2. dy = − 32 sec dt 3. Using the following diagram,

dy dt

=

√720 ft 5200 sec

143

ft ≈ 10.18 sec .

www.ck12.org

4. Using the following diagram,

dy dt

=

ft √ 6000 100000 sec

5. Using the following diagram,

ds dt

=

√17550 31300

www.ck12.org

ft ≈ 18.97 sec .

≈ 99.20 mph.

144

6. Using the following diagram,

dx dt

ft = − 16 . 5 sec

ft 7. dA = 140 min dt 8. The demand is increasing at a rate of 1/4 per thousand units, or 250 units per week. in3 9. dV = 27 min dt in 10. (a) dr = π2 min dt dp in (b) dt = 4 min

145

www.ck12.org

3.2

Extrema and the Mean Value Theorem

Learning Objectives A student will be able to: • Solve problems that involve extrema. • Study Rolle’s Theorem. • Use the Mean Value Theorem to solve problems.

Introduction In this lesson we will discuss a second application of derivatives, as a means to study extreme (maximum and minimum) values of functions. We will learn how the maximum and minimum values of functions relate to derivatives. Let’s start our discussion with some formal working deﬁnitions of the maximum and minimum values of a function. Deﬁnition A function f has a maximum at x = a if f (a) ≥ f (x) for all x in the domain of f. Similarly, f has a minimum at x = a if f (a) ≤ f (x) for all x in the domain of f. The values of the function for these x−values are called extreme values or extrema. Here is an example of a function that has a maximum at x = a and a minimum at x = d:

Observe the graph at x = b. While we do not have a minimum at x = b, we note that f (b) ≤ f (x) for all x near b. We say that the function has a local minimum at x = b. Similarly, we say that the function has a local maximum at x = c since f (c) ≥ f (x) for some x contained in open intervals of c. Let’s recall the Min-Max Theorem that we discussed in lesson on Continuity. Min-Max Theorem: If a function f (x) is continuous in a closed interval I, then f (x) has both a maximum value and a minimum value in I. In order to understand the proof for the www.ck12.org

146

Min-Max Theorem conceptually, attempt to draw a function on a closed interval (including the endpoints) so that no point is at the highest part of the graph. No matter how the function is sketched, there will be at least one point that is highest. We can now relate extreme values to derivatives in the following Theorem by the French mathematician Fermat. Theorem: If f (c) is an extreme value of f for some open interval of c, and if f ′ (c) exists, then f ′ (c) = 0. Proof: The theorem states that if we have a local max or local min, and if f ′ (c) exists, then we must have f ′ (c) = 0. Suppose that f has a local max at x = c. Then we have f (c) ≥ f (x) for some open interval (c − h, c + h) with h > 0. So f (c + h) − f (c) ≤ 0. Consider limh→0+

f (c+h)−f (c) . h

Since f (c + h) − f (c) ≤ 0, we have limh→0+ Since f ′ (c) exists, we have f ′ (c) = limh→0

f (c+h)−f (c) h

f (c+h)−f (c) h

If we take the left-hand limit, we get f ′ (c) = limh→0

≤ limh→0+ 0 = 0.

= limh→0+

f (c+h)−f (c) , h

f (c+h)−f (c) h

= limh→0−

and so f ′ (c) ≤ 0. f (c+h)−f (c) h

≥ 0.

Hence f ′ (c) ≥ 0 and f ′ (c) ≤ 0 it must be that f ′ (c) = 0. If x = c is a local minimum, the same argument follows.

Deﬁnition We will call x = c a critical value in [a, b] if f ′ (c) = 0 or f ′ (c) does not exist, or if x = c is an endpoint of the interval. We can now state the Extreme Value Theorem.

Extreme Value Theorem: If a function f (x) is continuous in a closed interval [a, b], with the maximum of f at x = c1 and the minimum of f at x = c2 , then c1 and c2 are critical values of f. Proof: The proof follows from Fermat’s theorem and is left as an exercise for the student. Example 1: Let’s observe that the converse of the last theorem is not necessarily true: If we consider f (x) = x3 and its graph, then we see that while f ′ (0) = 0 at x = 0, x = 0 is not an extreme point of the function.

147

www.ck12.org

Rolle’s Theorem: If f is continuous and diﬀerentiable on a closed interval [a, b] and if f (a) = f (b), then f has at least one value c in the open interval (a, b) such that f ′ (c) = 0. The proof of Rolle’s Theorem can be found at http://en.wikipedia.org/wiki/Rolle's_ theorem. Mean Value Theorem: If f is a continuous function on a closed interval [a, b] and if f ′ contains the open interval (a, b) in its domain, then there exists a number c in the interval (a, b) such that f (b) − f (a) = (b − a)f ′ (c). Proof: Consider the graph of f and secant line s as indicated in the ﬁgure.

By the Point-Slope form of line s we have y − f (a) = m(x − a) and y = m(x − a) + f (a). For each x in the interval (a, b), let g(x) be the vertical distance from line S to the graph of f. Then we have g(x) = f (x) − [m(x − a) + f (a)] for every x in (a, b). Note that g(a) = g(b) = 0. Since g is continuous in [a, b] and g ′ exists in (a, b), then Rolle’s

www.ck12.org

148

Theorem applies. Hence there exists c in (a, b) with g ′ (c) = 0. So g ′ (x) = f ′ (x) − m for every x in (a, b). In particular, g ′ (c) = f ′ (c) − m = 0 and

f ′ (c) = m f (b) − f (a) f ′ (c) = b−a f (b) − f (a) = (b − a)f ′ (c). The proof is complete. Example 2: Verify that the Mean Value Theorem applies for the function f (x) = x3 + 3x2 − 24x on the interval [1, 4]. Solution: We need to ﬁnd c in the interval (1, 4) such that f (4) − f (1) = (4 − 1)f ′ (c). Note that f ′ (x) = 3x2 + 6x − 24, and f (4) = 16, f (1) = −20. Hence, we must solve the following equation:

36 = 3f ′ (c) 12 = f ′ (c). By substitution, we have

12 = 3c2 + 6c − 24 3c2 + 6c − 36 = 0 c2 + 2c − 12 = 0 √ −2 ± 52 c= ≈ −4.61, 2.61. 2 Since we need to have c in the interval (1, 4), the positive root is the solution, c = 2.61.

149

√ −2+ 52 2

≈

www.ck12.org

Lesson Summary 1. We learned to solve problems that involve extrema. 2. We learned about Rolle’s Theorem. 3. We used the Mean Value Theorem to solve problems.

Review Questions In problems #1–3, identify the absolute and local minimum and maximum values of the function (if they exist); ﬁnd the extrema. (Units on the axes indicate 1 unit). 1. Continuous on [0, 9]

2. Continuous on [0, 9]

www.ck12.org

150

3. Continuous on [0, 4] ∪ (4, 9)

In problems #4–6, ﬁnd the extrema and sketch the graph.

f (x) = −x2 − 6x + 4, [−4, 1] f (x) = x3 − x4 , [0, 2] f (x) = −x2 + x42 , [−2, 0] Verify Rolle’s Theorem by ﬁnding values of x for which f (x) = 0 and f ′ (x) = 0. f (x) = 3x3 − 12x 2 8. Verify Rolle’s Theorem for f (x) = x2 − x−1 . 9. Verify that the Mean Value Theorem works for f (x) = (x+2) , [1, 2]. x 3 2 10. Prove that the equation x + a1 x + a2 x = 0 has a positive root at x = r, and that the equation 3x2 + 2a1 x + a2 = 0 has a positive root less than r. 4. 5. 6. 7.

Review Answers 1. Absolute max at x = 7, absolute minimum at x = 4, relative maximum at x = 2, Note: there is no relative minimum at x = 9 because there is no open interval around x = 9 since the function is deﬁned only on x = 9; the extreme values of f are f (7) = 7, f (4) = 0.

151

www.ck12.org

2. Absolute maximum at x = 7, absolute minimum at x = 9, relative minimum at x = 3, Note: there is no relative minimum at x = 0 because there is no open interval around x = 0 since the function is deﬁned only on [0, 9]; the extreme values of f are f (7) = 9, f (9) = 0.

3. Absolute minimum at x = 0, f (0) = 1; there is no maximum since the function is not continuous on a closed interval.

4. Absolute maximum at x = −3, f (−3) = 13, absolute minimum at x = 1, f (1) = −3 www.ck12.org

152

5. Absolute maximum at x = 34 , f ( 34 ) ≈ .1055, absolute minimum at x = 2, f (2) = −8.

√ √ 6. Absolute minimum at x = − 2, f (− 2) = 4.

153

www.ck12.org

√

7. f (x) = 0 at x = 0, ±2 f ′ (x) = 0, x = ± 2 3 3 ; (by Rolle’s Theorem, there is a critical √ value in each of the intervals (−2, 0) and (2, 0) and we found those to be x = ± 2. 8. f (x) = 2 at x = −1, 0 f ′ (x) = 0 at x ≈ −0.84 by Rolle’s Theorem, there is a critical value in the interval (−1, 0) and we found it to be x ≈ −0.48. √ 9. Need to ﬁnd c ∈ (1, 2) such that f (2) − f (1) = (2 − 1)f ′ (c); c = 2. 10. Let f (x) = x3 + a1 x2 + a2 x. Observe that f (0) = f (r) = 0. By Rolle’s Theorem, there exist c ∈ (0, r) such that f ′ (0) = 0.

3.3

The First Derivative Test

Learning Objectives A student will be able to: • Find intervals where a function is increasing and decreasing. • Apply the First Derivative Test to ﬁnd extrema and sketch graphs.

Introduction In this lesson we will discuss increasing and decreasing properties of functions, and introduce a method with which to study these phenomena, the First Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph. Note on notation: The symbol ϵ and ∈ are equivalent and denote that a particular element is contained within a particular set. www.ck12.org

154

Deﬁnition A function f is said to be increasing on [a, b] contained in the domain of f if f (x1 ) ≤ f (x2 ) whenever x1 ≤ x2 for all x1 , x2 ∈ [a, b]. A function f is said to be decreasing on [a, b] contained in the domain of f if f (x1 ) ≥ f (x2 ) whenever x1 ≥ x2 for all x1 , x2 ∈ [a, b]. If f (x1 ) < f (x2 ) whenever x1 < x2 for all x1 , x2 ∈ [a, b], then we say that f is strictly increasing on [a, b]. If f (x1 ) > f (x2 ) whenever x1 > x2 for all x1 , x2 ∈ [a, b], then we say that f is strictly decreasing on [a, b]. We saw several examples in the Lesson on Extreme and the Mean Value Theorem of functions that had these properties.

Example 1: The function f (x) = x3 is strictly increasing on (−∞, +∞):

Example 2: The function indicated here is strictly increasing on (0, a) and (b, c), and strictly decreasing on (a, b) and (c, d).

155

www.ck12.org

We can now state the theorems that relate derivatives of functions to the increasing/decreasing properties of functions. Theorem: If f is continuous on interval [a, b], then: 1. If f ′ (x) > 0 for every x ∈ [a, b], then f is strictly increasing in [a, b]. 2. If f ′ (x) < 0 for every x ∈ [a, b], then f is strictly decreasing in [a, b]. Proof : We will prove the ﬁrst statement. A similar method can be used to prove the second statement and is left as an exercise to the student. Consider x1 , x2 ∈ [a, b] with x1 < x2 . By the Mean Value Theorem, there exists c ∈ (x1 , x2 ) such that

f (x2 ) − f (x1 ) = (x2 − x1 )f ′ (c). By assumption, f ′ (x) > 0 for every x ∈ [a, b]; hence f ′ (c) > 0. Also, note that x2 − x1 > 0. Hence f (x2 ) − f (x1 ) > 0 and f (x2 ) > f (x1 ). We can observe the consequences of this theorem by observing the tangent lines of the following graph. Note the tangent lines to the graph, one in each of the intervals (0, a), (a, b), (b, +∞).

Note ﬁrst that we have a relative maximum at x = a and a relative minimum at x = b. The slopes of the tangent lines change from positive for x ∈ (0, a) to negative for x ∈ (a, b) and then back to positive for x ∈ (b, +∞). From this we example infer the following theorem: First Derivative Test Suppose that f is a continuous function and that x = c is a critical value of f. Then: 1. If f ′ changes from positive to negative at x = c, then f has a local maximum at x = c. 2. If f ′ changes from negative to positive at x = c, then f has a local minimum at x = c. www.ck12.org

156

3. If f ′ does not change sign at x = c, then f has neither a local maximum nor minimum at x = c.

Proof of these three conclusions is left to the reader. Example 3: Our previous example showed a graph that had both a local maximum and minimum. Let’s reconsider f (x) = x3 and observe the graph around x = 0. What happens to the ﬁrst derivative near this value?

Example 4: Let’s consider the function f (x) = x2 + 6x − 9 and observe the graph around x = −3. What happens to the ﬁrst derivative near this value?

157

www.ck12.org

We observe that the slopes of the tangent lines to the graph change from negative to positive at x = −3. The ﬁrst derivative test veriﬁes this fact. Note that the slopes of the tangent lines to the graph are negative for x ∈ (−∞, −3) and positive for x ∈ (−3, −∞).

Lesson Summary 1. We found intervals where a function is increasing and decreasing. 2. We applied the First Derivative Test to ﬁnd extrema and sketch graphs.

Review Questions In problems #1–2, identify the intervals where the function is increasing, decreasing, or is constant. (Units on the axes indicate single units). 1.

2.

3. Give the sign of the following quantities for the graph in #2. (a) (b) (c) (d)

f ′ (−3) f ′ (1) f ′ (3) f ′ (4)

For problems #4–6, determine the intervals in which the function is increasing and those in which it is decreasing. Sketch the graph. 4. f (x) = x2 − x1 5. f (x) = (x2 − 1)5 6. f (x) = (x2 − 1)4 www.ck12.org

158

(a) Use the First Derivative Test to ﬁnd the intervals where the function increases and/or decreases (b) Identify all max, mins, or relative max and mins (c) Sketch the graph

For problems #7–10, ﬁnd the following:

7. 8. 9. 10.

f (x) = −x2 − 4x − 1 f (x) = x3 + 3x2 − 9x + 1 2 f (x) = x 3 √ (x − 5) f (x) = 2x x2 + 1

Review Answers 1. 2. 3. 4.

Increasing on (0, 3), decreasing on (3, 6), constant on (6, +∞). Increasing on (−∞, 0) and (3, 7), decreasing on (0, 3). f ′ (−3) > 0, f ′ (1) < 0, f ′ (3) =√0, f ′ (4) > 0 √ 3 3 Relative minimum at x = − 0.5; increasing on (− 0.5, 0) and (0, +∞), decreasing √ 3 on (−∞, − 0.5).

5. Absolute minimum at x = 0; decreasing on (−∞, 0), increasing on (0, +∞).

159

www.ck12.org

6. Absolute minimum at x = ±1; relative maximum at x = 0; decreasing on (−∞, −1), (0, 1), increasing on (−1, 0), (1, +∞).

7. Absolute maximum at x = −2; increasing on (−∞, −2), decreasing on (−2, +∞).

www.ck12.org

160

8. Relative maximum at x = −3, f (−3) = 28; relative minimum at x = 1, f (1) = −4; increasing on (−∞, −3) and (1, +∞), decreasing on (−3, 1).

2

9. Relative maximum at x = 0, f (0) = 0; relative minimum at x = 2, f (2) = −3 · 2 3 = √ 3 −3 4, x = 1, f (1) = −4; increasing on (−∞, 0) and (2, +∞) decreasing on (0, 2).

161

www.ck12.org

10. There are no maximums or minimums; no relative maximums or minimums.

3.4

The Second Derivative Test

Learning Objectives A student will be able to:

• Find intervals where a function is concave upward or downward. • Apply the Second Derivative Test to determine concavity and sketch graphs. www.ck12.org

162

Introduction In this lesson we will discuss a property about the shapes of graphs called concavity, and introduce a method with which to study this phenomenon, the Second Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph. Deﬁnition A function f is said to be concave upward on [a, b] contained in the domain of f if f ′ is an increasing function on [a, b] and concave downward on [a, b] if f ′ is a decreasing function on [a, b]. Here is an example that illustrates these properties. Example 1: Consider the function f (x) = x3 − x:

√

The function has zeros at x = ±1, 0 and has a relative maximum at x = − 33 and a relative √ minimum at x = 33 . Note that the graph appears to be concave down for all intervals in (−∞, 0) and concave up for all intervals in (0, +∞). Where do you think the concavity of the graph changed from concave down to concave up? If you answered at x = 0 you would be correct. In general, we wish to identify both the extrema of a function and also points, the graph changes concavity. The following deﬁnition provides a formal characterization of such points. Deﬁnition A point on a graph of a function f where the concavity changes is called an inﬂection point. The example above had only one inﬂection point. But we can easily come up with examples of functions where there are more than one point of inﬂection. Example 2: Consider the function f (x) = x4 − 3x3 + x − 2.

163

www.ck12.org

We can see that the graph has two relative minimums, one relative maximum, and two inﬂection points (as indicated by arrows). In general we can use the following two tests for concavity and determining where we have relative maximums, minimums, and inﬂection points. Test for Concavity Suppose that f is continuous on [a, b] and that I is some open interval in the domain of f. 1. If f ′′ (x) > 0 for all x ∈ I, then the graph of f is concave upward on I. 2. If f ′′ (x) < 0 for all x ∈ I, then the graph of f is concave downward on I. A consequence of this concavity test is the following test to identify extreme values of f. Second Derivative Test for Extrema Suppose that f is a continuous function near c and that c is a critical value of f. Then 1. If f ′′ (c) > 0, then f has a relative maximum at x = c. 2. If f ′′ (c) < 0, then f has a relative minimum at x = c. 3. If f ′′ (c) = 0, then the test is inconclusive and x = c may be a point of inﬂection. Recall the graph f (x) = x3 . We observed that x = 0, and that there was neither a maximum nor minimum. The Second Derivative Test cautions us that this may be the case since at f ′′ (0) = 0 at x = 0. So now we wish to use all that we have learned from the First and Second Derivative Tests to sketch graphs of functions. The following table provides a summary of the tests and can be a useful guide in sketching graphs.

www.ck12.org

164

Table 3.1: Signs of ﬁrst and second derivatives

Information from applying Shape of the graphs First and Second Derivative Tests

f ′ (x) > 0 f ′′ (x) > 0

f is increasing f is concave upward

f ′ (x) > 0 f ′′ (x) < 0

f is increasing f is concave downward

f ′ (x) < 0 f ′′ (x) > 0

f is decreasing f is concave upward

f ′ (x) < 0 f ′′ (x) < 0

f is decreasing f is concave downward

Lets’ look at an example where we can use both the First and Second Derivative Tests to ﬁnd out information that will enable us to sketch the graph. Example 3: Let’s examine the function f (x) = x5 − 5x + 2. 1. Find the critical values for which f ′ (c) = 0. f ′ (x) = 5x4 − 5 = 0, or x4 − 1 = 0 at x = ±1. Note that f ′′ (x) = 20x3 = 0 when x = 0. 2. Apply the First and Second Derivative Tests to determine extrema and points of inﬂection. We can note the signs of f ′ and f ′′ in the intervals partioned by x = ±1, 0.

165

www.ck12.org

Table 3.2: Key intervals

f ′ (x)

f ′′ (x)

Shape of graph

x < −1

+

−

−1 < × < 0

−

−

0<×<1

−

+

x>1

+

+

Increasing, down Decreasing, down Decreasing, up Increasing, up

concave concave concave concave

Also note that f ′′ (−1) = −20 < 0. By the Second Derivative Test we have a relative maximum at x = −1, or the point (−1, 6). In addition, f ′′ (1) = 20 > 0. By the Second Derivative Test we have a relative minimum at x = 1, or the point (1, −2). Now we can sketch the graph.

Lesson Summary 1. We learned to identify intervals where a function is concave upward or downward. 2. We applied the First and Second Derivative Tests to determine concavity and sketch graphs.

Review Questions 1. Find all extrema using the Second Derivative Test. f (x) = 2. Consider f (x) = x2 + ax + b, with f (1) = 3.

x2 4

+

4 x

(a) Determine a and b so that x = 1 is a critical value of the function f. (b) Is the point (1, 3) a maximum, a minimum or neither? www.ck12.org

166

In problems #3–6, ﬁnd all extrema and inﬂection points. Sketch the graph. 3. 4. 5. 6. 7.

f (x) = x3 + x2 2 f (x) = x x+3 f (x) = x3 − 12x f (x) = − 14 x4 + 2x2 Use your graphing calculator to examine the graph of f (x) = x(x − 1)3 (Hint: you will need to change the y range in the viewing window) (a) Discuss the concavity of the graph in the interval (0, 12 ). (b) Use your calculator to ﬁnd the minimum value of the function in the interval.

8. True or False: f (x) = x4 + 4x3 has a relative minimum at x = −2 and a relative maximum at x = 0? 9. If possible, provide an example of a non-polynomial function that has exactly one relative minimum. 10. If possible, provide an example of a non-polynomial function that is concave downward everywhere in its domain.

Review Answers 1. There is a relative minimum at x = 2; the relative minimum is located at (2, 3). 2. f (1) = 3 suggests that a + b = 2 and f ′ (1) = 0 = 2 + a; solving this system we have that a = −2, b = 4; the point (1, 3) is an absolute max of f. 3. Relative maximum at x = −2 , relative minimum at x = 0; the relative maximum is 3 located at ( −2 , 0.15); the relative minimum is located at (0, 0). There is a point of 3 1 inﬂection at (− 3 , 0.07).

167

www.ck12.org

√ √ √ 4. Relative maximum at x = − 3, located at (− 3, −2 3); relative minimum at x = √ √ √ 3, located at ( 3, 2 3). There are no inﬂection points.

5. Relative maximum at x = −2, relative minimum at x = 2; the relative maximum is located at (−2, 16); the relative minimum is located at (2, 16). There is a point of inﬂection at (0, 0).

6. Relative maximums at x = ±2, relative minimum at x = 0; the relative maximums are located at (−2, 4) and (2, 4); the√ relative minimum is located at (0, 0). There are two √ 2 3 20 2 3 20 inﬂection points, located at (− 3 , 9 ) and ( 3 , 9 ).

www.ck12.org

168

7. (a) The graph is concave up in the interval; (b) There is a relative minimum at (0.25, −0.10) 8. False: there are inﬂection points at x = 0 and x = −2. There is a relative minimum at x = −3.

1 ) 9. f (x) = x2 + ( x−1

169

www.ck12.org

10. f (x) =

√

x on (0, +∞)

Also, f (x) = − x12

3.5

Limits at Inﬁnity

Learning Objectives A student will be able to: • Examine end behavior of functions on inﬁnite intervals. www.ck12.org

170

• • • •

Determine horizontal asymptotes. Examine indeterminate forms of limits of rational functions. Apply L’Hospital’s Rule to ﬁnd limits. Examine inﬁnite limits at inﬁnity.

Introduction In this lesson we will return to the topics of inﬁnite limits and end behavior of functions and introduce a new method that we can use to determine limits that have indeterminate forms. Examine End Behavior of Functions on Inﬁnite Intervals Suppose we are trying to analyze the end behavior of rational functions. In Lesson on 2 −1 Inﬁnite Limits we looked at some rational functions such as f (x) = 2x and showed that x2 −1 limx→+∞ f (x) = 2 and limx→−∞ f (x) = 2. We required an analysis of the end behavior of f since computing the limit by direct substitution yielded the indeterminate form ∞ . Our ∞ approach to compute the inﬁnite limit was to look at actual values of the function f (x) as x approached ±∞. We interpreted the result graphically as the function having a horizontal asymptote at f (x) = 2.

We were then able to ﬁnd inﬁnite limits of more complicated rational functions such as 4 −2x2 +3x+1 limx→∞ 3x = 32 using the fact that limx→∞ x1p = θ, p > θ. Similarly, we used such 2x4 −2x2 +x−3 an approach to compute limits whenever direct substitution resulted in the indeterminate 2 −1 form θθ , such as limx→1 xx−1 = 2. Now let’s consider other functions of the form (f (x)/g(x)) where we get the indeterminate forms θθ and θθ and determine an appropriate analytical method for computing the limits. Example 1: Consider the function f (x) = We note the following:

ln(x+1) x

and suppose we wish to ﬁnd limx→0

ln(x+1) x

and limx→∞

ln(x+1) x

1. Direct substitution leads to the indeterminate forms 00 and ∞ . ∞ 2. The function in the numerator is not a polynomial function, so we cannot use our

171

www.ck12.org

previous methods such as applying limx→∞

1 xp

= θ.

Let’s examine both the graph and values of the function for appropriate x values, to see if they cluster around particular y values. Here is a sketch of the graph and a table of extreme values. We ﬁrst note that domain of the function is (−1, 0) ∪ (0, +∞) and is indicated in the graph as follows:

So, limx→0

ln(x+1) x

appears to approach the value 1 as the following table suggests.

Note: Please see Diﬀerentiation and Integration of Logarithmic and Exponential Functions in Chapter 6 for more on derivatives of Logarithmic functions.

www.ck12.org

172

ln(x + 1)/x 1.05361 1.0005 undef 0.9995 0.953102

x − 0.1 − 0.001 0 0.001 0.1 So we infer that limx→0

ln(x+1) x

= 1.

For the inﬁnite limit, limx→∞ ln(x+1) = 1, the inference of the limit is not as obvious. The x function appears to approach the value 0 but does so very slowly, as the following table suggests.

ln(x + 1)/x 0.23979 0.078637 0.046151 0.006909 0.000921

x 10 50 100 1000 10000

This unpredictable situation will apply to many other functions of the form. Hence we need (x) another method that will provide a diﬀerent tool for analyzing functions of the form fg(x) . L’Hospital’s Rule: Let functions f and g be diﬀerentiable at every number other than c in some interval, with g ′ (x) ̸= 0 if x ̸= c. If limx→c f (x) = limx→c g(x) = 0, or if limx→c f (x) = ±∞ and limx→c g(x) = ±∞, then: ′

(x) (x) 1. limx→c fg(x) = limx→c fg′ (x) as long as this latter limit exists or is inﬁnite. 2. If f and g are diﬀerentiable at every number x greater than some number a, with ′ (x) (x) g ′ (x) ̸= 0, then limx→∞ fg(x) = limx→∞ fg′ (x) as long as this latter limit exists or is inﬁnite.

Let’s look at applying the rule to some examples. Example 2: We will start by reconsidering the previous example, f (x) = limits using L’Hospital’s Rule:

173

ln(x+1) , x

and verify the following

www.ck12.org

ln(x + 1) = 1. x→0 x ln(x + 1) lim = 0. x→∞ x lim

Solution: Since limx→0 ln(x + 1) = limx→0 x = 0, L’Hospital’s Rule applies and we have 1

ln(x + 1) 1 lim = lim x+1 = = 1. x→0 x→0 x 1 1 Likewise,

1

ln(x + 1) 0 lim = lim x+1 = = 0. x→∞ x→∞ x 1 1 Now let’s look at some more examples. Example 3: Evaluate limx→0

ex −1 . x

Solution: Since limx→0 (ex − 1) = limx→0 x = 0, L’Hospital’s Rule applies and we have ex 1 ex − 1 = lim = = 1. x→0 1 x→0 x 1 lim

Let’s look at an example with trigonometric functions. Example 4: Evaluate limx→0

1−cos x . x2

Solution: Since limx→0 (1 − cos x) = limx→0 x2 = 0, L’Hospital’s Rule applies and we have sin x cos x 1 1 − cos x = lim = lim = . 2 x→0 2x x→0 x→0 x 2 2 lim

www.ck12.org

174

Example 5: Evaluate limx→+∞

x2 ex

Solution: Since limx→+∞ x2 = limx→+∞ ex = +∞, L’Hospital’s Rule applies and we have x2 2x = lim x . x x→+∞ e x→+∞ e lim

Here we observe that we still have the indeterminate form again to ﬁnd the limit as follows:

∞ . ∞

So we apply L’Hospital’s Rule

x2 2x 2 lim = lim x = lim x = 0 x→+∞ ex x→+∞ e x→+∞ e L’Hospital’s Rule can be used repeatedly on functions like this. It is often useful because polynomial functions can be reduced to a constant.

Lesson Summary 1. 2. 3. 4. 5.

We We We We We

learned to examine end behavior of functions on inﬁnite intervals. determined horizontal asymptotes of rational functions. examined indeterminate forms of limits of rational functions. applied L’Hospital’s Rule to ﬁnd limits of rational functions. examined inﬁnite limits at inﬁnity.

Review Questions 1. Use your graphing calculator to estimate limx→+∞ [x[ln(x + 3) − ln(x)]] . x 2. Use your graphing calculator to estimate limx→+∞ ln(1+2e x) . In problems #3–10, use L’Hospital’s Rule to compute the limits, if they exist. x2 −9 x−3 √ √ limx→0 1+x−x 1−x √ limx→+∞ ln(x) x limx→+∞ x2 e−2x

3. limx→3 4. 5. 6. 7. 8. 9. 10.

1

limx→0 (1 − x) x x limx→0 e −1−x xx2 limx→−∞ e −1−x x2 1 limx→∞ x 4 ln(x)

175

www.ck12.org

Review Answers 1. limx→+∞ [x[ln(x + 3) − ln(x)]] = 3 x 2. limx→+∞ ln(1+2e x) = 1 x2 −9 =6 x−3 √ √ 1+x− 1−x limx→0 =1 x ln(x) limx→+∞ √x = 0 limx→+∞ x2 e−2x = 0 1 limx→0 (1−x) x = 1e :Hint: limx→0 x1 ln(1−x)

3. limx→3 4. 5. 6. 7.

1

1

1

1

Let (1−x) x = eln(1−x) x , so limx→0 (1−x) x = limx→0 eln(1−x) x =

e x = 12 8. limx→0 e −1−x x2 ex −1−x 9. limx→−∞ x2 = 0 1 10. limx→∞ x 4 ln(x) = 0

3.6

Analyzing the Graph of a Function

Learning Objectives A student will be able to: • Summarize the properties of function including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inﬂection, limits at inﬁnity. • Apply the First and Second Derivative Tests to sketch graphs.

Introduction In this lesson we summarize what we have learned about using derivatives to analyze the graphs of functions. We will demonstrate how these various methods can be applied to help us examine a function’s behavior and sketch its graph. Since we have already discussed the various techniques, this lesson will provide examples of using the techniques to analyze the examples of representative functions we introduced in the Lesson on Relations and Functions, particularly rational, polynomial, radical, and trigonometric functions. Before we begin our work on these examples, it may be useful to summarize the kind of information about functions we now can generate based on our previous discussions. Let’s summarize our results in a table like the one shown because it provides a useful template with which to organize our ﬁndings.

www.ck12.org

176

Table 3.3: Table Summary Analysis

f (x) Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points

Example 1: Analyzing Rational Functions Consider the function f (x) =

x2 −4 . x2 −2x−8

General Properties: The function appears to have zeros at x = ±2. However, once we factor the expression we see x2 − 4 (x + 2)(x − 2) x−2 = = . 2 x − 2x − 8 (x − 4)(x + 2) x−4 Hence, the function has a zero at x = 2, there is a hole in the graph at x = −2, the domain is (−∞, −2) ∪ (−2, 4) ∪ (4, +∞), and the y−intercept is at (0, 12 ). Asymptotes and Limits at Inﬁnity Given the domain, we note that there is a vertical asymptote at x = 4. To determine other asymptotes, we examine the limit of f as x → ∞ and x → −∞. We have x2 − 4 = lim lim 2 x→∞ x − 2x − 8 x→∞ Similarly, we see that limx→−∞

x2 −4 x2 −2x−8

x2 x2

x2 x2

−

4 x2

−

2x x2

−

8 x2

1 − x42 = lim = 1. x→∞ 1 − 2 − 82 x x

= 1. We also note that y ̸= 1 since x ̸= −2.

Hence we have a horizontal asymptote at y = 1. Diﬀerentiability −8x−8 −2 f ′ (x) = −2x = (x−4) 2 < 0. Hence the function is diﬀerentiable at every point of its (x2 −2x−8) ′ domain, and since f (x) < 0 on its domain, then f is decreasing on its domain, (−∞, −2) ∪ (−2, 4) ∪ (4, +∞). 2

177

www.ck12.org

f ′′ (x) =

4 . (x − 4)3

f ′′ (x) ̸= 0 in the domain of f. Hence there are no relative extrema and no inﬂection points. So f ′′ (x) > 0 when x > 4. Hence the graph is concave up for x > 4. Similarly, f ′′ (x) < 0 when x < 4. Hence the graph is concave down for x < 4, x ̸= −2. Let’s summarize our results in the table before we sketch the graph. Table 3.4: Table Summary f (x) =

x2 −4 x2 −2x−8

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points

Analysis D { = (−∞, −2) } ∪ (−2, 4) ∪ (4, +∞) R = all reals ̸= 1 ( ) zero at x = 2, y−intercept at 0, 21 VA at x = 4, HA at y = 1, hole in the graph at x = −2 diﬀerentiable at every point of its domain nowhere (−∞, −2) ∪ (−2, 4) ∪ (4, +∞) none concave up in (4, +∞), concave down in (−∞, −2) ∪ (−2, 4) none

Finally, we sketch the graph as follows:

Let’s look at examples of the other representative functions we introduced in Lesson 1.2. www.ck12.org

178

Example 2: Analyzing Polynomial Functions Consider the function f (x) = x3 + 2x2 − x − 2. General Properties The domain of f is (−∞, +∞) and the y−intercept at (0, −2). The function can be factored f (x) = x3 + 2x2 − x − 2 = x2 (x + 2) − 1(x + 2) = (x2 − 1)(x + 2) = (x − 1)(x + 1)(x + 2) and thus has zeros at x = ±1, −2.

Asymptotes and limits at inﬁnity Given the domain, we note that there are no vertical asymptotes. We note that limx→∞ f (x) = +∞ and limx→−∞ f (x) = −∞. Diﬀerentiability √

√

f ′ (x) = 3x2 + 4x − 1 = 0 if x = −4±6 28 = −2±3 7 . These are the critical values. We note that the function is diﬀerentiable at every point of its domain. ( ( ) √ ) √ f ′ (x) > 0 on −∞, −2−3 7 and −2+3 7 , +∞ ; hence the function is increasing in these intervals. ( √ √ ) −2− 7 −2+ 7 ′ Similarly, f (x) < 0 on , 3 and thus is f decreasing there. 3

179

www.ck12.org

f ′′ (x) = 6x + 4 = 0 if x = − 23 , where there is an inﬂection point. ( √ ) In addition, f ′′ −2−3 7 < 0. Hence the graph has a relative maximum at x = located at the point (−1.55, 0.63). ( ) We note that f ′′ (x) < 0 for x < − 23 . The graph is concave down in −∞, − 23 . ( √ ) And we have f ′′ −2+3 7 > 0; hence the graph has a relative minimum at x = located at the point (0.22, −2.11). ( ) We note that f ′′ (x) > 0 for x > − 32 . The graph is concave up in − 32 , +∞ .

√ −2− 7 3

and

√ −2+ 7 3

and

Table 3.5: Table Summary f (x) = x3 + 2x2 − x − 2

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability

D = (−∞, +∞), R = { all reals } zeros at x = ±1, −2, y, intercept at (0, −2) no asymptotes diﬀerentiable ( )at every ( point of it’s ) domain

Intervals where f is increasing Intervals where f is decreasing Relative extrema

(

−∞, −2−3

√

7

√ √ −2− 7 −2+ 7 , 3 3

and )

√ −2+ 7 , +∞ 3 √

relative maximum at x = −2−3 7 and located at the point (−1.55, 0.63); √ relative minimum at x = −2+3 7 and located at the point (0.22, −2.11).

Concavity

) ( concave up in −(32 , +∞ . ) concave down in −∞, − 32 .

Inﬂection points

( ) x = − 23 , located at the point − 23 , −.74

Here is a sketch of the graph:

www.ck12.org

180

Example 3: Analyzing Radical Functions √ Consider the function f (x) = 2x − 1. General Properties ( ) The domain of f is 21 , +∞ , and it has a zero at x = 21 . Asymptotes and Limits at Inﬁnity Given the domain, we note that there are no vertical asymptotes. We note that limx→∞ f (x) = +∞. Diﬀerentiability 1 f ′ (x) = √2x−1 > 0 for the entire domain of f. Hence f is increasing everywhere in its domain. ′ f (x) is not deﬁned at x = 12 , so x = 12 is a critical value. ( ) ( ) f ′′ (x) = √ −1 3 < 0 everywhere in 12 , +∞ . Hence f is concave down in 12 , +∞ . f ′ (x) (2x−1)

is not deﬁned at x = 12 , so x =

1 2

is an absolute minimum.

Table 3.6: Table Summary f (x) =

√

2x − 1

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema

Concavity Inﬂection points

Analysis ( ) D = 12 , +∞ , R = {y ≥ 0} zeros at x = 21 , no y−intercept no asymptotes ( ) diﬀerentiable in 12 , (+∞ ) everywhere in D = 21 , +∞ nowhere none absolute minimum at x = 12 , located at (1 ) ,0 2 concave down in none

(1 2

, +∞

)

Here is a sketch of the graph:

181

www.ck12.org

Example 4: Analyzing Trigonometric Functions We will see that while trigonometric functions can be analyzed using what we know about derivatives, they will provide some interesting challenges that we will need to address. Consider the function f (x) = x − 2 sin x on the interval [−π, π]. General Properties We note that f is a continuous function and thus attains an absolute maximum and minimum in [−π, π]. Its domain is [−π, π] and its range is R = {−π ≤ y ≤ π} . Diﬀerentiability f ′ (x) = 1 − 2 cos x = 0 at x = − π3 , π3 . ) ( ) ( ) ( Note that) f ′ (x) > 0 on π3 , π and −π, π3 ; therefore the function is increasing in π3 , π and ( −π, − π3 . ) ( ) ( Note that f ′ (x) < 0 on − π3 , π3 ; therefore the function is decreasing in − π3 , π3 . f ′′ (x) = 2 sin x = 0 if x = 0, π, −π. Hence the critical values are at x = −π, − π3 , 0, π3 , π. ( ) f ′′ π3 > 0; hence there is a relative minimum at x = π3 . ( ) f ′′ − π3 < 0; hence there is a relative maximum at x = − π3 . f ′′ (x) > 0 on (0, π) and f ′′ (x) < 0 on (−π, 0). Hence the graph is concave up and decreasing on (0, π) and concave down on (−π, 0). There is an inﬂection point at x = 0, located at the point (0, 0). Finally, there is absolute minimum at x = −π, located at (−π, −π), and an absolute maximum at x = π, located at (π, π). Table 3.7: Table Summary f (x) = x − 2 sin x

Analysis

Domain and Range

D = [−π, π], R = {−π ≤ y ≤ π}

www.ck12.org

182

Table 3.7: (continued) f (x) = x − 2 sin x

Analysis

Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema

x = − π3 , π3 no asymptotes diﬀerentiable in D = [−π, π] (π ) π ( 3 ,ππ π and ) (−π, − 3 ) −3, 3 relative maximum at x = −π/3 relative minimum at x = π/3 absolute maximum at x = π absolute minimum at x = 21 , located at (1 ) ,0 2

Concavity Inﬂection points

concave up in (0, π) x = 0, located at the point (0, 0)

Multimedia Link Each of the problems above started with a function and then we analyzed it’s zeros, derivative, and concavity. Even without the function deﬁnition it is possible to sketch the graph if you know some key pieces of information. In the following video the narrator illustrates how to use information about the derivative of a function and given points on the function graph to sketch the function. Khan Academy Graphing with Calculus (9:43) . Another approach to this analysis is to look at a function, its derivative, and its second derivative on the same set of axes. This interactive applet called Curve Analysis allows you to trace function points on a graph and its ﬁrst and second derivative. You can also enter new functions (including the ones from the examples above) to analyze the functions and their derivatives.

183

www.ck12.org

Figure 3.2: More graphing with calculus. ojcp0GJKluM&feature=youtube_gdata

http://www.youtube.com/watch?v=

Lesson Summary 1. We summarized the properties of function, including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inﬂection, and limits at inﬁnity. 2. We applied the First and Second Derivative Tests to sketch graphs.

Review Questions 1. Summarize each of the following functions by ﬁlling out the table. Use the information to sketch a graph of the function. 2. f (x) = x3 + 3x2 − x − 3 Table 3.8: f (x) = x3 + 3x2 − x − 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points 2. f (x) = −x4 + 4x3 − 4x2 www.ck12.org f (x) = −x4 + 4x3 − 4x2 Domain and Range Intercepts and Zeros

184

Table 3.9: Analysis

3. f (x) =

2x−2 x2

Table 3.10: f (x) =

2x−2 x2

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points 1

4. f (x) = x − x 3 Table 3.11: 1

f (x) = x − x 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points √ 5. f (x) = − 2x − 6 + 3 Table 3.12: √ f (x) = − 2x − 6 + 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity

185

www.ck12.org

Table 3.12: (continued) √ f (x) = − 2x − 6 + 3

Analysis

Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points √ 6. f (x) = x2 − 2 x Table 3.13: √ f (x) = x2 − 2 x

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points 7. f ′ (x) = 1 + cos x = 0 on [−π, π] Table 3.14: f ′ (x) = 1 + cos x

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points www.ck12.org

186

Review Answers Table 3.15: f (x) = x3 + 3x2 − x − 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability

D = (−∞, +∞), R = {all reals} zeros at x = ±1, −3, y−intercept at (0, −3) no asymptotes diﬀerentiable ( )at every ( point of its)domain

Intervals where f is increasing

√

(

Intervals where f is decreasing

−∞, −3−2 3

3

and √ )

√ −3−2 3 −3+2 3 , 3 3

√ −3+2 3 , +∞ 3 √

Relative extrema

3 relative maximum at x = −3−2 , located 3 at the point (−2.15, 3.07); √ 3 relative minimum at x = −3+2 , located at 3 the point (0.15, −3.07)

Concavity

concave up in (−1, +∞) concave down in (−∞, −1)

Inﬂection points

x = −1, located at the point (−1, 0)

1.

Table 3.16: f (x) = −x4 + 4x3 − 4x2

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema

D = (−∞, +∞), R = {y ≤ 0} zeros at x = 0, 2, y−intercept at (0, 0) no asymptotes diﬀerentiable at every point of its domain (−∞, 0) and (1, 2) (0, 1) and (2, +∞) relative maximum at x = 0, located at the point (0, 0); and at at x = 2 located at the point (2, 0) relative minimum at x = 1, located at the point (1, −1)

187

www.ck12.org

Table 3.16: (continued) f (x) = −x4 + 4x3 − 4x2

Analysis

( ) concave up in 23 ,(53 ) ( ) concave down in −∞, 23 and 53 , +∞

Concavity

x = (23 , 53 , located at the points ) and 35 , −0.30

Inﬂection points

(2

) , −0.79 3

2.

Table 3.17: f (x) =

2x−2 x2

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity

D = (−∞, 0) ∪ (0, +∞), R = {y ̸= 0} zeros at x = 1, no y−intercept HA y = 0 diﬀerentiable at every point of its domain (0, 2) (−∞, 0) and (2, +∞) relative maximum at x = 2, located at the point (2, 0.5) concave up in (3, +∞) concave down in (−∞, 0) and (0, 3) x = 3, located at the point (3, 94 )

Inﬂection points 3.

Table 3.18: 1

f (x) = x − x 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability

D = (−∞, +∞), R = {all reals} zeros at x = ±1, 0, y−intercept at (0, 0) no asymptotes diﬀerentiable ) in (∞, ( 0) ∪ (0, ) +∞) (

Intervals where f is increasing Intervals where f is decreasing

www.ck12.org

√

−∞, − 3 and ( √ √ 9) − 3 , 93 9

188

√

3 , +∞ 9

Table 3.18: (continued) 1

f (x) = x − x 3

Analysis

Relative extrema

relative maximum at )x = ( √ − 3 the point , 0.384 9 relative at x = ( √minimum ) 3 point 9 , −0.384

√ − 3 , 9

√ 3 , 9

located at

located at the

Concavity

concave up in (0, +∞) concave down in (−∞, 0)

Inﬂection points

x = 0, located at the point (0, 0)

4.

Table 3.19: √ f (x) = − 2x − 6 + 3

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema Concavity Inﬂection points

D = (3, +∞), R = {y ≤ 3} zero at x = 15 , no y−intercept 2 no asymptotes diﬀerentiable in (3, +∞) nowhere everywhere in D = (3, +∞) none absolute maximum at x = 3, located at (3, 3) concave up in (3, +∞) none

5.

Table 3.20: √ f (x) = x2 − 2 x

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability

D = (0, +∞), R = {y ≥ −1.19} √ 3 zero at x = 4, no y−intercept no asymptotes diﬀerentiable in (0, +∞)

189

www.ck12.org

Table 3.20: (continued) √ f (x) = x2 − 2 x

Analysis ) (√ 3 16 , +∞ ( 4√ ) 3 0, 416

Intervals where f is increasing Intervals where f is decreasing Relative extrema

relative minimum at x ) = ( √ 3

√ 3

16 , 4

located at the

16 , −1.19 point x = 4 concave up in (0, +∞) none

Concavity Inﬂection points

Table 3.21: f ′ (x) = 1 + cos x

Analysis

Domain and Range Intercepts and Zeros Asymptotes and limits at inﬁnity Diﬀerentiability Intervals where f is increasing Intervals where f is decreasing Relative extrema

D = [−π, π], R = {0 ≤ y ≤ 2} zeros at x = −π, π, y−intercept at (0, 2) no asymptotes; limx↔∞ f (x) does not exist diﬀerentiable at every point of its domain (−π, 0) (0, π) absolute max at x = 0, located at the point (0, 2) absolute minimums at x = ±π, located at the points (−π, 0) and (π, 0)

Concavity Inﬂection points

concave down in (−π, π) ( π ) π x = ± , located at the points − 2 , 1 and 2 (π ) ,1 2

6.

3.7

Optimization

Learning Objectives A student will be able to: • Use the First and Second Derivative Tests to ﬁnd absolute maximum and minimum values of a function. www.ck12.org

190

• Use the First and Second Derivative Tests to solve optimization applications.

Introduction In this lesson we wish to extend our discussion of extrema and look at the absolute maximum and minimum values of functions. We will then solve some applications using these methods to maximize and minimize functions. Absolute Maximum and Minimum We begin with an observation about ﬁnding absolute maximum and minimum values of functions that are continuous on a closed interval. Suppose that f is continuous on a closed interval [a, b]. Recall that we can ﬁnd relative minima and maxima by identifying the critical numbers of f in (a, b) and then applying the Second Derivative Test. The absolute maximum and minimum must come from either the relative extrema of f in (a, b) or the value of the function at the endpoints, f (a) or f (b). Hence the absolute maximum or minimum values of a function f that is continuous on a closed interval [a, b] can be found as follows:

1. Find the values of f for each critical value in (a, b); 2. Find the values of the function f at the endpoints of [a, b]; 3. The absolute maximum will be the largest value of the numbers found in 1 and 2; the absolute minimum will be the smallest number.

The optimization problems we will solve will involve a process of maximizing and minimizing functions. Since most problems will involve real applications that one ﬁnds in everyday life, we need to discuss how the properties of everyday applications will aﬀect the more theoretical methods we have developed in our analysis. Let’s start with the following example. Example 1: A company makes high-quality bicycle tires for both recreational and racing riders. The number of tires that the company sells is a function of the price charged and can be modeled by the formula T (x) = −x3 + 36.5x2 + 50x + 250, where x is the priced charged for each tire in dollars. At what price is the maximum number of tires sold? How many tires will be sold at that maximum price? Solution: Let’s ﬁrst look at a graph and make some observations. Set the viewing window ranges on your graphing calculator to [−10, 50] for x and [−500, 10000] for y. The graph should appear as follows:

191

www.ck12.org

We ﬁrst note that since this is a real-life application, we observe that both quantities, x and T (x), are positive or else the problem makes no sense. These conditions, together with the fact that the zero of T (x) is located at x = 37.9, suggest that the actual domain of this function is 0 < × < 37. This domain, which we refer to as a feasible domain, illustrates a common feature of optimization problems: that the real-life conditions of the situation under study dictate the domain values. Once we make this observation, we can use our First and Second Derivative Tests and the method for ﬁnding absolute maximums and minimums on a closed interval (in this problem, [0, 37]), to see that the function attains an absolute maximum at x = 25, at the point (25, 8687.5). So, charging a price of $25 will result in a total of 8687 tires being sold. In addition to the feasible domain issue illustrated in the previous example, many optimization problems involve other issues such as information from multiple sources that we will need to address in order to solve these problems. The next section illustrates this fact. Primary and Secondary Equations We will often have information from at least two sources that will require us to make some transformations in order to answer the questions we are faced with. To illustrate this, let’s return to our Lesson on Related Rates problems and recall the right circular cone volume problem.

1 V = πr2 h. 3 www.ck12.org

192

We started with the general volume formula V = 13 πr2 h , but quickly realized that we did not have suﬀicient information to ﬁnd dh since we had no information about the radius when dt the water level was at a particular height. So we needed to employ some indirect reasoning to ﬁnd a relationship between r and h, r(t) = 2h(t) . We then made an appropriate substitution 5 ( 2h )2 1 4π 3 in the original formula (V = 3 π 5 h = 75 h ) and were able to ﬁnd the solution. We started with a primary equation, V = 13 πr2 h, that involved two variables and provided a general model of the situation. However, in order to solve the problem, we needed to generate a secondary equation, r(t) = 2h(t) , that we then substituted into the primary 5 equation. We will face this same situation in most optimization problems. Let’s illustrate the situation with an example. Example 2: Suppose that Mary wishes to make an outdoor rectangular pen for her pet chihuahua. She would like the pen to enclose an area in her backyard with one of the sides of the rectangle made by the side of Mary’s house as indicated in the following ﬁgure. If she has 90 ft of fencing to work with, what dimensions of the pen will result in the maximum area?

193

www.ck12.org

Solution: The primary equation is the function that models the area of the pen and that we wish to maximize,

A = xy. The secondary equation comes from the information concerning the fencing Mary has to work with. In particular,

2x + y = 90. Solving for y we have

y = 90 − 2x. We now substitute into the primary equation to get A = xy = x(90 − 2x), or

A = 90x − 2x2 . It is always helpful to view the graph of the function to be optimized. Set the viewing window ranges on your graphing calculator to [−10, 100] for x and [−500, 1200] for y. The graph should appear as follows: www.ck12.org

194

The feasible domain of this function is 0 < × < 45, which makes sense because if x is 45 feet, then the ﬁgure will be two 45-foot-long fences going away from the house with 0 feet left for the width, y. Using our First and Second Derivative Tests and the method for ﬁnding absolute maximums and minimums on a closed interval (in this problem, [0, 45]), we see that the function attains an absolute maximum at x = 22.5, at the point (22.5, 1012.5). So the dimensions of the pen should be x = 22.5, y = 45; with those dimensions, the pen will enclose an area of 1012.5 ft2 . Recall in the Lesson Related Rates that we solved problems that involved a variety of geometric shapes. Let’s consider a problem about surface areas of cylinders. Example 3: A certain brand of lemonade sells its product in 16−ounce aluminum cans that hold 473 ml (1 ml = 1 cm3 ). Find the dimensions of the cylindrical can that will use the least amount of aluminum. Solution: We need to develop the formula for the surface area of the can. This consists of the top and bottom areas, each πr2 , and the surface area of the side, 2πrh (treating the side as a rectangle, the lateral area is (circumference of the top) × (height)). Hence the primary equation is

A = 2πr2 + 2πrh. We observe that both our feasible domains require r, h > 0. In order to generate the secondary equation, we note that the volume for a circular cylinder is given by V = πr2 h. Using the given information we can ﬁnd a relationship between (r and ) 473 473 2 h, h = πr , 2 . We substitute this value into the primary equation to get A = 2πr + 2πr πr 2 946 2 or A = 2πr + r .

195

www.ck12.org

= 4πr −

√

2

≈ 9.06 cm. We note that ddrA2 > 0 since r > 0. Hence we √ 946 473 have a minimum surface area when r = 3 4π0 ≈ 4.22 cm and h = √ = 8.44 cm. 3 946 2 dA dr

946 r2

= 0 when r =

3

946 4π

π(

4π

)

Multimedia Link to see more examples of worked out problems involving ﬁnding minima and maxima on an interval, see the video at Khan Academy Minimum and Maximum Values on an Interval (11:41) . This video shows the process of applying the ﬁrst derivative test

Figure 3.3: 2 examples of ﬁnding the maximum and minimum points on an interval. http: //www.youtube.com/watch?v=gzmSKrwiG3g&feature=youtube_gdata to problems with no context, just a given function and a domain. A classic problem in calculus involves maximizing the volume of an open box made by cutting squares from a rectangular sheet and folding up the edges. This very cool calculus applet shows one solution to this problem and multiple representations of the problem as well. Calculus Applet on Optimization

Lesson Summary 1. We used the First and Second Derivative Tests to ﬁnd absolute maximum and minimum values of a function. www.ck12.org

196

2. We used the First and Second Derivative Tests to solve optimization applications.

Review Questions In problems #1–4, ﬁnd the absolute maximum and absolute minimum values, if they exist. f (x) = 2x2 − 6x + 6 on [0, 5] f (x) = x3 + 3x2 on [−2, 3] 2 f (x) = 3x 3 − 6x + x on [1, 8] f (x) = x4 − x3 on [−2, 2] Find the dimensions of a rectangle having area 2000 ft2 whose perimeter is as small as possible. 6. Find two numbers whose product is 50 and whose sum is a minimum. 7. John is shooting a basketball from half-court. It is approximately 45 ft from the half court line to the hoop. The function s(t) = −0.025x2 + x + 15 models the basketball’s height above the ground s(t) in feet, when it is t feet from the hoop. How many feet from John will the ball reach its highest height? What is that height? 8. The height of a model rocket t seconds into ﬂight is given by the formula h(t) = − 31 t3 + 4t2 + 25t + 4. 1. 2. 3. 4. 5.

(a) How long will it take for the rocket to attain its maximum height? (b) What is the maximum height that the rocket will reach? (c) How long will the ﬂight last? 9. Show that of all rectangles of a given perimeter, the rectangle with the greatest area is a square. 10. Show that of all rectangles of a given area, the rectangle with the smallest perimeter is a square.

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

( ) Absolute minimum at x = 32 , f 23 = 32 . Absolute maximum at x = 5, f (5) = 26. Absolute minimum at x = 0, f (0) = 0. Absolute maximum at x = 3, f (3) = 54. Absolute minimum at x = 8, f (8) = −30. Absolute maximum at x = 1, f (1) = 3. Absolute minimum at x = 0.75, f (0.75) = −0.105. Absolute maximum at x = −2, f (−2) = 24. √ x = y = 20√ 5 x=y=5 2 At t = 20 ft, the basketball will reach a height of s(t) = 25 ft. The rocket will take approximately t ≈ 10.4 sec to attain its maximum height of 321.7 ft. the rocket will hit the ground at t ≈ 16.6 sec. . .

197

www.ck12.org

3.8

Approximation Errors

Learning Objectives A student will be able to:

• • • •

Extend the Mean Value Theorem to make linear approximations. Analyze errors in linear approximations. Extend the Mean Value Theorem to make quadratic approximations. Analyze errors in quadratic approximations.

Introduction In this lesson we will use the Mean Value Theorem to make approximations of functions. We will apply the Theorem directly to make linear approximations and then extend the Theorem to make quadratic approximations of functions. Let’s consider the tangent line to the graph of a function f at the point (a, f (a)). The equation of this line is y = f (a) + f ′ (a)(x − a). We observe from the graph that as we consider x near a, the value of f (x) is very close to f (a).

In other words, for x values close to a, the tangent line to the graph of a function f at the point (a, f (a)) provides an approximation of f (x) or f (x) ≈ f (a) + f ′ (a)(x − a). We call this the linear or tangent line approximation of f at a and indicate it by the formula L(x) = f (a) + f ′ (a)(x − a). www.ck12.org

198

The linear approximation can be used to approximate functional values that deviate slightly from known values. The following example illustrates this process. Example 1: Use the linear approximation of the function f (x) =

√

x − 2 at a = 6 to approximate

√

3.95.

Solution:

√ We know that f (6) = 6 − 2 = 2. So we will ﬁnd the linear approximation of the function and substitute x values close to 6.

L(x) = f (6) + f ′ (6)(x − 6).

We note that f ′ (x) =

√1 , f ′ (6) 2 x−2

= 14 .

We also know that f (6) = 2. By substitution, we have f (x) ≈ f (6) + f ′ (6)(x − 6) for x near 6. Hence L(x) = 2 + 41 (x − 6) =

1 2

+ 14 x. √ We observe that to approximate 3.95 we need to evaluate the linear approximation at 5.95, and we have L(5.95) = 12 + 14 (5.95) = 1.9875. If we were to compare this approximation to the actual √ value, 3.95 ≈ 1.9874, we see that it is a very good approximation. If we observe a table of x values close to 6, we see how the approximations compare to the actual value.

199

www.ck12.org

f (x) = √ 3.95 √ 3.99 √ 4 √ 4.1 √ 4.05

√

x−2

1 1 + x 2 4

Actual

x

L(x) =

5.95

1.9875

1.9874

5.99

1.9975

1.9974

6

2

2

6.01

2.0025

2.0024

6.05

2.0125

2.0124

Setting Error Estimates We would like to have conﬁdence in the approximations we make. We therefore can choose the x values close to a to ensure that the errors are within acceptable boundaries. For the previous example, we saw that the values of L(x) close to a = 6, gave very good approximations, all within 0.0001 of the actual value. Example 2: Let’s suppose that for the previous example, we did not require such precision. Rather, suppose we wanted to ﬁnd the range of x values close to 6 that we could choose to ensure that our approximations lie within 0.01 of the actual value. Solution: The easiest way for us to ﬁnd the proper range of x values is to√use the graphing ( 1 x ) calculator. We ﬁrst note that our precision requirement can be stated as | x − 2 − 2 + 4 | < 0.01. √ If we enter the functions f (x) = x − 2 and L(x) = 12 + 14 x into the Y = menu as Y1 and Y2 , respectively, we will be able to view the function values of the functions using the [TABLE] feature of the calculator. In order to view the diﬀerences between the actual and approximate values, we can enter into the Y = menu the diﬀerence function Y3 = Y1 − Y2 as follows:

1. Go to the Y = menu and place cursor on the Y3 line. 2. Press the following sequence of key strokes: [VARS] [FUNCTION] [Y1 ]. This will copy the function Y1 onto the Y3 line of the Y = menu. 3. Press [-] to enter the subtraction operation onto the Y3 line of the Y = menu. 4. Repeat steps 1 - 2 and choose Y2 to copy Y2 onto the Y3 line of the Y = menu.

Your screen should now appear as follows: www.ck12.org

200

Now let’s setup the [TABLE] function so that we ﬁnd the required accuracy. 1. Press 2ND followed by [TBLSET] to access the Table Setup screen. 2. Set the [TBLStart] value to 5 and △ Tbl to 0.1. Your screen should now appear as follows:

Now we are ready to ﬁnd the required accuracy. Access the [TABLE] function, scroll through the table, and ﬁnd those x values that ensure Y3 . At x = 5.24 we At x = 6.84, we see that Y3 = 0.00911. Hence if ) √ see that(Y3 = 0.01115. 5.24 ≤ × ≤ 6.84, | x − 2 − 12 + x4 | < 0.01.

Non-Linear Approximations It turns out that the linear approximations we have discussed are not the only approximations that we can derive using derivatives. We can use non-linear functions to make approximations. These are called Taylor Polynomials and are deﬁned as

Tn (x) = f (a) + f ′ (a)(x − a) +

f ′′ (a) f n (a) (x − a)2 + . . . + (x − a)n . 2! n!

201

www.ck12.org

We call this the Taylor Polynomial of fcentered at a. For our discussion, we will focus on the quadratic case. The Taylor Polynomial corresponding to n = 2 is given by

1 T2 (x) = f (a) + f ′ (a)(x − a) + f ′′ (a)(x − a)2 . 2

Note that this is just our linear approximation with an added term. Hence we can view it as an approximation of f for x values close to a. Example 3: Find the quadratic approximation of the function f (x) = them to the linear approximations from the ﬁrst example.

√

x − 2 at a = 6 and compare

Solution: Recall that L(x) =

1 2

+ 41 x.

Hence T2 (x) = L(x) + 12 f ′′ (6)(x − 6)2 . 1 ′′ √ f ′′ (x) = − 4√4(x−2) 3 ; so f (6) = − 4

Hence T2 (x) = L(x) −

1 (x 64

− 6)2 =

1 (6−2)2

1 2

1 . = − 32

+ 14 x −

1 (x 64

1 2 − 6)2 = − 64 x +

7 x 16

−

1 . 16

7 1 1 2 x + 16 x − 16 . If we update our table from the ﬁrst example we can see how So T2 (x) = − 64 the quadratic approximation compares with the linear approximation.

f (x) = √ 3.95 √ 3.99 √ 4 √ 4.1 √ 4.05

√

x−2

1 1 + x 2 4

T2 (x) = −

1 2 7 1 x + x− 64 16 16

Actual

x

L(x) =

5.95

1.9875

1.9874

1.9874

5.99

1.9975

1.9974

1.9974

6

2

2

2

6.01

2.0025

2.0024

2.0024

6.05

2.0125

2.0124

2.0124

As you can see from the graph below, T (x) is an excellent approximation of f (x) near x = 6. www.ck12.org

202

We get a slightly better approximation for the quadratic than for the linear. If we reﬂect on this a bit, the ﬁnding makes sense since the shape and properties of quadratic functions more closely approximate the shape of radical functions. Finally, as in the ﬁrst example, we wish to determine the range of x values that will ensure that our approximations are within 0.01 of the actual value. Using the [TABLE] feature of √ the calculator, we ﬁnd that if 4.444 ≤ × ≤ 7.87, then | x − 2 − T (x)| < 0.01.

Lesson Summary 1. 2. 3. 4.

We We We We

extended the Mean Value Theorem to make linear approximations. analyzed errors in linear approximations. extended the Mean Value Theorem to make quadratic approximations. analyzed errors in quadratic approximations.

Review Questions In problems #1–4, ﬁnd the linearization L(x) of the function at x = a. 1. f (x) = 2x4 − 6x3 near a = −2 2 2. f (x) = x 3 on a = 27 √ 3. Find the linearization of the function f (x) = 5 − x near a = 1 and use it to approx√ imate 4.01. 4. Based on using linear approximations, is the following approximation reasonable? 1.0014 = 1.004 5. Use a linear approximation to approximate the following: 3

16.08 4

203

www.ck12.org

6. Verify the the following linear approximation at a = 1. Determine the values of x for which the linear approximation is accurate to 0.01. √ 3

2−x≈

x 2 + 3 3

√ 7. Find the quadratic approximation for the function in #3, f (x) = 5 − x near a = 1. 8. Determine the values of x for which the quadratic approximation found in #7 is accurate to 0.01. 9. Determine the quadratic approximation for f (x) = 2x4 − 6x3 near a = −2. Do you expect that the quadratic approximation is better or worse than the linear approximation? Explain your answer.

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9.

f (−2) = 80, f ′ (−2) = −136; L(x) = −192 − 136x f (27) = 9, f ′ (27) = 2/9; L(x) = −45 +√2x f (1) = 2, f ′ (1) = − 14 ; L(x) = 49 − 41 x; 4.01 ≈ 2.0025 Yes; using linear approximation on f (x) = x4 near a = 1 we ﬁnd that L(x) = 4x − 3; L(1.001) = 1.004. 3 Using linear approximation on f (x) = x 4 near a = 16 we ﬁnd L(x) = 8 + 83 (x − 16) = 2 + 38 x; L(16.08) = 8.03. 0.84 < × < 1.14 1 2 7 T (x) = − 64 x − 32 x + 143 64 0.87 ≤ × ≤ 2.56 T (x) = 48x2 + 56x; we would expect it to be a better approximation since the graph of f (x) = 2x4 − 6x3 is similar to the graph of a quadratic function. f (x) = 2x4 − 6x3 T (x) = 48x2 + 56x

www.ck12.org

204

Chapter 4 Integration 4.1

Indeﬁnite Integrals Calculus

Learning Objectives A student will be able to: • • • • • •

Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve diﬀerential equations. Use basic antidiﬀerentiation techniques. Use basic integration rules.

Introduction In this lesson we will introduce the idea of the antiderivative of a function and formalize as indeﬁnite integrals. We will derive a set of rules that will aid our computations as we solve problems. Antiderivatives Deﬁnition A function F (x) is called an antiderivative of a function f if F ′ (x) = f (x) for all x in the domain of f. Example 1:

205

www.ck12.org

Consider the function f (x) = 3x2 . Can you think of a function F (x) such that F ′ (x) = f (x)? (Answer: F (x) = x3 , F (x) = x3 − 6, many other examples.) Since we diﬀerentiate F (x) to get f (x), we see that F (x) = x3 + C will work for any constant C. Graphically, we can think the set of all antiderivatives as vertical transformations of the graph of F (x) = x3 . The ﬁgure shows two such transformations.

With our deﬁnition and initial example, we now look to formalize the deﬁnition and develop some useful rules for computational purposes, and begin to see some applications. Multimedia Link The following applet shows a graph, f (x) and it’s derivative, f ′ (x). This is similar to other applets we’ve explored with a function and it’s derivative graphed sidebu-side, but this time f (x) is on the right, and f ′ (x) is on the left. If you edit the deﬁnition of f ′ (x), you will see the graph of f (x) change as well. The c parameter adds a constant to f (x). Notice that you can change the value of c without aﬀecting f ′ (x). Why is this? Antiderivative Applet. Notation and Introduction to Indeﬁnite Integrals The process of ﬁnding antiderivatives is called antidiﬀerentiation, more commonly referred to as integration. We have a particular sign and set of symbols we use to indicate integration: ∫ f (x)dx = F (x) + C. We refer to the left side of the equation as “the indeﬁnite integral of f (x) with respect to x.” The function f (x) is called the integrand and the constant C is called the constant of integration. Finally the symbol dx indicates that we are to integrate with respect to x. Using this notation, we would summarize the last example as follows: www.ck12.org

206

∫ 3x2 dx = x3 + C Using Derivatives to Derive Basic Rules of Integration As with diﬀerentiation, there are several useful rules that we can derive to aid our computations as we solve problems. The ﬁrst of these is a rule for integrating power functions, f (x) = xn [n ̸= −1], and is stated as follows: ∫ xn dx =

1 xn+1 + C. n+1

We can easily prove this rule. Let F (x) = respect to x and we have:

1 xn+1 n+1

+ C, n ̸= −1 . We diﬀerentiate with

( ) ( ) 1 1 d d d n+1 n+1 F (x) = x +C = x + (C) dx n + 1 dx n + 1 dx ) ( d ( n+1 ) 1 d x + (C) = n + 1 dx dx ( ) n+1 = xn + 0 n+1 = xn . ′

The rule holds for f (x) = xn [n ̸= −1].∫ What happens in the case where we have a power ∫ function to integrate with n = −1, say x−1 dx = x1 dx. We can see that the rule does not work since it would result in division by 0. However, if we pose the problem as ﬁnding F (x) such that F ′ (x) = x1 , we recall that the derivative of logarithm functions had this form. In d particular, dx ln x = x1 . Hence ∫

1 dx = ln x + C. x

In addition to logarithm functions, we recall that the basic exponentional function, f (x) = ex , was special in that its derivative was equal to itself. Hence we have ∫ ex dx = ex + C.

207

www.ck12.org

Again we could easily prove this result by diﬀerentiating the right side of the equation above. The actual proof is left as an exercise to the student. As with diﬀerentiation, we can develop several rules for dealing with a ﬁnite number of integrable functions. They are stated as follows: If f and g are integrable functions, and C is a constant, then ∫

∫ [f (x) + g(x)] dx =

∫

∫ f (x)dx +

∫ [f (x) − g(x)]dx = f (x)dx − ∫ ∫ [Cf (x)]dx = C f (x)dx.

g(x)dx, ∫ g(x)dx,

Example 2: Compute the following indeﬁnite integral. ] ∫ [ 1 3 3 dx. 2x + 2 − x x Solution: Using our rules we have ] ∫ [ ∫ ∫ ∫ 3 1 1 1 3 3 2x + 2 − − dx = 2 x dx + 3 dx − dx x x x2 x ( 4) ( −1 ) x x =2 +3 − ln x + C 4 −1 x4 3 = − − ln x + C. 2 x Sometimes our rules need to be modiﬁed slightly due to operations with constants as is the case in the following example. Example 3: Compute the following indeﬁnite integral: ∫ e3x dx. www.ck12.org

208

Solution: We ﬁrst note that our rule for integrating exponential functions does not work here since d 3x e = 3e3x . However, if we remember to divide the original function by the constant then dx we get the correct antiderivative and have ∫ e3x dx =

e3x + C. 3

We can now re-state the rule in a more general form as ∫ ekx dx =

ekx + C. k

Diﬀerential Equations We conclude this lesson with some observations about integration of functions. First, recall that the integration process allows us to start with function f from which we ﬁnd another function F (x) such that F ′ (x) = f (x). This latter equation is called a diﬀerential equation. This characterization of the basic situation for which integration applies gives rise to a set of equations that will be the focus of the Lesson on The Initial Value Problem. Example 4: Solve the general diﬀerential equation f ′ (x) = x 3 + 2

√

x.

Solution: We solve the equation by integrating the right side of the equation and have ∫

∫

′

f (x) =

f (x)dx =

2 3

x dx +

∫

√

xdx.

We can integrate both terms using the power rule, ﬁrst noting that ∫ f (x) =

2 3

x dx +

∫

√

1

x = x 2 , and have

1 3 5 2 3 x 2 dx = x 3 + x 2 + C. 5 3

Lesson Summary 1. We learned to ﬁnd antiderivatives of functions.

209

www.ck12.org

2. 3. 4. 5. 6.

We We We We We

learned to represent antiderivatives. interpreted constant of integration graphically. solved general diﬀerential equations. used basic antidiﬀerentiation techniques to ﬁnd integration rules. used basic integration rules to solve problems.

Review Questions In problems #1–3, ﬁnd an antiderivative of the function 1. f (x) = 1 − 3x2 − 6x 2 2. f (x) = x − x3 √ 3. f (x) = 5 2x + 1 In #4–7, ﬁnd the indeﬁnite integral √ ∫ 4. ∫ (2 + 5)dx 5. ∫ 2(x − 3)3 dx √ 2 3 6. (x · x)dx) ∫( x + x41√x dx 7. 8. Solve the diﬀerential equation f ′ (x) = 4x3 − 3x2 + x − 3 2x 9. Find the antiderivative F (x) of the ∫ function f (x) = 2e +x−2 that satisﬁes F (0) = 5. 10. Evaluate the indeﬁnite integral |x|dx (Hint: Examine the graph of f (x) = |x|.)

Review Answers F (x) = x − x3 − 3x2 + C 5 2 F (x) = x2 − 35 x 3 + C 6 5 F (x) = 12 (2x + 1) 5 + C √ ) √ ∫( 2 + 5 dx = 2x + 5x + C ∫ 4 2(x − 3)3 dx = (x−3) +C 2 ∫ 2 √ 3 10 3 3 (x · x) dx ) = 10 x 2 + C ∫( 4 x + x41√x dx = x2 − √ +C 7. 4 4 1. 2. 3. 4. 5. 6.

2

8. f (x) = x4 − x3 + x2 − 3x + C 2 9. F (x) = e2x + xx − 2x + 4 ∫ 2 10. |x|dx = x2 + C www.ck12.org

210

4.2

The Initial Value Problem

Learning Objectives • Find general solutions of diﬀerential equations • Use initial conditions to ﬁnd particular solutions of diﬀerential equations

Introduction In the Lesson on Indeﬁnite Integrals Calculus we discussed how ﬁnding antiderivatives can be thought of as ﬁnding solutions to diﬀerential equations: F ′ (x) = f (x). We now look to extend this discussion by looking at how we can designate and ﬁnd particular solutions to diﬀerential equations. Let’s recall that a general diﬀerential equation will have an inﬁnite number of solutions. We will look at one such equation and see how we can impose conditions that will specify exactly one particular solution. Example 1: Suppose we wish to solve the following equation: √ f ′ (x) = e3x − 6 x. Solution: We can solve the equation by integration and we have

3 1 f (x) = e3x − 4x 2 + C. 3

We note that there are an inﬁnite number of solutions. In some applications, we would like to designate exactly one solution. In order to do so, we need to impose a condition on the function f. We can do this by specifying the value of f for a particular value of x. In this problem, suppose that add the condition that f (0) = 1. This will specify exactly one value of C and thus one particular solution of the original equation: 3

Substituting f (0) = 1 into our general solution f (x) = 31 e3x − 4x 2 + C gives 1 = 31 e3(0) − 3 3 = 31 e3x − 4x 2 + 23 is the particular 4(0) 2 + C or C = 1 − 13 = 23 . Hence the solution f (x) √ solution of the original equation f ′ (x) = e3x − 6 x satisfying the initial condition f (0) = 1.

211

www.ck12.org

We now can think of other problems that can be stated as diﬀerential equations with initial conditions. Consider the following example. Example 2: Suppose the graph of f includes the point (2, 6) and that the slope of the tangent line to f at any point x is given by the expression 3x + 4. Find f (−2). Solution: We can re-state the problem in terms of a diﬀerential equation that satisﬁes an initial condition. f ′ (x) = 3x + 4 with f (2) = 6. By integrating the right side of the diﬀerential equation we have

3 f (x) = x2 + 4x + C as the general solution. Substituting the condition thatf (2) = 6 gives 2 3 2 6 = (2) + 4(2) + C, 2 6 = 6 + 8 + C, C = −8. Hence f (x) = 23 x2 +4x−8 is the particular solution of the original equation f ′ (x) = 3x+4 satisfying the initial condition f (2) = 6. Finally, since we are interested in the value f (−2), we put −2 into our expression for f and obtain:

f (−2) = −10 Multimedia Link Another way to look at an initial value problem is with a slope ﬁeld. The following applet allows you to set the initial equation for f ′ (x) and then the slope ﬁeld for that equation is displayed. In magenta you’ll see one possible solution for f (x). If you move the magenta point to the initial value, then you will see the graph of the solution to the initial value problem. Follow the directions on the page with the applet to explore this idea, and then try redoing the examples from this section on the applet. Slope Fields Applet.

Lesson Summary 1. We found general solutions of diﬀerential equations. 2. We used initial conditions to ﬁnd particular solutions of diﬀerential equations. www.ck12.org

212

Review Questions In problems #1–3, solve the diﬀerential equation for f (x). √ 1. f ′ (x) = 2e2x − 2 x 1 2. f ′ (x) = sin x − √ ex 3. f ′′ (x) = (2 + x) x In problems #4–7, solve the diﬀerential equation for f (x) given the initial condition. 4. 5. 6. 7. 8.

f ′ (x) = 6x5 − 4x2 + 73 and f (1) = 4 2 f ′ (x) = 3x + e2x and f (0) = 3. √ 3 ′ f (x) = x2 − x12 and f (1) = 3 ( ) √ f ′ (x) = (2 cos x − sin x), − π2 ≤ x ≤ π2 , and f π3 = 3 + 12 Suppose the graph of f includes the point (−2, 4) and that the slope of the tangent line to f at x is −2x + 4. Find f (5).

In problems #9–10, ﬁnd the function f that satisﬁes the given conditions. 9. f ′′ (x) = sin x − e−2x with f ′ (0) = 52 and f (0) = 0 10. f ′′ (x) = √1x with f ′ (4) = 7 and f (4) = 25

Review Answers 3

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

f (x) = 22x − 43 x 2 + C f (x) = − cos x + e1x + C 8 52 4 72 x + 35 x +C f (x) = 15 4 3 6 f (x) = x − 3 x + 73 x + 2 2x f (x) = x3 + e2 + 52 √ 3 f (x) = 35 x5 + x1 + 75 f (x) = 2 sin x + cos x f (x) = −x2 + 4x + 16; f (5) = 11 f (x) = − sin x − 14 e2x + 4x + 14 3 f (x) = 43 x 2 + 3x + 37

4.3

The Area Problem

Learning Objectives • Use sigma notation to evaluate sums of rectangular areas

213

www.ck12.org

• Find limits of upper and lower sums • Use the limit deﬁnition of area to solve problems

Introduction In The Lesson The Calculus we introduced the area problem that we consider in integral calculus. The basic problem was this: f (x) = x2 . Suppose we are interested in ﬁnding the area between the x−axis and the curve of f (x) = x2 , from x = 0 to x = 1.

We approximated the area by constructing four rectangles, with the height of each rectangle equal to the maximum value of the function in the sub-interval.

We then summed the areas of the rectangles as follows: www.ck12.org

214

R1 = R2 = R3 = R4 = and R1 + R2 + R3 + R4 =

30 64

=

15 32

1 4 1 4 1 4 1 4

( ) 1 1 ·f = , 4 64 ( ) 1 1 ·f = , 2 16 ( ) 3 9 ·f = , 4 64 1 · f (1) = , 4

≈ 0.46.

We call this the upper sum since it is based on taking the maximum value of the function within each sub-interval. We noted that as we used more rectangles, our area approximation became more accurate. We would like to formalize this approach for both upper and lower sums. First we note that the lower sums of the area of the rectangles results in R1 + R2 + R3 + R4 = 13/64 ≈ 0.20 Our intuition tells us that the true area lies somewhere between these two sums, or 0.20 < Area < 0.46 and that we will get closer to it by using more and more rectangles in our approximation scheme. In order to formalize the use of sums to compute areas, we will need some additional notation and terminology. Sigma Notation In The Lesson The Calculus we used a notation to indicate the upper we increased ∑ sum when 195 our rectangles to N = 16 and found that our approximation A = 16 R = ≈ 0.38 . The i 1 512 notation we used to enabled us to indicate the sum without the need to write out all of the individual terms. We will make use of this notation as we develop more formal deﬁnitions of the area under the curve. ∑ Let’s be more precise with the notation. For example, the quantity A = Ri was found by summing the areas of N = 16 rectangles. We want to indicate this process, and we can do so by providing indices to the symbols used as follows:

A=

16 ∑

Ri = R1 + R2 + R3 + ... + R15 + R16 .

i=1

The sigma symbol with these indices tells us how the rectangles are labeled and how many terms are in the sum. Useful Summation Formulas

215

www.ck12.org

We can use the notation to indicate useful formulas that we will have occasion to use. For example, you may recall that the sum of the ﬁrst n integers is n(n + 1)/2. We can indicate this formula using sigma notation. The formula is given here along with two other formulas that will become useful to us. n ∑

i=

i=1 n ∑

n(n + 1) , 2

n(n + 1)(2n + 1) , 6 i=1 [ ]2 n ∑ n(n + 1) 3 i = . 2 i=1 i2 =

We can show from associative, commutative, and distributive laws for real numbers that ∑n ∑n ∑n i=1 (ai + bi ) = i=1 (ai ) + i=1 (bi ) and ∑n ∑n i=1 (kai ) = k i=1 (ai ). Example 1: Compute the following quantity using the summation formulas: 10 ∑

2i(i − 6i).

i=1

Solution: 10 ∑ i=1

2i(i − 6i) =

10 ∑

(2i − 12i) = 2 2

(

i=1

10 ∑

i − 12 2

i=1

(10)(10 + 1)(2 · 10 + 1) =2 6 = 770 − 660 = 110.

)

10 ∑

i

i=1

− 12

(

(10)(11) 2

)

Another Look at Upper and Lower Sums We are now ready to formalize our initial ideas about upper and lower sums. Let f be a bounded function in a closed interval [a, b] and P = [x0 , . . . , xn ] the partition of [a, b] into n subintervals. www.ck12.org

216

We can then deﬁne the lower and upper sums, respectively, over partition P , by

S(P ) = T (P ) =

n ∑ 1 n ∑

mi (xi − xi−1 ) = m1 (x1 − x0 ) + m2 (x2 − x1 ) + . . . + mn (xn − xn−1 ), Mi (xi − xi−1 ) = M1 (x1 − x0 ) + M2 (x2 − x1 ) + . . . + Mn (xn − xn−1 ).

1

where mi is the minimum value of f in the interval of length xi −xi−1 and Mi is the maximum value of f in the interval of length xi − xi−1 . The following example shows how we can use these to ﬁnd the area. Example 2: Show that the upper and lower sums for the function f (x) = x2 , from x = 0 to x = 1, approach the value A = 1/3. Solution: Let P be a partition of n equal sub intervals over [0, 1]. We will show the result for the upper sums. By our deﬁnition we have

T (P ) =

n ∑

Mi (xi − xi−1 ) = M1 (x1 − x0 ) + M2 (x2 − x1 ) + . . . + Mn (xn − xn−1 ).

1

We note that each rectangle will have width indicated:

1 , n

217

and lengths

( 1 )2 ( 2 )2 ( 3 )2 ( )2 , n , n , . . . , nn as n

www.ck12.org

T (P ) =

n ∑

Mi (xi − xi−1 ) = M1 (x1 − x0 ) + M2 (x2 − x1 ) + . . . + Mn (xn − xn−1 )

1

( )2 ( )2 ( )2 1 1 2 1 3 1 ( n )2 + + + ... + n n n n n n n ( )2 1 1 = (12 + 22 + 32 + . . . + n2 ) n n ) ( ( ) ( )( ) (n + 1)(2n + 1) 1 1 n(n + 1)(2n + 1) 2 2 2 2 = = (1 + 2 + 3 + . . . + n ) = . n3 n3 6 6n2

1 = n

We can re-write this result as:

(n + 1)(2n + 1) 1 = 2 6n 6

(

n+1 n

)(

2n + 1 n

)

1 = 6

(

1 1+ n

)(

1 2+ n

) .

We observe that as

1 x → +∞, 6

(

1 1+ n

)(

1 2+ n

)

1 → . 3

We now are able to deﬁne the area under a curve as a limit.

Deﬁnition Let f be a continuous function on a closed interval [a, b]. Let P be a partition of n equal sub intervals over [a, b]. Then the area under the curve of f is the limit of the upper and lower sums, that is

A = lim S(P ) = lim T (P ). n→+∞

n→+∞

Example 3: Use the limit deﬁnition of area to ﬁnd the area under the function f (x) = 4 − x from 1 to x = 3. www.ck12.org

218

Solution: If we partition the interval [1, 3] into n equal sub-intervals, then each sub-interval will have length 3−1 = n2 and height 3 − i△x as i varies from 1 to n. So we have △x = n2 and n

S(P ) =

n ∑

(3 − i△x)△x =

n ∑

1

(3△x) −

1

n ∑

i(△x)2

1

n(n + 1) = (3△x)n − (△x)2 . n Since △x =

2 n

, we then have by substitution ( ) 2 2 (3△x)n − n(n+1) (△x) = 6 − 2 + = 4 + n2 → 4 as n → ∞ . Hence the area is A = 4. n n This example may also be solved with simple geometry. It is left to the reader to conﬁrm that the two methods yield the same area.

Lesson Summary 1. We used sigma notation to evaluate sums of rectangular areas. 2. We found limits of upper and lower sums. 3. We used the limit deﬁnition of area to solve problems.

Review Questions In problems #1–2 , ﬁnd the summations.

219

www.ck12.org

∑ 1. ∑10 i=1 i(2i − 3) n 2. i=1 (3 − i)(2 + i) In problems #3–5, ﬁnd S(P ) and T (P ) under the partition P. { } 3. f (x) = 1 − x2 , P {= 0, 12 , 1, 32 , 2 } 4. f (x) = 2x2 , P = −1, − 12 , 0, 12 , 1 5. f (x) = x1 , P = {−4, −3, −2, −1} In problems #6–8, ﬁnd the area under the curve using the limit deﬁnition of area. 6. f (x) = 3x + 5 from x = 2 to x = 6. 7. f (x) = x2 from x = 1 to x = 3. 8. f (x) = x1 from x = 1 to x = 4. In problems #9–10, state whether the function is integrable in the given interval. Give a reason for your answer. 9. f (x) = |x { − 2| on the interval [1, 4] 1 if x is rational 10. f (x) = on the interval [0, 1] −1 if x is irrational

Review Answers ∑ 1. ∑10 i=1 i(2i − 3) = 605 n 1 2 2. i=1 (3 − i)(2 + i) = 3 (19 − n ) 3. S(P ) = −1.75; T (P ) = 0.25(note that we have included areas under the x−axis as negative values.) 4. S(P ) = 0.5; T (P ) = 2.5 5. S(P ) = −1.83; T (P ) = −1.08 6. Area = 68 7. Area = 26 3 8. Area = 15 16 9. Yes, since f (x) = |x − 2| is continuous on [1, 4] 10. No, since S(P ) = −1; T (P ) = 1

4.4

Deﬁnite Integrals

Learning Objectives • Use Riemann Sums to approximate areas under curves www.ck12.org

220

• Evaluate deﬁnite integrals as limits of Riemann Sums

Introduction In the Lesson The Area Problem we deﬁned the area under a curve in terms of a limit of sums.

A = lim S(P ) = lim T (P ) n→+∞

n→+∞

where

S(P ) =

n ∑

mi (xi − xi−1 ) = m1 (x1 − x0 ) + m2 (x2 − x1 ) + . . . + mn (xn − xn−1 ),

1

T (P ) =

n ∑

Mi (xi − xi−1 ) = M1 (x1 − x0 ) + M2 (x2 − x1 ) + . . . + Mn (xn − xn−1 ),

1

S(P of Riemann Sums. In general, Riemann Sums are of form ∑n ), and∗ T (P ) were examples ∗ i=1 f (xi )△x where each xi is the value we use to ﬁnd the length of the rectangle in the ith sub-interval. For example, we used the maximum function value in each sub-interval to ﬁnd the upper sums and the minimum function in each sub-interval to ﬁnd the lower sums. But since the function is continuous, we could have used any points within the sub-intervals to ﬁnd the limit. Hence we can deﬁne the most general situation as follows: Deﬁnition If f is continuous on [a, b], we divide the interval [a, b] into n sub-intervals of equal width with △x = b−a . We let x0 = a, x1 , x2 , . . . , xn = b be the endpoints of these n ∗ ∗ sub-intervals and let x1 , x2 , . . . , x∗n be any sample points in these sub-intervals. Then the deﬁnite integral of f from x = a to x = b is ∫

b

f (x)dx = lim a

n ∑

n→∞

f (x∗i )△x.

i=1

Example 1: Evaluate the Riemann Sum for f (x) = x3 from x = 0 to x = 3 using n = 6 sub-intervals and taking the sample points to be the midpoints of the sub-intervals. Solution:

221

www.ck12.org

If we partition the interval [0, 3] into n = 6 equal sub-intervals, then each sub-interval will have length 3−0 = 12 . So we have △x = 12 and 6

R6 =

6 ∑

f (x∗i )△x = f (0.25)△x + f (0.75)△x + f (1.25)△x + f (1.75)△x + f (2.25)△x + f (2.75)△x

(1

)( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 27 1 125 1 343 1 729 1 1331 1 = + + + + + 64 2 64 2 64 2 64 2 64 2 64 2 2556 = = 39.93. 64 Now let’s compute the deﬁnite integral using our deﬁnition and also some of our summation formulas. Example 2: Use the deﬁnition of the deﬁnite integral to evaluate

∫3 0

x3 dx.

Solution: Applying our deﬁnition, we need to ﬁnd ∫

3 3

x dx = lim 0

n→∞

n ∑

f (x∗i )△x.

i=1

We will use right endpoints to compute the integral. We ﬁrst need to divide [0, 3] into n sub-intervals of length △x = 3−0 = n3 . Since we are using right endpoints, x0 = 0, x1 = n 3 , x2 = n6 , . . . xi = 3i . n n ∫3 ∑ ∑ ∑ So 0 x3 dx = limn→∞ ni=1 f ( 3i )( n3 ) = limn→∞ n3 ni=1 ( 3i )3 = limn→∞ n3 ni=1 ( n273 )i3 = limn→∞ n n www.ck12.org

222

81 n4

∑n i=1

i3 .

Recall that ∫

∑n

3 1 i =

3

[

n(n+1) 2

]2

. By substitution, we have

[ ]2 [ ]2 81 1 81 81 n(n + 1) x dx = lim = 4 = lim 1+ → as n → ∞. n→∞ 4 n→∞ n 2 n 4 3

0

Hence ∫

3

x3 dx = 0

81 . 4

Before we look to try some problems, let’s make a couple of observations. First, we will soon not need to rely on the summation formula and Riemann Sums for actual computation of deﬁnite integrals. We will develop several computational strategies in order to solve a variety of problems that come up. Second, the idea of deﬁnite integrals as approximating the area under a curve can be a bit confusing since we may sometimes get results that do not make∫ sense when interpreted as areas. For example, if we were to compute the deﬁnite 3 integral −3 x3 dx, then due to the symmetry of f (x) = x3 about the origin, we would ﬁnd ∫3 that −3 x3 dx = 0. This is because for every sample point x∗j , we also have −x∗j is also a ∫3 sample point with f (−x∗j ) = −f (x∗j ). Hence, it is more accurate to say that −3 x3 dx gives us the net area between x = −3 and x = 3. If we the total area bounded by the ∫ 3wanted 3 . graph and the x−axis, then we would compute 2 0 x dx = 81 2 Multimedia Link The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number of partitions. Follow the examples given on the page, and then use the applet to explore on your own. Riemann Sums Applet. Note: On this page the author uses Left- and Right- hand sums. These are similar to the sums S(P ) and T (P ) that you have learned, particularly in the case of an increasing (or decreasing) function. Left-hand and Right-hand sums are frequently used in calculations of numerical integrals because it is easy to ﬁnd the left and right endpoints of each interval, and much more diﬀicult to ﬁnd the max/min of the function on each interval. The diﬀerence is not always important from a numerical approximation standpoint; ss you increase the number of partitions, you should see the Left-hand and Right-hand sums converging to the same value. Try this in the applet to see for yourself.

Lesson Summary 1. We used Riemann Sums to approximate areas under curves. 2. We evaluated deﬁnite integrals as limits of Riemann Sums.

223

www.ck12.org

Review Questions In problems #1–7 , use Riemann Sums to approximate the areas under the curves. 1. Consider f (x) = 2 − x from x = 0 to x = 2. Use Riemann Sums with four subintervals of equal lengths. Choose the midpoints of each subinterval as the sample points. 2. Repeat problem #1 using geometry to calculate the exact area of the region under the graph of f (x) = 2 − x from x = 0 to x = 2. (Hint: Sketch a graph of the region and see if you can compute its area using area measurement formulas from geometry.) 3. Repeat problem #1 using the deﬁnition of the deﬁnite integral to calculate the exact area of the region under the graph of f (x) = 2 − x from x = 0 to x = 2. 4. f (x) = x2 − x from x = 1 to x = 4. Use Riemann Sums with ﬁve subintervals of equal lengths. Choose the left endpoint of each subinterval as the sample points. 5. Repeat problem #4 using the deﬁnition of the deﬁnite intergal to calculate the exact area of the region under the graph of f (x) = x2 − x from x = 1 to x = 4. 6. Consider f (x) = 3x2 . Compute the Riemann Sum of f on [0, 1] under each of the following situations. In each case, use the right endpoint as the sample points. (a) (b) (c) (d)

Two sub-intervals of equal length. Five sub-intervals of equal length. Ten sub-intervals of equal length. Based on your answers above, try to guess the exact area under the graph of f on [0, 1].

7. Consider f (x) = ex . Compute the Riemann Sum of f on [0, 1] under each of the following situations. In each case, use the right endpoint as the sample points. (a) (b) (c) (d)

Two sub-intervals of equal length. Five sub-intervals of equal length. Ten sub-intervals of equal length. Based on your answers above, try to guess the exact area under the graph of f on [0, 1].

8. Find the net area under the graph of f (x) = x3 − x; x = −1 to x = 1. (Hint: Sketch the graph and check for symmetry.) 9. Find the total area bounded by the graph of f (x) = x3 − x and the x−axis, from to x = −1 to x = 1. 10. Use your knowledge of geometry to evaluate the following deﬁnite integral: ∫3√

9 − x2 dx (Hint: set y = 0 region from geometry.)

√

9 − x2 and square both sides to see if you can recognize the

Review Answers 1. Area = 2 www.ck12.org

224

2. 3. 4. 5. 6.

7.

8. 9. 10.

4.5

Area = 2 Area = 2 Area = 10.08 Area = 15.5 (a) Area = 1.875 (b) Area = 1.32 (c) Area = 1.15 (d) Area = 1 (a) Area = 2.18 (b) Area = 1.89 (c) Area = 1.80 (d) Area = e1 − 1 ≈ 1.71 The graph is symmetric about the origin; hence Area = 0. Area = 12 The graph is that of a quarter circle of radius 3; hence Area =

9π . 4

Evaluating Deﬁnite Integrals

Learning Objectives • Use antiderivatives to evaluate deﬁnite integrals • Use the Mean Value Theorem for integrals to solve problems • Use general rules of integrals to solve problems

Introduction In the Lesson on Deﬁnite Integrals, we evaluated deﬁnite integrals using the limit deﬁnition. This process was long and tedious. In this lesson we will learn some practical ways to evaluate deﬁnite integrals. We begin with a theorem that provides an easier method for evaluating deﬁnite integrals. Newton discovered this method that uses antiderivatives to calculate deﬁnite integrals. Theorem: If f is continuous on the closed interval [a, b], then ∫

b

f (x)dx = F (b) − F (a), a

where F is any antiderivative of f.

225

www.ck12.org

We sometimes use the following shorthand notation to indicate

∫ a

b

∫b a

f (x)dx = F (b) − F (a) :

]b f (x)dx = F (x) a .

The proof of this theorem is included at the end of this lesson. Theorem 4.1 is usually stated as a part of the Fundamental Theorem of Calculus, a theorem that we will present in the Lesson on the Fundamental Theorem of Calculus. For now, the result provides a useful and eﬀicient way to compute deﬁnite integrals. We need only ﬁnd an antiderivative of the given function in order to compute its integral over the closed interval. It also gives us a result with which we can now state and prove a version of the Mean Value Theorem for integrals. But ﬁrst let’s look at a couple of examples. Example 1: Compute the following deﬁnite integral:

∫

3

x3 dx. 0

Solution: Using the limit deﬁnition we found that theorem as follows:

∫3 0

x3 dx =

81 . 4

We now can verify this using the

We ﬁrst note that x4 /4 is an antiderivative of f (x) = x3 . Hence we have

∫

3

x4 x dx = 4

]3

3

0

= 0

81 0 81 − = . 4 4 4

We conclude the lesson by stating the rules for deﬁnite integrals, most of which parallel the rules we stated for the general indeﬁnite integrals. www.ck12.org

226

∫

a

f (x)dx = 0 a

∫

∫

b

a

f (x)dx = − f (x)dx a b ∫ b ∫ b k · f (x)dx = k f (x)dx a a ∫ b ∫ b ∫ b [f (x) ± g(x)]dx = f (x)dx ± g(x)dx a a a ∫ b ∫ c ∫ b f (x)dx where a < c < b. f (x)dx + f (x)dx = c

a

a

Given these rules together with Theorem 4.1, we will be able to solve a great variety of deﬁnite integrals. Example 2: ∫2 √ Compute −2 (x − x)dx. Solution: ∫

4

(x −

√

1

∫

∫

4

xdx −

x)dx = 1

1

4

√

x2 ] 4 2 3 ] 4 xdx = − x2 = 2 1 3 1

(

1 8− 2

)

2 15 14 17 − (8 − 1) = − = . 3 2 3 6

Example 3: ∫π Compute 02 (x + cos x)dx. Solution: ∫

π 2

∫ (x + cos x)dx =

0

∫

π 2

(x)dx = 0

π 2

(cos x)dx = 0

x2 ] π2 sin x ] π2 π2 π2 + 4 + = +1= . 2 0 1 0 4 4

Lesson Summary 1. We used antiderivatives to evaluate deﬁnite integrals. 2. We used the Mean Value Theorem for integrals to solve problems. 3. We used general rules of integrals to solve problems. Proof of Theorem 4.1

227

www.ck12.org

We ﬁrst need to divide [a, b] into n sub-intervals of length △x = a, x1 , x2 , . . . , xn = b be the endpoints of these sub-intervals.

b−a n

. We let x0 =

Let F be any antiderivative of f. Consider F (b) − F (a) = F (xn ) − F (x0 ). We will now employ a method that will express the right side of this equation as a Riemann Sum. In particular,

F (b) − F (a) = F (xn ) − F (x0 ) = F (xn ) − F (xn−1 ) + F (xn+1 ) − F (xn−2 ) + F (xn−2 ) − . . . − F (x1 ) + F (x1 ) n ∑ = [F (xi ) − F (xi−1 )]. 1

Note that F is continuous. Hence, by the Mean Value Theorem, there exist ci ∈ [xi − 1, xi ] such that F (xi ) − F (xi−1 ) = F ′ (ci )(xi − xi−1 ) = f (ci )△x. Hence

F (b) − F (a) =

n ∑

′

F (ci )(xi − xi−1 ) =

n ∑

1

f (ci )△x.

1

Taking the limit of each side as n → ∞ we have

lim [F (b) − F (a)] = lim

n→ ∞

n→ ∞

n ∑

f (ci )△x.

1

We note that the left side is a constant and the right side is our deﬁnition for Hence

F (b) − F (a) = lim

n→ ∞

n ∑

∫ f (ci )△x =

1

Proof of Theorem 4.2 ∫x Let F (x) = a f (x)dx. www.ck12.org

228

b

f (x)dx. a

∫b a

f (x)dx.

By the Mean Value Theorem for derivatives, there exists c ∈ [a, b] such that

F ′ (c) =

F (b) − F (a) . b−a

From Theorem 4.1 we have that F is an antiderivative of f. Hence, F ′ (x) = f (x) and in particular, F ′ (c) = f (c). Hence, by substitution we have

f (c) = Note that F (a) =

∫a a

F (b) − F (a) . b−a

f (x)dx = 0. Hence we have

f (c) =

F (b) − 0 F (b) = , b−a b−a

and by our deﬁnition of F (x) we have

1 1 f (c) = F (b) = b−a b−a

∫

b

f (x)dx. a

This theorem allows us to ﬁnd for positive functions a rectangle that has base [a, b] and ∫b height f (c) such that the area of the rectangle is the same as the area given by a f (x)dx. In other words, f (c) is the average function value over [a, b].

Review Questions In problems #1–8, use antiderivatives to compute the deﬁnite integral. 1. 2. 3. 4. 5. 6. 7.

∫9

( √3 )dx ∫41 x 2 (t − t )dt ∫05 1 ( √ + √12 )dx ∫21 x 2 4(x − 1)(x2 + 1)dx ∫08 4 ( + x2 + x)dx ∫24 x3x (e )dx ∫24 2 dx 1 x+3

229

www.ck12.org

√ 8. Find the average value ∫of f (x) = x over [1, 9]. 4 9. If f is continuous and 1 f (x)dx = 9, show that f takes on the value 3 at least once on the interval [1, 4]. 10. Your friend states ∫ 2π that there is no area under the curve of f (x) = sin x on [0, 2π] since he computed 0 sin xdx = 0. Is he correct? Explain your answer.

Review Answers 1. 2. 3.

∫9

( √3 )dx = 6 ∫41 x 2 1 (t − t )dt = 30 0 √ √ √ ∫5 1 √ + √1 )dx = 2 5 − 2 2 + 3 2 ( 2 2 ∫21 x 2 16 2 4(x − 1)(x + 1)dx = − 5 ∫08 4 ( + x2 + x)dx = 9417 ≈ 196.19 48 ∫24 x3x e12 −e6 (e )dx = 3 ∫24 2 dx = 2 ln 7 − 2 ln 4 1 x+3

4. 5. 6. 7. 8. 13 6 9. Apply the Mean Value Theorem for integrals.∫ 2π 10. He is partially correct. The deﬁnite integral 0 sin x dx computes the net area under the curve. However, the area between the curve and the x-axis is given by ∫ π ]π A=2 sin x dx = − cos x 0 = 2. 0

4.6

The Fundamental Theorem of Calculus

Learning Objectives • Use the Fundamental Theorem of Calculus to evaluate deﬁnite integrals

Introduction In the Lesson on Evaluating Deﬁnite Integrals, we evaluated deﬁnite integrals using antiderivatives. This process was much more eﬀicient than using the limit deﬁnition. In this lesson we will state the Fundamental Theorem of Calculus and continue to work on methods for computing deﬁnite integrals. Fundamental Theorem of Calculus: Let f be continuous on the closed interval [a, b]. www.ck12.org

230

∫x 1. If function F is deﬁned by F (x) = a f (t)dx, on [a, b] , then F ′ (x) = f (x) on [a, b]. 2. If g is any antiderivative of f on [a, b], then ∫

b

f (t)dt = g(b) − g(a). a

We ﬁrst note that we have already proven part 2 as Theorem 4.1. The proof of part 1 appears at the end of this lesson. Think about this Theorem. Two of the major unsolved problems in science and mathematics turned out to be solved by calculus which was invented in the seventeenth century. These are the ancient problems: 1. Find the areas deﬁned by curves, such as circles or parabolas. 2. Determine an instantaneous rate of change or the slope of a curve at a point. With the discovery of calculus, science and mathematics took huge leaps, and we can trace the advances of the space age directly to this Theorem. Let’s continue to develop our strategies for computing deﬁnite integrals. We will illustrate how to solve the problem of ﬁnding the area bounded by two or more curves. Example 1: Find the area between the curves of f (x) = x and g(x) = x3 . Solution: We ﬁrst observe that there are no limits of integration explicitly stated here. Hence we need to ﬁnd the limits by analyzing the graph of the functions.

231

www.ck12.org

We observe that the regions of interest are in the ﬁrst and third quadrants from x = −1 to x = 1. We also observe the symmetry of the graphs about the origin. From this we see that the total area enclosed is ∫

[∫

1

∫

1

(x − x3 )dx = 2

2 0

1

xdx − 0

0

1 1 ] [ ] [ ] 2 4 x x 1 1 1 1 3 x dx = 2 − =2 = . − = =2 2 4 2 4 4 2 0

0

Example 2: Find the area between the curves of f (x) = |x − 1| and the x−axis from x = −1 to x = 3. Solution: We observe from the graph that we will have to divide the interval [−1, 3] into subintervals [−1, 1] and [1, 3].

Hence the area is given by ∫

∫

1

(x − 1)dx =

(−x + 1)dx + −1

(

3

1

) +1 ( 2 ) +3 x x2 − +x + − x = 2 + 2 = 4. 2 2 −1 +1

Example 3: Find the area enclosed by the curves of f (x) = x2 + 2x + 1 and

g(x) = −x2 − 2x + 1. Solution: www.ck12.org

232

The graph indicates the area we need to focus on.

∫

∫

0

−2

(−x − 2x + 1)dx −

(

0

2

2

(x + 2x + 1)dx = −2

) 0 ( 2 ) 0 x3 10 2 8 x 2 3 − −x +x + − x + x = − = . 3 3 3 3 3 −2 −2

Before providing another example, let’s look∫ back at the ﬁrst part of the Fundamental x Theorem. If function F is deﬁned by F (x) = a f (t)dt, on [a, b] then F ′ (x) = f (x) on [a, b]. Observe that if we diﬀerentiate the integral with respect to x, we have d dx

∫

x

f (t)dt = F ′ (x) = f (x).

a

This fact enables us to compute derivatives of integrals as in the following example. Example 4: Use the Fundamental Theorem to ﬁnd the derivative of the following function: ∫ g(x) =

x

(1 +

√ 3

t)dt.

0

Solution: While we could easily integrate the right side and then diﬀerentiate, the Fundamental Theorem enables us to ﬁnd the answer very routinely. d g (x) = dx ′

∫

x

(1 +

√ 3

t)dt = 1 +

√ 3

x.

0

233

www.ck12.org

This application of the Fundamental Theorem becomes more important as we encounter functions that may be more diﬀicult to integrate such as the following example. Example 5: Use the Fundamental Theorem to ﬁnd the derivative of the following function: ∫

x

(t2 cos t)dt.

g(x) = 2

Solution: In this example, the integral is more diﬀicult to evaluate. The Fundamental Theorem enables us to ﬁnd the answer routinely.

d g (x) = dx ′

∫

x

(t2 cos t)dt = x2 cos x. 2

Lesson Summary 1. We used the Fundamental Theorem of Calculus to evaluate deﬁnite integrals. Fundamental Theorem of Calculus Let f be continuous on the closed interval [a, b]. ∫x 1. If function F is deﬁned by F (x) = a f (t)dt, on [a, b], then F ′ (x) = f (x), on [a, b]. 2. If g is any antiderivative of f on [a, b], then ∫

b

f (t)dt = g(b) − g(a). a

We ﬁrst note that we have already proven part 2 as Theorem 4.1. Proof of Part 1. 1. Consider F (x) =

∫x a

f (t)dt, on [a, b].

2. x, c ∈ [a, b], c < x. ∫x ∫c ∫x Then a f (t)dt = a f (t)dt + c f (t)dt by our rules for deﬁnite integrals. ∫x ∫c ∫x ∫x 3. Then a f (t)dt − a f (t)dt = c f (t)dt . Hence F (x) − F (c) = c f (t)dt. www.ck12.org

234

4. Since f is continuous on [a, b] and x, c ∈ [a, b], c < x then we can select u, v ∈ [c, v] such that f (u) is the minimum value of and f (v) is the maximum value of f in [c, x]. Then we can consider f (u)(x − c) as a lower sum and f (v)(x − c) as an upper sum of f from c to x. Hence ∫x 5. f (u)(x − c) ≤ c f (t)dt ≤ f (v)(x − c). 6. By substitution, we have:

f (u)(x − c) ≤ F (x) − F (c) ≤ f (v)(x − c). 7. By division, we have

f (u) ≤

F (x) − F (c) ≤ f (v). x−c

8. When x is close to c, then both f (u) and f (v) are close to f (c) by the continuity of f (c) 9. Hence limx→c+ F (x)−F = f (c). Similarly, if x < c, then limx→c− x−c F (x)−F (c) limx→c x−c = f (c).

F (x)−F (c) x−c

= f (c). Hence,

10. By the deﬁnition of the derivative, we have that F ′ (c) = limx→c

F (x)−F (c) x−c

= f (c) for every c ∈ [a, b]. Thus, F is an antiderivative of f on [a, b].

Review Questions In problems #1–4, sketch the graph of the function f (x) in the interval [a, b]. Then use the Fundamental Theorem of Calculus to ﬁnd the area of the region bounded by the graph and the x−axis. 1. 2. 3. 4.

f (x) = 2x + 3, [0, 4] f (x) = ex , [0, 2] f (x) = x2 + x, [1, 3] f (x) = x2 − x, [0, 2]

(Hint: Examine the graph of the function and divide the interval accordingly.) In problems #5–7 use antiderivatives to compute the deﬁnite integral. 5.

∫ +1 −1

|x|dx

235

www.ck12.org

6.

∫3 0

|x3 − 2|dx

(Hint: Examine the graph of the function and divide the interval accordingly.) 7.

∫ +4 −2

[|x − 1| + |x + 1|] dx

(Hint: Examine the graph of the function and divide the interval accordingly.) In problems #8–10, ﬁnd the area between the graphs of the functions. √ 8. f (x) = x, g(x) = x, [0, 2] 9. f (x) = x2 , g(x) = 4, [0, 2] 10. f (x) = x2 + 1, g(x) = 3 − x, [0, 3]

Review Answers 1. Area = 28

2. Area = e2 − 1

www.ck12.org

236

3. Area =

38 3

4. Area = 1

5.

∫ +1 −1

∫3

|x|dx = 1

∫

√ 3

2

|x − 2|dx = 0 (2 − x3 )dx + 0 [ ] ∫ +4 7. −2 |x − 1| + |x + 1| dx = 22 6.

3

8. Area =

∫3

√ 3

(x3 − 2)dx = 3 · 2

√ 3

2+

57 4

√ 7−4 2 3

237

www.ck12.org

9. Area =

16 3

10. Area =

10 3

4.7

Integration by Substitution

Learning Objectives • Integrate composite functions • Use change of variables to evaluate deﬁnite integrals • Use substitution to compute deﬁnite integrals

Introduction In this lesson we will expand our methods for evaluating deﬁnite integrals. We ﬁrst look at a couple of situations where ﬁnding antiderivatives requires special methods. These involve ﬁnding antiderivatives of composite functions and ﬁnding antiderivatives of products of functions. www.ck12.org

238

Antiderivatives of Composites Suppose we needed to compute the following integral: ∫

√ 3x2 1 + x3 dx.

Our rules of integration are of no help here. We √ note that the integrand is of the form ′ 3 f (g(x)) ∗ g (x) where g(x) = 1 + x and f (x) = x. Since we are looking for an antiderivative F of f, and we know that F ′ = f, we can re-write our integral as ∫ √

3 2 √ 1 + x3 · 3x2 dx = ( 1 + x3 ) 2 + C. 3

In practice, we use the following substitution scheme to verify that we can integrate in this way: 1. Let u = 1 + x3 . 2. Diﬀerentiate both sides so du = 3x2 dx. 3. integral in x to an integral in u: ∫ √Change the2original ∫√ 3 1 + x · 3x dx = udu, where u = 1 + x3 and du = 3x2 dx. 4. Integrate with respect to u: ∫

√

∫ udu =

1 2 3 u 2 du = u 2 + C. 3

5. Change the answer back to x: ∫

√

3 2 3 2 √ udu = u 2 + C = ( 1 + x3 ) 2 + C. 3 3

While this method of substitution is a very powerful method for solving a variety of problems, we will ﬁnd that we sometimes will need to modify the method slightly to address problems, as in the following example. Example 1: Compute the following indeﬁnite integral:

239

www.ck12.org

∫ 3

x2 ex dx. Solution: 3 2 ∫We ′note that′ the derivative of x is 3x ; hence, the current problem is not of the form F (g(x)) · g (x)dx. But we notice that the derivative is oﬀ only by a constant of 3 and we know that constants are easy to deal with when diﬀerentiating and integrating. Hence

Let u = x3 . Then du = 3x2 dx. Then 13 du = x2 dx. and we are ready to change the original integral from x to an integral in u and integrate: ∫

(

∫ 2 x3

x e dx =

e

u

1 du 3

)

1 = 3

∫

1 eu du = eu + C. 3

Changing back to x, we have ∫

1 3 3 x2 ex dx = ex + C. 3

We can also use this substitution method to evaluate deﬁnite integrals. If we attach limits of integration to our ﬁrst example, we could have a problem such as ∫

4

√

1 + x3 · 3x2 dx.

1

The method still works. However, we have a choice to make once we are ready to use the Fundamental Theorem to evaluate the integral. ∫√ ∫√ Recall that we found that 1 + x3 · 3x2 dx = udu for the indeﬁnite integral. At this point, we could evaluate the integral by changing the answer back to x or we could evaluate the integral in u. But we need to be careful. Since the original limits of integration were in x, we need to change the limits of integration for the equivalent integral in u. Hence, √ ∫4√ ∫ 3 · 3x2 dx = 65 1 + x udu, where u = 1 + x3 1 u=2 ∫ 1

www.ck12.org

4

√

∫ 1+

x3

65

· 3x dx = 2

u=2

√

u=65 √ 2 √ 2 3 = ( 653 − 8). udu = u 2 3 3 u=2

240

Integrating Products of Functions

∫ We are not able to state a rule for integrating products of functions, f (x)g(x)dx but we can get a relationship that is almost as eﬀective. Recall how we diﬀerentiated a product of functions:

d f (x)g(x) = f (x)g ′ (x) + g(x)f ′ (x). dx So by integrating both sides we get ∫ [f (x)g ′ (x) + g(x)f ′ (x)]dx = f (x)g(x), or ∫

∫

′

f (x)g (x)dx = f (x)g(x) −

g(x)f ′ (x).

In order to remember the formula, we usually write it as ∫

∫ udv = uv −

vdu.

We refer to this method as integration by parts. The following example illustrates its use. Example 2: Use integration by parts method to compute ∫ xex dx. Solution: We note that our other substitution method is not applicable here. But our integration by parts method will enable us to reduce the integral down to one that we can easily evaluate. Let u = x and dv = ex dx then du = dx and v = ex By substitution, we have ∫

∫ xe dx = xe − x

x

241

ex dx. www.ck12.org

We can easily evaluate the integral and have ∫

∫ xe dx = xe − x

ex dx = xex − ex + C.

x

And should we wish to evaluate deﬁnite integrals, we need only to apply the Fundamental Theorem to the antiderivative.

Lesson Summary 1. We integrated composite functions. 2. We used change of variables to evaluate deﬁnite integrals. 3. We used substitution to compute deﬁnite integrals.

Review Questions Compute the integrals in problems #1–10. ∫ 1. x ln xdx ∫3√ 2. ∫1 x ln xdx x 3. √2x+1 dx ∫ 1 3√ 4. ∫0 x 1 − x2 dx 5. x cos xdx ∫1 √ 6. 0(x2 x3 ) + 9dx ∫ 1 1 7. · e x dx x2 ∫ 3 x2 8. ∫ x e dx 9. lnx5 dx ∫ ex 2 10. 1 x1 dx

Review Answers 1. 2. 3. 4. 5.

∫

2

x ln xdx = x (2 ln4(x−1) + C ) ∫3√ 3 3 2 2 2 2 − 2 + x ln xdx = ln 3 · 3 3 3 3 1 √ ∫ x (x−1) 2x+1 √ dx = +c 3 √ ∫ 1 2x+1 2 3 2 ∫0 x 1 − x dx = 15 x cos xdx = x sin x + cos x + c

www.ck12.org

242

6. 7. 8. 9. 10.

4.8

[ 3 ] √ 2 3 + 9dx = 2 10 2 − 27 x x 9 0 ) ∫(1 1 1 · e x dx = −e x + c x2 ∫ 3 x2 2 x e dx = 12 e√x (x2 − 1) + c ∫ lnx −4ln x − 43 + c 5 dx = 3 2 3x 2 9x 2 ∫ ex1 dx = 1 1 x ∫1

Numerical Integration

Learning Objectives • • • •

Use the Trapezoidal Rule to solve problems Estimate errors for the Trapezoidal Rule Use Simpson’s Rule to solve problems Estimate Errors for Simpson’s Rule

Introduction Recall that we used diﬀerent ways to approximate the value of integrals. These included Riemann Sums using left and right endpoints, as well as midpoints for ﬁnding the length of each rectangular tile. In this lesson we will learn two other methods for approximating integrals. The ﬁrst of these, the Trapezoidal Rule, uses areas of trapezoidal tiles to approximate the integral. The second method, Simpson’s Rule, uses parabolas to make the approximation. Trapezoidal Rule Let’s recall how we would use the midpoint rule with n = 4 rectangles to approximate the area under the graph of f (x) = x2 + 1 from x = 0 to x = 1.

243

www.ck12.org

If instead of using the midpoint value within each sub-interval to ﬁnd the length of the corresponding rectangle, we could have instead formed trapezoids by joining the maximum and minimum values of the function within each sub-interval:

The area of a trapezoid is A = h(b12+b2 ) , where b1 and b2 are the lengths of the parallel sides and h is the height. In our trapezoids the height is △x and b1 and b2 are the values of the function. Therefore in ﬁnding the areas of the trapezoids we actually average the left and right endpoints of each sub-interval. Therefore a typical trapezoid would have the area

A=

To approximate www.ck12.org

∫b a

△x (f (xi−1 ) + f (xi )) . 2

f (x)dx with n of these trapezoids, we have

244

∫ a

b

[ n ] n ∑ 1 ∑ f (x)dx ≈ f (xi−1 )△x + f (xi )△x 2 i−1 i−1 △x [f (x0 ) + f (x1 ) + f (x1 ) + f (x2 ) + f (x2 ) + . . . + f (xn−1 )f (xn )] 2 △x b−a [f (x0 ) + 2f (x1 ) + 2f (x2 ) + . . . + 2f (xn−1 )f (xn )] , △x = . = 2 n

=

Example 1: Use the Trapezoidal Rule to approximate

∫3 0

x2 dx with n = 6.

Solution: We ﬁnd △x = ∫

3

b−a n

=

3−0 6

= 12 .

] 1[ f (0) + 2f ( 12 ) + 2f (1) + 2f ( 23 ) + 2f (2) + 2f ( 52 ) + f (3) 4 ] 1[ 0 + (2 · 14 ) + (2 · 1) + (2 · 94 ) + (2 · 4) + (2 · 25 )+9 = 4 4 1 [ 73 ] 73 = = 9.125. = 4 2 8

x2 dx ≈ 0

Of course, this estimate is not nearly as accurate as we would like. For functions such as f (x) = x2 , we can easily ﬁnd an antiderivative with which we can apply the Fundamental ]3 ∫3 2 3 Theorem that 0 x dx = x3 = 9. But it is not always easy to ﬁnd an antiderivative. 0

Indeed, for many integrals it is impossible to ﬁnd an antiderivative. Another issue concerns the questions about the accuracy of the approximation. In particular, how large should we ∫3 take n so that the Trapezoidal Estimate for 0 x2 dx is accurate to within a given value, say 0.001? As with our Linear Approximations in the Lesson on Approximation Errors, we can state a method that ensures our approximation to be within a speciﬁed value. Error Estimates for Simpson’s Rule We would like to have conﬁdence in the approximations we make. Hence we can choose n to ensure that the errors are within acceptable boundaries. The following method illustrates how we can choose a suﬀiciently large n. Suppose |f ′′ (x)| ≤ k for a ≤ x ≤ b. Then the error estimate is given by

|ErrorT rapezoidal | ≤

245

k(b − a)3 . 12n2 www.ck12.org

Example 2: Find n so that the Trapezoidal Estimate for

∫3 0

x2 dx is accurate to 0.001.

Solution: We need to ﬁnd n such that |ErrorT rapezoidal | ≤ 0.001. We start by noting that |f ′′ (x)| = 2 for 0 ≤ x ≤ 3. Hence we can take K = 2 to ﬁnd our error bound.

2(3 − 0)3 54 |ErrorT rapezoidal | ≤ = . 2 12n 12n2

We need to solve the following inequality for n:

54 < 0.001, 12n2 54 n2 > , 12(0.001) √ 54 n> ≈ 67.08. 12(0.001)

Hence we must take n = 68 to achieve the desired accuracy. From the last example, we see one of the weaknesses of the Trapezoidal Rule—it is not very accurate for functions where straight line segments (and trapezoid tiles) do not lead to a good estimate of area. It is reasonable to think that other methods of approximating curves might be more applicable for some functions. Simpson’s Rule is a method that uses parabolas to approximate the curve. Simpson’s Rule: As was true with the Trapezoidal Rule, we divide the interval [a, b] into n sub-intervals of . We then construct parabolas through each group of three consecutive length △x = b−a n points on the graph. The graph below shows this process for the ﬁrst three such parabolas for the case of n = 6 sub-intervals. You can see that every interval except the ﬁrst and last contains two estimates, one too high and one too low, so the resulting estimate will be more accurate. www.ck12.org

246

Using parabolas in this way produces the following estimate of the area from Simpson’s Rule: ∫

△x [f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + 2f (x4 ) . . . + 2f (xn−2 ) + 4f (xn−1 ) + f (xn )] . 3

b

f (x)dx ≈ a

We note that it has a similar appearance to the Trapezoidal Rule. However, there is one distinction we need to note. The process of using three consecutive xi to approximate parabolas will require that we assume that n must always be an even number. Error Estimates for the Trapezoidal Rule As with the Trapezoidal Rule, we have a formula that suggests how we can choose n to ensure that the errors are within acceptable boundaries. The following method illustrates how we can choose a suﬀiciently large n. Suppose |f 4 (x)| ≤ k for a ≤ x ≤ b. Then the error estimate is given by

|Errorsimpson | ≤ Example 3: a. Use Simpson’s Rule to approximate

∫4

1 dx 1 x

k(b − a)5 . 180n4

with n = 6.

Solution: We ﬁnd △x = ∫ 1

4

b−a n

=

4−1 6

= 12 .

] 1[ 1 f (1) + 4f ( 32 ) + 2f (2) + 4f ( 52 ) + 2f (3) + 4f ( 27 ) + f (4) dx ≈ x 6 ] 1[ 1 + (4 · 23 ) + (2 · 21 ) + (4 · 25 ) + (2 · 31 ) + (4 · 27 ) + 41 = 6 1 [ 3517 ] = = 1.3956. 6 420

247

www.ck12.org

This turns out to be a pretty good estimate, since we know that ∫ 1

4

1 dx = ln x x

]4 = ln(4) − ln(1) = 1.3863. 1

Therefore the error is less than 0.01. b. Find n so that the Simpson Rule Estimate for

∫4

1 dx 1 x

is accurate to 0.001.

Solution:

We need to ﬁnd n such that |Errorsimpson | ≤ 0.001. We start by noting that |f 4 (x)| = x245 for 1 ≤ x ≤ 4. Hence we can take K = 24 to ﬁnd our error bound:

|Errorsimpson | ≤

24(4 − 1)5 5832 = . 4 180n 180n4

Hence we need to solve the following inequality for n:

5832 < 0.001. 180n4 We ﬁnd that

5832 , 180(0.001) √ 5832 n>4 ≈ 13.42. 180(0.001)

n4 >

Hence we must take n = 14 to achieve the desired accuracy. Technology Note: Estimating a Deﬁnite Integral with a TI-83/84 Calculator ∫4 We will estimate the value of 1 x1 dx . 1. 2. 3. 4.

Graph the function f (x) = x1 with the [WINDOW] setting shown below. The graph is shown in the second screen. Press 2nd [CALC] and choose option 7 (see menu below) When the fourth screen appears, press [1] [ENTER] then [4] [ENTER] to enter the lower and upper limits.

www.ck12.org

248

5. The ﬁnal screen gives the estimate, which is accurate to 7 decimal places.

Lesson Summary 1. 2. 3. 4.

We We We We

used the Trapezoidal Rule to solve problems. estimated errors for the Trapezoidal Rule. used Simpson’s Rule to solve problems. estimated Errors for Simpson’s Rule.

Review Question 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

∫1 Use the Trapezoidal Rule to approximate 0 x2 e−x dx with n = 8. ∫4 √ Use the Trapezoidal Rule to approximate 1 ln xdx with n = 6. ∫1√ Use the Trapezoidal Rule to approximate 0 1 + x4 dx with n = 4. ∫3 Use the Trapezoidal Rule to approximate 1 x1 dx with n = 8. ∫3 How large should you take n so that the Trapezoidal Estimate for 1 x1 dx is accurate to within 0.001? ∫1 Use Simpson’s Rule to approximate 0 x2 e−x dx with n = 8. ∫4√ Use Simpson’s Rule to approximate 1 x ln xdx with n = 6. ∫2 Use Simpson’s Rule to approximate 0 √x14 +1 dx with n = 6. ∫1√ Use Simpson’s Rule to approximate 0 1 + x4 dx with n = 4. ∫2 How large should you take n so that the Simpson Estimate for 0 edx is accurate to within 0.00001?

249

www.ck12.org

Review Answers ∫ 1 2 −x x e dx ≈ 0.16 ∫04 √ ln xdx ≈ 1.26 ∫11 √ 1 + x4 dx ≈ 1.10 ∫03 1 dx ≈ 1.10 1 x Take ∫ 1 2 n−x= 19 x e dx ≈ 0.16 ∫04 √ xln xdx ≈ 4.28 ∫12 1 √ dx ≈ 1.36 4 ∫01 √x +1 9. 0 1 + x4 dx ≈ 1.09 10. Take n = 9 1. 2. 3. 4. 5. 6. 7. 8.

www.ck12.org

250

Chapter 5 Applications of Deﬁnite Integrals In this chapter, we will explore some of the many applications of the deﬁnite integral by using it to calculate areas between two curves, volumes, length of curves, and several other applications from real life such as calculating the work done by a force, the pressure a liquid exerts on an object, and basic statistical concepts.

5.1

Area Between Two Curves

Learning Objectives A student will be able to:

• Compute the area between two curves with respect to the x− and y−axes.

In the last chapter, we introduced the deﬁnite integral to ﬁnd the area between a curve and the x− axis over an interval [a, b]. In this lesson, we will show how to calculate the area between two curves. Consider the region bounded by the graphs f and g between x = a and x = b, as shown in the ﬁgures below. If the two graphs lie above the x−axis, we can interpret the area that is sandwiched between them as the area under the graph of g subtracted from the area under the graph f. Therefore, as the graphs show, it makes sense to say that [Area under f (Fig. 1a)] − [Area under g (Fig. 1b)] = [Area between f and g (Fig. 1c)],

251

www.ck12.org

Figure 5.1: (32)

Figure 5.2: (35)

www.ck12.org

252

Figure 5.3: (3)

∫

∫

b

f (x)dx − a

∫

b

a

b

[f (x)| − g(x)]dx.

g(x) = a

This relation is valid as long as the two functions are continuous and the upper function f (x) ≥ g(x) on the interval [a, b]. The Area Between Two Curves (With respect to the x−axis) If f and g are two continuous functions on the interval [a, b] and f (x) ≥ g(x) for all values of x in the interval, then the area of the region that is bounded by the two functions is given by ∫

b

[f (x) − g(x)]dx.

A= a

Example 1: Find the area of the region enclosed between y = x2 and y = x + 6. Solution: We ﬁrst make a sketch of the region (Figure 2) and ﬁnd the end points of the region. To do so, we simply equate the two functions,

x2 = x + 6,

253

www.ck12.org

Figure 5.4: (29) and then solve for x. x2 − x − 6 = 0 (x + 2)(x − 3) = 0 from which we get x = −2 and x = 3. So the upper and lower boundaries intersect at points (−2, 4) and (3, 9). As you can see from the graph, x + 6 ≥ x2 and hence f (x) = x + 6 and g(x) = x2 in the interval [−2, 3]. Applying the area formula, ∫

b

[f (x) − g(x)]dx

A= a

∫

3

= −2

[(x + 6) − (x2 )]dx.

Integrating, [

x2 x3 + 6x − A= 2 3 125 . = 6

]3 −2

So the area between the two curves f (x) = x + 6 and g(x) = x2 is 125/6. Sometimes it is possible to apply the area formula with respect to the y−coordinates instead of the x−coordinates. In this case, the equations of the boundaries will be written in such a way that y is expressed explicitly as a function of x (Figure 3). www.ck12.org

254

Figure 5.5: (5) The Area Between Two Curves (With respect to the y−axis) If w and v are two continuous functions on the interval [c, d] and w(y) ≥ v(y) for all values of y in the interval, then the area of the region that is bounded by x = v(y) on the left, x = w(y) on the right, below by y = c, and above by y = d, is given by ∫

d

[w(y) − v(y)]dy.

A= c

Example 2: Find the area of the region enclosed by x = y 2 and y = x − 6. Solution:

Figure 5.6: (14) As you can see from Figure 4, the left boundary is x = y 2 and the right boundary is y = x−6. The region extends over the interval −2 ≤ y ≤ 3. However, we must express the equations in terms of y. We rewrite

255

www.ck12.org

x = y2 x=y+6 Thus ∫

3

A= [

−2 2

[y + 6 − y 2 ]dy

y y3 = + 6y − 2 3 125 = . 6

]3 −2

Review Questions In problems #1 - 7, sketch the region enclosed by the curves and ﬁnd the area. √ y = x2 , y = x, on the interval [0.25, [ π π1]] y = 0, y = cos 2x, on the interval 4 , 2 y = | − 1 + x| + 2, y = −1 x+7 5 y = cos x, y = sin x, x = 0, x = 2π x = y 2 , y = x − 2, integrate with respect to y y 2 − 4x = 4, 4x − y = 16 y = 8 cos x, y = sec2 x, −π/3 ≤ x ≤ π/3 Find the area enclosed by x = y 3 and x = y. If the area enclosed by the two functions y = k cos x and y = kx2 is 2, what is the value of k? 10. Find the horizontal line y = k that divides the region between y = x2 and y = 9 into two equal areas. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Review Answers 1. 2. 3. 4. 5. 6.

49/192 1/2 24√ 4 2 9/2 30 38

www.ck12.org

256

√ 7. 6 3 8. 12 9. k ≈ √ 1.83 10. 9/ 3 4

5.2

Volumes

Learning Objectives • • • •

Learn Learn Learn Learn

the basic concepts of volume and how to compute it with a given cross-section how to compute volume by the disk method how to compute volume by the washer method how to compute volume by cylindrical shells

In this section, we will use deﬁnite integrals to ﬁnd volumes of diﬀerent solids.

The Volume Formula A circular cylinder can be generated by translating a circular disk along a line that is perpendicular to the disk (Figure 5). In other words, the cylinder can be generated by moving the cross-sectional area A (the disk) through a distance h. The resulting volume is called the volume of solid and it is deﬁned to be

V = Ah. The volume of solid does not necessarily have to be circular. It can take any arbitrary shape. One useful way to ﬁnd the volume is by a technique called “slicing.” To explain the idea, suppose a solid S is positioned on the x−axis and extends from points x = a to x = b (Figure 6). Let A(x) be the cross-sectional area of the solid at some arbitrary point x. Just like we did in calculating the deﬁnite integral in the previous chapter, divide the interval [a, b] into n sub-intervals and with widths

△x1 , △x2 , △x3 , . . . , △xn . Eventually, we get planes that cut the solid into n slices

S1 , S 2 , S 3 , . . . , S n .

257

www.ck12.org

Figure 5.7: (15)

Figure 5.8: (24)

www.ck12.org

258

Take one slice, Sk . We can approximate slice Sk to be a rectangular solid with thickness △xk and cross-sectional area A(xk ). Thus the volume Vk of the slice is approximately

Vk ≈ A(xk )△xk . Therefore the volume V of the entire solid is approximately

V = V1 + V2 + . . . + Vn n ∑ ≈ A(xk )△xk . k=1

If we use the same argument to derive a formula to calculate the area under the curve, let us increase the number of slices in such a way that △xk → 0. In this case, the slices become thinner and thinner and, as a result, our approximation will get better and better. That is,

V = lim = △x→0

n ∑

A(xk )△xk .

k=1

Notice that the right-hand side is just the deﬁnition of the deﬁnite integral. Thus

V = lim = △x→0

∫ =

n ∑

A(xk )△xk

k=1

b

A(x)dx. a

The Volume Formula (Cross-section perpendicular to the x−axis) Let S be a solid bounded by two parallel planes perpendicular to the x−axis at x = a and x = b. If each of the cross-sectional areas in [a, b] are perpendicular to the x− axis, then the volume of the solid is given by ∫ V =

b

A(x)dx. a

where A(x) is the area of a cross section at the value of x on the x−axis.

259

www.ck12.org

The Volume Formula (Cross-section perpendicular to the y−axis) Let S be a solid bounded by two parallel planes perpendicular to the y−axis at y = c and y = d. If each of the cross-sectional areas in [c, d] are perpendicular to the y−axis, then the volume of the solid is given by ∫ V =

d

A(y)dy. c

where A(y) is the area of a cross section at the value of y on the y−axis. Example 1: Derive a formula for the volume of a pyramid whose base is a square of sides a and whose height (altitude) is h.

Figure 5.9: (10) Solution: Let the y−axis pass through the apex of the pyramid, as shown in Figure (7a). At any point y in the interval [0, h], the cross-sectional area is a square. If b is the length of the sides of any arbitrary square, then, by similar triangles (Figure 7b), h−y , h a b = (h − y). h

1 b 2 1 a 2

www.ck12.org

=

260

Figure 5.10: (2) Since the cross-sectional area at y is A(y) = b2 ,

A(y) = b2 =

a2 (h − y)2 . h2

Using the volume formula, ∫

d

V =

A(y)dy c

∫

h

a2 (h − y)2 dy 2 0 h ∫ a2 h = 2 (h − y)2 dy. h 0 =

Using u−substitution to integrate, we eventually get [ ]h a2 1 3 V = 2 − (h − y) h 3 0 1 2 = a h. 3 Therefore the volume of the pyramid is V = 13 a2 h, which agrees with the standard formula.

Volumes of Solids of Revolution The Method of Disks

261

www.ck12.org

Figure 5.11: (22) Figure 8a

Figure 5.12: (28) Suppose a function f is continuous and non-negative on the interval [a, b], and suppose that R is the region between the curve f and the x−axis (Figure 8a). If this region is revolved about the x−axis, it will generate a solid that will have circular cross-sections (Figure 8b) with radii of f (x) at each x. Each cross-sectional area can be calculated by

A(x) = π[f (x)]2 . Since the volume is deﬁned as ∫ V =

b

A(x)dx, a

www.ck12.org

262

the volume of the solid is ∫

b

π[f (x)]2 dx.

V = a

Volumes by the Method of Disks (revolution about the x−axis) ∫

b

π[f (x)]2 dx.

V = a

Because the shapes of the cross-sections are circular or look like the shapes of disks, the application of this method is commonly known as the method of disks. Example 2 Calculate the volume of the solid that is obtained when the region under the curve revolved about the x−axis over the interval [1, 7].

√

x is

Solution: As Figures 9a and 9b show, the volume is ∫

b

π[f (x)]2 dx

V = ∫a 7

√ 2 π[ x] dx 1 [ 2 ]7 x =π 2 1 = 24π. =

Example 3: Derive a formula for the volume of the sphere with radius r. Solution: One way to ﬁnd the formula is to use the disk method. From your algebra, a circle of radius r and center at the origin is given by the formula

x2 + y 2 = r 2

263

www.ck12.org

Figure 5.13: (27)

Figure 5.14: (11)

www.ck12.org

264

If we revolve the circle about the x−axis, we will get a sphere. Using the disk method, we will obtain a formula for the volume. From the equation of the circle above, we solve for y:

f (x) = y =

√

r 2 − x2 ,

thus ∫

b

π[f (x)]2 dx

V =

∫a +r [√ ]2 2 2 = π r − x dx −r [ ]r x3 2 =π r x− 3 −r 4 = πr3 . 3 This is the standard formula for the volume of the sphere. The Method of Washers To generalize our results, if f and g are non-negative and continuous functions and

f (x) ≥ g(x) for

a ≤ x ≤ b, Then let R be the region enclosed by the two graphs and bounded by x = a and x = b. When this region is revolved about the x−axis, it will generate washer-like cross-sections (Figures 10a and 10b). In this case, we will have two radii: an inner radius g(x) and an outer radius f (x). The volume can be given by: ∫ V (x) =

b

( ) π [f (x)]2 − [g(x)]2 dx.

a

265

www.ck12.org

Figure 5.15: (16)

Figure 5.16: (4)

www.ck12.org

266

Volumes by the Method of Washers (revolution about the x−axis)

∫ V (x) =

b

( ) π [f (x)]2 − [g(x)]2 dx.

a

Example 4: Find the volume generated when the region between the graphs f (x) = x2 + 1 and g(x) = x over the interval [0, 3] is revolved about the x−axis. Solution:

Figure 5.17: (36)

From the formula above,

267

www.ck12.org

Figure 5.18: (17)

∫

b

V (x) = ∫a 3 = ∫0 3 =

( ) π [f (x)]2 − [g(x)]2 dx ( ) π (x2 + 1)2 − (x)2 dx ( ) π x4 + x2 + 1 dx

0

303π . = 5 The methods of disks and washers can also be used if the region is revolved about the y−axis. The analogous formulas can be easily deduced from the above formulas or from the volumes of solids generated. Disks: ∫

d

π[u(y)]2 dy.

V = c

Washers: www.ck12.org

268

∫

d

V =

( ) π [w(y)]2 − [v(y)]2 dy.

c

Example 5: What is the volume of the solid generated when the region enclosed by y = x = 0 is revolved about the y−axis?

√

x, y = 3, and

Solution: Since the solid generated is revolved about the y−axis (Figure 12), we must rewrite y = as x = y 2 .

√

x

Thus u(y) = y 2 . The volume is ∫

d

π[u(y)]2 dy

V = c

∫

3

π[y 2 ]2 dy

= ∫0 3

πy 4 dx 0 [ 5 ]3 y =π 5 [ 5 0 ] 3 =π −0 5 243π = . 5 =

Multimedia Link The following applet allows you to try out solids of revolution about the x-axis for any two functions. You can try inputting the examples above to test it out, and then experiment with new functions and changing the bounds. Volumes of Revolution Applet. In the following video the narrator walks trough the steps of setting up a volume integration. Khan Academy Solids of Revolution (10:04) .

Volume By Cylindrical Shells The method of computing volumes so far depended upon computing the cross-sectional area of the solid and then integrating it across the solid. What happens when the cross-sectional area cannot be found or the integration is too diﬀicult to solve? Here is where the shell method comes along.

269

www.ck12.org

Figure 5.19: (1)

Figure 5.20: (13)

www.ck12.org

270

Figure 5.21: Figuring out the volume of a function rotated about the x-axis. http://www. youtube.com/watch?v=R_aqSL-q6_8&feature=youtube_gdata To show how diﬀicult it sometimes is to use the disk or the washer methods to compute volumes, consider the region enclosed by the function f (x) = x − x2 . Let us revolve it about the line x = −1 to generate the shape of a doughnut-shaped cake. What is the volume of this solid?

Figure 5.22: (23) If we wish to integrate with respect to the y−axis, we have to solve for x in terms of y. That would not be easy (try it!). An easier way is to integrate with respect to the x−axis by using the shell method. Here is how: A cylindrical shell is a solid enclosed by two concentric

271

www.ck12.org

Figure 5.23: (33)

www.ck12.org

272

cylinders. If the inner radius is r1 and the outer one is r2 , with both of height h, then the volume is (Figure 14)

V = [area of the cross-section] · [height] = π(r22 − r12 )h = π(r2 + r1 )(r2 − r1 )h [ ] 1 = 2π · (r2 + r1 ) · h · (r2 − r1 ). 2

Figure 5.24: (8) Notice however that (r2 − r1 ) is the thickness of the shell and 12 (r2 + r1 ) is the average radius of the shell. Thus

V = 2π · [average radius] · [height] · [thickness]. Replacing the average radius with a single variable r and using h for the height, we have

V = 2π · r · h · [thickness].

273

www.ck12.org

In general the shell’s thickness will be dx or dy depending on the axis of revolution. This discussion leads to the following formulas for rotation about an axis. We will then use this formula to compute the volume V of the solid of revolution that is generated by revolving the region about the x−axis. Volume By Cylindrical Shell about the y−Axis Suppose f is a continuous function in the interval [a, b] and the region R is bounded above by y = f (x) and below by the x−axis, and on the sides by the lines x = a and x = b. If R is rotated around the y−axis, then the cylinders are vertical, with r = x and h = f (x). The volume of the solid is given by

∫ V =

∫

b

a

a

2πxf (x)dx.

2πrhdx = b

Volume By Cylindrical Shell about the x−Axis Equivalently, if the volume is generated by revolving the same region about the x−axis, then the cylinders are horizontal with

∫ v=

d

2πrhdy, c

where c = f −1 (a) and d = f −1 (b). The values of r and h are determined in context, as you will see in Example 6. Note: Example 7 shows what to do when the rotation is not about an axis. Example 6: A solid ﬁgure is created by rotating the region R (Figure 15) around the x−axis. R is bounded by the curve y = x2 and the lines x = 0 and x = 2. Use the shell method to compute the volume of the solid. Solution: From Figure 15 we can identify the limits of integration: y runs from 0 to 4. A horizontal √ strip of this region would generate a cylinder with height 2 − y and radius y. Thus the volume of the solid will be www.ck12.org

274

Figure 5.25: (25)

∫

d

V =

2πrhdy ∫

c

√ 2πy(2 − y)dy 0 ∫ 4 = 2π (2y − y 3/2 )dy 0 ]4 [ 2 5/2 2 = 2π y − y 5 0 32π = . 5 4

=

Note: The alert reader will have noticed that this example could be worked with a simpler integral using disks. However, the following example can only be solved with shells. Example 7: Find the volume of the solid generated by revolving the region bounded by y = x3 + 12 x + 14 , y = 14 , and x = 1, about x = 3. Solution: As you can see, the equation y = x3 + 12 x +

1 4

cannot be easily solved for x and therefore

275

www.ck12.org

Figure 5.26: (34) it will be necessary to solve the problem by the shell method. We are revolving the region about a line parallel to the y−axis and thus integrate with respect to x. Our formula is ∫

b

V =

2πrhdx. a

In this case, the radius is 3 − x and the height is x3 + 12 x + 14 − 14 . Substituting, ) 1 1 1 V = 2π (3 − x) x + x + − dx 2 4 4 0 ) ∫ 1( 1 2 3 4 3 = 2π −x + 3x − x + x dx 2 2 0 [ ]1 −1 5 3 4 1 3 3 2 = 2π x + x − x + x 5 4 6 4 0 [ ] −1 3 1 3 + − + = 2π 5 4 6 4 [ ] 17 = 2π 15 34π . = 15 ∫

1

(

3

Multimedia Links Sometimes the same volume problem can be solved in two diﬀerent ways. In these two videos, the narrator ﬁrst ﬁnds a volume using shells Khan Academy www.ck12.org

276

Solid of Revolution (Part 5) (9:29) , and then he does the same volume problem using disks.

Figure 5.27: Using the disk method around the y-axis. http://www.youtube.com/watch? v=F2psxMnGdUw&feature=youtube_gdata Khan Academy Solid of Revolution (Part 6) (9:18) . Together these videos show how both

Figure 5.28: Using the disk method around the y-axis. http://www.youtube.com/watch? v=F2psxMnGdUw&feature=youtube_gdata methods can be used to solve the same problem (though it’s not always done this easily!).

Review Questions In problems #1 - 4, ﬁnd the volume of the solid generated by revolving the region bounded by the curves about the x−axis. 1. 2. 3. 4.

y y y y

√ = 9 − x2 , y = 0 = 3 + x, y =√ 1 + x2 = sec x, y = 2, −π/4 ≤ x ≤ π/4 = 1, y = x, x = 0

277

www.ck12.org

In problems #5–8, ﬁnd the volume of the solid generated by revolving the region bounded by the curves about the y−axis. 5. 6. 7. 8.

y = x3 , x = 0, y = 1 x = y2, y = x − 2 x = csc y, y√= π/4, y = 3π/4, x = 0 y = 0, y = x, x = 4

In problems #9–12, use cylindrical shells to ﬁnd the volume generated when the region bounded by the curves is revolved about the axis indicated. 9. 10. 11. 12. 13.

y = x1 , y = 0, x = 1, x = 3, about the y−axis y = x2 , x = 1, y = 0, about the x−axis y = 2x − 1, y = −2x + 3, x = 2, about the y−axis y 2 = x, y = 1, x = 0, about the x−axis. Use the cylindrical shells method to ﬁnd the volume generated when the region is bounded by y = x3 , y = 1, x = 0 is revolved about the line y = 1.

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

5.3

36π 117π 5 2

π − 2π 2 π 3 3π 5 72π 5

2π 128π 5

4π π 5

20π/3 π 2 9π 14

The Length of a Plane Curve

Learning Objectives A student will be able to: www.ck12.org

278

• Learn how to ﬁnd the length of a plane curve for a given function. In this section will consider the problem of ﬁnding the length of a plane curve. Formulas for ﬁnding the arcs of circles appeared in early historical records and they were known to many civilizations. However, very little was known about ﬁnding the lengths of general curves, such as the length of the curve y = x2 in the interval [0, 2], until the discovery of calculus in the seventeenth century. In calculus, we deﬁne an arc length as the length of a plane curve y = f (x) over an interval [a, b] (Figure 17). When the curve f (x) has a continuous ﬁrst derivative f ′ on [a, b], we say that f is a smooth function (or smooth curve) on [a, b].

Figure 5.29: (37) The Arc Length Problem If y = f (x) is a smooth curve on the interval [a, b], then the arc length L of this curve is deﬁned as ∫ b√ ∫ L= 1 + [f ′ (x)]2 dx = a

a

b

√ 1+

(

dy dx

)2 dx.

Example 1: Find the arc length of the curve y = x3 /2 on [1, 3] (Figure 18). Solution: Since y = x3 /2,

279

www.ck12.org

Figure 5.30: (21)

dy 3 = x1/2 . dx 2 Using the formula above, we get ∫ b√ ∫ 1 + [f ′ (x)]2 dx = a

3

1

∫ =

1

3

√ √

[

]2 3 1/2 1+ x dx 2 9 1 + xdx. 4

Using u−substitution by letting u = 1 + 94 x, then du = 49 dx. Substituting, and remembering to change the limits of integration, ∫ 4 31/4 √ L= udu 9 13/4 8 [ 3/2 ]31/4 = u 13/4 27 ≈ 4.65. Multimedia Link The formula you just used to ﬁnd the length of a curve was derived by using line segments to approximate the curve. The derivation of that formula can be found www.ck12.org

280

at Wikipedia Entry on Arc Length. In the following applet you can explore this further. Experiment with various curves and change the number of segments to see how changing the number of segments is related to approximating the arc length. Arc Length Applet.

Review Questions 1. Find the arc length of the curve y=

(x2 + 2)3/2 3

on [0, 3]. 2. Find the arc length of the curve 1 1 x = y3 + 6 2y on y ∈ [1, 2]. 3. Integrate ∫ x=

y

√

sec4 t − 1dt, −

0

π π ≤y≤ . 4 4

4. Find the length of the curve shown in the ﬁgure below. The shape of the graph is called the astroid because it looks like a star. The equation of its graph is x2/3 + y 2/3 = 1.

281

www.ck12.org

5. The ﬁgure below shows a suspension bridge. The cable has the shape of a parabola with equation kx2 = y. The suspension bridge has a total length of 2S and the height of the cable is h at each end. Show that the total length of the cable is ∫ s√ 4h2 L=2 1 + 4 x2 dx. S 0

Review Answers 1. 2. 3. 4. 5.

5.4

11 (17/12) 2 6 .

Area of a Surface of Revolution

Learning Objectives A student will be able to: • Learn how to ﬁnd the area of a surface that is generated by revolving a curve about an axis or a line. In this section we will deal with the problem of ﬁnding the area of a surface that is generated by revolving a curve about an axis or a line. For example, the surface of a sphere can be generated by revolving a semicircle about its diameter (Figure 19) and the circular cylinder www.ck12.org

282

Figure 5.31: (18)

Figure 5.32: (31)

283

www.ck12.org

can be generated by revolving a line segment about any axis that is parallel to it (Figure 20). Area of a Surface of Revolution If f is a smooth and non-negative function in the interval [a, b], then the surface area S generated by revolving the curve y = f (x) between x = a and x = b about the x−axis is deﬁned by

∫

b

S=

∫ b √ ′ 2 2πf (x) 1 + [f (x)] dx = 2πy

a

√

(

1+

a

dy dx

)2 dx.

Equivalently, if the surface is generated by revolving the curve about the y−axis between y = c and y = d, then

∫

d

S=

∫ √ ′ 2 2πg(y) 1 + [g (y)] dy =

c

√

d

2πx c

( 1+

dx dy

)2 dy.

Example 1: Find the surface area that is generated by revolving y = x3 on [0, 2] about the x−axis (Figure 21). Solution: The surface area S is

∫

√

b

S=

2πy

1+

a

∫

(

dy dx

)2 dx

√ 2πx3 1 + (3x2 )2 dx 0 ∫ 2 = 2π x3 (1 + 9x4 )1/2 dx. 2

=

0

Using u−substitution by letting u = 1 + 9x4 , www.ck12.org

284

Figure 5.33: (12)

285

www.ck12.org

∫

145

S = 2π 1

du u1/2 36 ]145

[ 2π 2 3/2 = u 36 3 1 ] 2π 2 [ = · (145)3/2 − 1 36 3 4π ≈ [1745] 108 ≈ 203 Example 2: 2 Find √ the area of the surface generated by revolving the graph of f (x) = x on the interval [0, 3] about the y−axis (Figure 22).

Solution:

Figure 5.34: (9) Since the curve is revolved about the y−axis, we apply

∫

√

d

S=

2πx

( 1+

c

www.ck12.org

286

dx dy

)2 dy.

So we write y = x2 as x = Thus

√

y. In addition, the interval on the x−axis [0,

∫

3

S=

√

√

2π y

( 1+

0

1 √ 2 y

√

3] becomes [0, 3].

)2 dy.

Simplifying, ∫

3

S=π

√ 4y + 1dy.

0

With the aid of u−substitution, let u = 4y + 1, ∫ π 13 1/2 u du S= 4 1 ] π[ = (13)3/2 − 1 6 π = [46.88 − 1] 6 ≈ 24

Review Questions In problems #1 - 3 ﬁnd the area of the surface generated by revolving the curve about the x−axis. 1. y = 3x, √ 0≤x≤1 2. y = √x, 1 ≤ x ≤ 9 3. y = 4 − x2 , −1 ≤ x ≤ 1 In problems #4–6 ﬁnd the area of the surface generated by revolving the curve about the y−axis. 4. 5. 6. 7. 8.

x = 7y + 2, 0 ≤ y ≤ 3 x = y√3 , 0 ≤ y ≤ 8 x = 9 − y 2 , −2 ≤ y ≤ 2 Show that the surface area of a sphere of radius r is 4πr2 . Show that the lateral area S of a right circular cone of height h and base radius r is √ S = πr r2 + h2 .

287

www.ck12.org

Review Answers 1. 2. 3. 4. 5. 6. 7. 8.

5.5

√ 3π 10 ≈ 112 8π √ 75π 50 ≈ 823165.5 24π . .

Applications from Physics, Engineering, and Statistics

Learning Objectives A student will be able to: • Learn how to apply deﬁnite integrals to several applications from physics, engineering, and applied mathematics such as work, ﬂuids statics, and probability. In this section we will show how the deﬁnite integral can be used in diﬀerent applications. Some of the concepts may sound new to the reader, but we will explain what you need to comprehend as we go along. We will take three applications: The concepts, of work from physics, ﬂuid statics from engineering, and the normal probability from statistics.

Work Work in physics is deﬁned as the product of the force and displacement. Force and displacement are vector quantities, which means they have a direction and a magnitude. For example, we say the compressor exerts a force of 200 Newtons (N ) upward. The magnitude here is 200 N and the direction is upward. Lowering a book from an upper shelf to a lower one by a distance of 0.5 meters away from its initial position is another example of the vector nature of the displacement. Here, the magnitude is 0.5 m and the direction is downward, usually indicated by a minus sign, i.e., a displacement of −0.5 m. The product of those two vector quantities (called the inner product, see Chapter 10) gives the work done by the force. Mathematically, we say

W = F d, www.ck12.org

288

where F is the force and d is the displacement. If the force is measured in Newtons and distance is in meters, then work is measured in the units of energy which is in joules (J). Example 1: You push an empty grocery cart with a force of 44 N for a distance of 12 meters. How much work is done by you (the force)? Solution: Using the formula above,

W = Fd = (44)(12) = 528 J. Example 2: A librarian displaces a book from an upper shelf to a lower one. If the vertical distance between the two shelves is 0.5 meters and the weight of the book is 5 Newtons . How much work is done by the librarian? Solution: In order to be able to lift the book and move it to its new position, the librarian must exert a force that is at least equal to the weight of the book. In addition, since the displacement is a vector quantity, then the direction must be taken into account. So,

d = −0.5 meters. Thus

W = Fd = (5)(−0.5) = −2.5 J. Here we say that the work is negative since there is a loss of gravitational potential energy rather than a gain in energy. If the book is lifted to a higher shelf, then the work is positive, since there will be a gain in the gravitational potential energy. Example 3:

289

www.ck12.org

A bucket has an empty weight of 23 N. It is ﬁlled with sand of weight 80 N and attached to a rope of weight 5.1 N/m. Then it is lifted from the ﬂoor at a constant rate to a height 32 meters above the ﬂoor. While in ﬂight, the bucket leaks sand grains at a constant rate, and by the time it reaches the top no sand is left in the bucket. Find the work done:

1. 2. 3. 4.

by by by by

lifting the lifting the lifting the the lifting

empty bucket; sand alone; rope alone; the bucket, the sand, and the rope together.

Solution: 1. The empty bucket. Since the bucket’s weight is constant, the worker must exert a force that is equal to the weight of the empty bucket. Thus

W = Fd = (23)(+32) = 736 J.

2. The sand alone. The weight of the sand is decreasing at a constant rate from 80 N to 0 N over the 32 − meter lift. When the bucket is at x meters above the ﬂoor, the sand weighs

F (x) = [original weight of sand][proportion left at elevation x] ( ) 32 − x = 80 32 ( x) = 80 1 − 32 = 80 − 2.5x N. The graph of F (x) = 80 − 2.5x represents the variation of the force with height x (Figure 23). The work done corresponds to computing the area under the force graph. Thus the work done is www.ck12.org

290

Figure 5.35: (20)

∫ W =

b

F (x)dx a

∫

32

[80 − 2.5x]dx ]32 [ 2.5 2 x = 80x − 2 0 = 1280 J. =

0

3. The rope alone. Since the weight of the rope is 5.1 N/m and the height is 32 meters , the total weight of the rope from the ﬂoor to a height of 32 meters is

(5.1)(32) = 163.2 N.

But since the worker is constantly pulling the rope, the rope’s length is decreasing at a constant rate and thus its weight is also decreasing as the bucket being lifted. So at x meters, the (32 − x) meters there remain to be lifted of weight F (x) = (5.1)(32 − x) N. Thus the work done to lift the weight of the rope is

291

www.ck12.org

∫

∫

32

35

F (x)dx = (5.1)(32 − x)dx 0 [ ]32 x2 W = (5.1) 32x − 2 0 = 2611.2 J.

W =

0

4. The bucket, the sand, and the rope together. Here we are asked to sum all the work done on the empty bucket, the sand, and the rope. Thus

Wtotal = 736 + 1280 + 2611.2 = 4627.2 J.

Fluid Statics: Pressure You have probably studied that pressure is deﬁned as the force per area

P =

F , A

which has the units of Pascals (Pa) or Newtons per meter squared, Pa = N/m2 . In the study of ﬂuids, such as water pressure on a dam or water pressure in the ocean at a depth h, another equivalent formula can be used. It is called the liquid pressure P at depth h:

P = wh. where w is the weight density, which is the weight of the column of water per unit volume. For example, if you are diving in a pool, the pressure of the water on your body can be measured by calculating the total weight that the column of water is exerting on you times your depth. Another way to express this formula, the weight density w, is deﬁned as

w = ρg, where ρ is the density of the ﬂuid and g is the acceleration due to gravity (which is g = 9.8 m/sec2 on Earth). The pressure then can be written as www.ck12.org

292

P = wh = ρgh. Example 4: What is the total pressure experienced by a diver in a swimming pool at a depth of 2 meters ? Solution First we calculate the ﬂuid pressure the water exerts on the diver at a depth of 2 meters :

P = ρgh. The density of water is ρ = 1000 kg/m3 , thus

P = (1000)(9.8)(2) = 19600 Pa. The total pressure on the diver is the pressure due to the water plus the atmospheric pressure. If we assume that the diver is located at sea-level, then the atmospheric pressure at sea level is about 105 Pa . Thus the total pressure on the diver is

Ptotal = Pwater + Patm = 19600 + 105 = 119600 = 1.196 × 105 Pa. Example 5: What is the ﬂuid pressure (excluding the air pressure) and force on the top of a ﬂat circular plate of radius 3 meters that is submerged horizontally at a depth of 10 meters ? Solution: The density of water is ρ = 1000 kg/m3 . Then

P = ρgh = (1000)(9.8)(10) = 98000 Pa.

293

www.ck12.org

Since the force is F = P A, then

F = PA = P · πr2 = (98000)(π)(3)2 = 2.77 × 106 N. As you can see, it is easy to calculate the ﬂuid force on a horizontal surface because each point on the surface is at the same depth. The problem becomes a little complicated when we want to calculate the ﬂuid force or pressure if the surface is vertical. In this situation, the pressure is not constant at every point because the depth is not constant at each point. To ﬁnd the ﬂuid force or pressure on a vertical surface we must use calculus. The Fluid Force on a Vertical Surface Suppose a ﬂat surface is submerged vertically in a ﬂuid of weight density w and the submerged portion of the surface extends from x = a to x = b along the vertical x−axis, whose positive direction is taken as downward. If L(x) is the width of the surface and h(x) is the depth of point x, then the ﬂuid force F is deﬁned as ∫

b

F =

wh(x)L(x)dx. a

Example 6: A perfect example of a vertical surface is the face of a dam. We can picture it as a rectangle of a certain height and certain width. Let the height of the dam be 100 meters and of width of 300 meters. Find the total ﬂuid force exerted on the face if the top of the dam is level with the water surface (Figure 24). Solution: Let x = the depth of the water. At an arbitrary point x on the dam, the width of the dam is L(x) = 300 m and the depth is h(x) = xm . The weight density of water is

wwater = ρg = (1000)(9.8) = 9800 N/m2 . Using the ﬂuid force formula above, www.ck12.org

294

Figure 5.36: (7)

∫ F =

b

wh(x)L(x)dx a

∫

100

=

(9800)(x)(300)dx ∫ 100 6 = 2.94 × 10 xdx 0 [ 2 ]100 6 x = 2.94 × 10 2 0 10 = 1.47 × 10 N. 0

Normal Probabilities If you were told by the postal service that you will receive the package that you have been waiting for sometime tomorrow, what is the probability that you will receive it sometime between 3:00 PM and 5:00 PM if you know that the postal service’s hours of operations are between 7:00 AM to 6:00 PM? If the hours of operations are between 7 AM to 6 PM, this means they operate for a total of 11 hours. The interval between 3 PM and 5 PM is 2 hours, and thus the probability that

295

www.ck12.org

your package will arrive is

2 hours = 0.182 11 hours = 18.2%

P =

So there is a probability of 18.2% that the postal service will deliver your package sometime between the hours of 3 PM and 5 PM (or during any 2 − hour interval). That is easy enough. However, mathematically, the situation is not that simple. The 11 − hour interval and the 2 − hour interval contain an inﬁnite number of times. So how can one inﬁnity over another inﬁnity produce a probability of 18.2%? To resolve this issue, we represent the total probability of the 11 − hour interval as a rectangle of area 1 (Figure 25). Looking at the 2 2 − hour interval, we can see that it is equal to 11 of the total rectangular area 1. This is why it is convenient to represent probabilities as areas. But since areas can be deﬁned by deﬁnite integrals, we can also deﬁne the probability associated with an interval [a, b] by the deﬁnite integral

Figure 5.37: (19)

∫ P =

b

f (x)dx, a

where f (x) is called the probability density function (pdf). One of the most useful probability density functions is the normal curve or the Gaussian curve (and sometimes the bell curve) (Figure 26). This function enables us to describe an entire population based on statistical measurements taken from a small sample of the population. The only www.ck12.org

296

Figure 5.38: (38) measurements needed are the mean (µ) and the standard deviation (σ). Once those two numbers are known, we can easily ﬁnd the normal curve by using the following formula. The Normal Probability Density Function The Gaussian curve for a population with mean µ and standard deviation σ is given by 1 2 2 f (x) = √ e−(x−µ) /(2σ ) , σ 2π √ where the factor 1/(σ 2π) is called the normalization constant. It is needed to make the probability over the entire space equal to 1. That is, ∫ P (−∞ < x < ∞) =

+∞

−∞

1 2 2 √ e−(x−µ) /(2σ ) = 1. σ 2π

Example 7: Suppose that boxes containing 100 tea bags have a mean weight of 10.2 ounces each and a standard deviation of 0.1 ounce. 1. What percentage of all the boxes is expected to weigh between 10 and 10.5 ounces ? 2. What is the probability that a box weighs less than 10 ounces ? 3. What is the probability that a box will weigh exactly 10 ounces ? Solution: 1. Using the normal probability density function, 1 2 2 f (x) = √ e−(x−µ) /(2σ ) . σ 2π Substituting for µ = 10.2 and σ = 0.1, we get

297

www.ck12.org

f (x) =

1 2 2 √ e−(x−10.2) /(2(0.1) ) . (0.1) 2π

The percentage of all the tea boxes that are expected to weight between 10 and 10.5 ounces can be calculated as ∫

10.5

P (10 ≤ x ≤ 10.5) = 10

1 2 2 √ e−(x−10.2) /(2(0.1) ) dx. (0.1) 2π

2

The integral of ex does not have an elementary anti-derivative and therefore cannot be evaluated by standard techniques. However, we can use numerical techniques, such as The Simpson’s Rule or The Trapeziod Rule, to ﬁnd an approximate (but very accurate) value. Using the programing feature of a scientiﬁc calculator or, mathematical software, we eventually get ∫

10.5

10

1 2 2 √ e−(x−10.2) /(2(0.1) ) dx ≈ 0.976. (0.1) 2π

That is,

P (10 ≤ x ≤ 10.2) ≈ 97.6%. Technology Note: To make this computation with a graphing calculator of the TI-83/84 family, do the following: • From the [DISTR] menu (Figure 27) select option 2, which puts the phrase ”normalcdf” in the home screen. Add lower bound, upper bound, mean, standard deviation, separated by commas, close the parentheses, and press [ENTER]. The result is shown in Figure 28. 2. For the probability that a box weighs less than 10.2 ounces, we use the area under the curve to the left of x = 10.2. Since the value of f (9) is very small (less than a billionth), 1 2 2 √ e−(9−10.2) /(2(0.1) ) dx (0.1) 2π = 1.35 × 10−32 ,

f (9) =

www.ck12.org

298

Figure 5.39: (26)

Figure 5.40: (30)

299

www.ck12.org

getting the area between 9 and 10 will yield a fairly good answer. Integrating numerically, we get ∫

10

1 2 2 √ e−(x−10.2) /(2(0.1) ) dx (0.1) 2π 9 P (9 ≤ x ≤ 10.2) ≈ 0.02275 = 2.28%, P (9 ≤ x ≤ 10) =

which says that we would expect 2.28% of the boxes to weigh less than 10 ounces. 3. Theoretically the probability here will be exactly zero because we will be integrating from 10 to 10, which is zero. However, since all scales have some error (call it ϵ), practically we would ﬁnd the probability that the weight falls between 10 − ϵ and 10 + ϵ. Example 8: An Intelligence Quotient or IQ is a score derived from diﬀerent standardized tests attempting to measure the level of intelligence of an adult human being. The average score of the test is 100 and the standard deviation is 15. 1. What is the percentage of the population that has a score between 85 and 115? 2. What percentage of the population has a score above 140? Solution: 1. Using the normal probability density function,

1 2 2 f (x) = √ e−(x−µ) /(2σ ) , σ 2π and substituting µ = 100 and σ = 15,

f (x) =

1 2 2 √ e−(x−100) /(2(15) ) . 15 2π

The percentage of the population that has a score between 85 and 115 is ∫

115

P (85 ≤ x ≤ 115) = 85

www.ck12.org

1 2 2 √ e−(x−100) /(2(15) ) . 15 2π

300

Again, the integral of e−x does not have an elementary anti-derivative and therefore cannot be evaluated. Using the programing feature of a scientiﬁc calculator or a mathematical computer software, we get 2

∫

115 85

1 2 2 √ e−(x−100) /(2(15) ) dx ≈ 0.68. 15 2π

That is,

P (85 ≤ x ≤ 115) ≈ 68%. Which says that 68% of the population has an IQ score between 85 and 115. 2. To measure the probability that a person selected randomly will have an IQ score above 140, ∫ P (x ≥ 140) =

∞

140

1 2 2 √ e−(x−100) /(2(15) ) dx. 15 2π

This integral is even more diﬀicult to integrate since it is an improper integral. To avoid the messy work, we can argue that since it is extremely rare to meet someone with an IQ score of over 200, we can approximate the integral from 140 to 200. Then ∫

200

P (x ≥ 140) ≈ 140

1 2 2 √ e−(x−100) /(2(15) ) dx. 15 2π

Integrating numerically, we get

P (x ≥ 140) ≈ 0.0039. So the probability of selecting at random a person with an IQ score above 140 is 0.39%. That’s about one person in every 250 individuals!

Review Questions 1. A particle moves along the x−axis by a force F (x) =

301

x2

1 . +1 www.ck12.org

If the particle has already moved a distance of 10 meters from the origin, what is the work done by the ( πx )force? 2. A force of cos 2 acts on an object when it is x meters away from the origin. How much work is done by this force in moving the object from x = 1 to x = 5 meters ? 3. In physics, if the force on an object varies with distance then work done by the force is deﬁned as (see Example 5.15) ∫ b W = F (r)dr. a

That is, the work done corresponds to computing the area under the force graph. For example, the strength of the gravitational ﬁeld varies with the distance r from the Earth’s center. If a satellite of mass m is to be launched into space, then the force experienced by the satellite during and after launch is

F (r) = G

mM , r2 2

where M = 6 × 1024 kg is the mass of the Earth and G = 6.67 × 10−11 Nm is the kg2 Universal Gravitational Constant. If the mass of the satellite is 1000 kg and we wish to lift it to an altitude of 35, 780 km above the Earth’s surface, how much work is needed to lift it? (Radius of Earth is 6370 km.) 4. Hook’s Law states that when a spring is stretched x units beyond its natural length it pulls back with a force F (x) = kx, where k is called the spring constant or the stiﬀness constant. To calculate the work required to stretch the spring a length x we use ∫

b

W =

F (x)dx, a

where a is the initial displacement of the spring (a = 0 if the spring is initially unstretched) and b is the ﬁnal displacement. A force of 5 N is exerted on a spring and stretches it 1 m beyond its natural length. (a) Find the spring constant k. (b) How much work is required to stretch the spring 1.8 m beyond its natural length? 5. When a force of 30 N is applied to a spring, it stretches it from a length of 12 cm to 15 cm. How much work will be done in stretching the spring from 12 cm to 20 cm? (Hint: read the ﬁrst part of problem #4 above.) 6. A ﬂat surface is submerged vertically in a ﬂuid of weight density w. If the weight density w is doubled, is the force on the plate also doubled? Explain. www.ck12.org

302

Figure 5.41: (6) 7. The bottom of a rectangular swimming pool, whose bottom is an inclined plane, is shown below. Calculate the ﬂuid force on the bottom of the pool when it is ﬁlled completely with water. 8. Suppose f (x) is the probability density function for the lifetime of a manufacturer’s light bulb, where x is measured in hours. Explain the meaning of each integral. ∫ 5000 (a) ∫1000 f (x)dx ∞ (b) 3000 f (x)dx 9. The length of time a customer spends waiting until his/her entre is served at a certain restaurant is modeled by an exponential density function with an average time of 8 minutes. (a) What is the probability that a customer is served in the ﬁrst 3 minutes? (b) What is the probability that a customer has to wait more than 10 minutes ? 10. The average height of an adult female in Los Angeles is 63.4 inches (5 feet 3.4 inches) with a standard deviation of 3.2 inches . (a) (b) (c) (d)

What What What What

is is is is

the the the the

probability probability probability probability

that that that that

a a a a

female’s female’s female’s female’s

height height height height

is is is is

less than 63.4 inches ? between 63 and 65 inches ? more than 6 feet ? excatly 5 feet ?

Review Answers 1. 2. 3. 4.

1.471 J 0J 5 × 1010 J (a) k = 5 N/m (b) 8.1 J

303

www.ck12.org

5. 6. 7. 8.

3.2 J Yes. To explain why, ask how w and F are mathematically related. 63, 648 N (a) The probability that a randomly chosen light bulb will have a lifetime between 1000 and 5000 hours. (b) The probability that a randomly chosen light bulb will have a lifetime of at least 3000 hours. 9. (a) 31% (b) 29% 10. (a) 50% (b) 24% (c) 0.36% (d) almost 0%

Image Sources (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . (11) . (12) . (13) . (14) . (15) . www.ck12.org

304

(16) . (17) . (18) . (19) . (20) . (21) . (22) . (23) . (24) . (25) . (26) . (27) . (28) . (29) . (30) . (31) . (32) . (33) . (34) . (35) . (36) . (37) . (38) .

305

www.ck12.org

www.ck12.org

306

Chapter 6 Transcendental Functions 6.1

Inverse Functions

Functions such as logarithms, exponential functions, and trigonometric functions are examples of transcendental functions. If a function is transcendental, it cannot be expressed as a polynomial or rational function. That is, it is not an algebraic function. In this chapter, we will begin by developing the concept of an inverse of a function and how it is linked to its original numerically, algebraically, and graphically. Later, we will take each type of elementary transcendental function—logarithmic, exponential, and trigonometric—individually and see the connection between them and their respective inverses, derivatives, and integrals.

Learning Objectives A student will be able to: • Understand the basic properties of the inverse of a function and how to ﬁnd it. • Understand how a function and its inverse are represented graphically. • Know the conditions of invertabilty of a function.

One-to-One Functions A function, as you know from your previous mathematics background, is a rule that assigns a single value in its range to each point in its domain. In other words, for each output number, there is one or more input numbers. However, a function never produces more than a single output for one input. A function is said to be a one-to-one function if each output

307

www.ck12.org

is associated with only one single input. For example, f (x) = x2 assigns the output 9 for both 3 and −3, and thus it is not a one-to-one function. One-to-One Function The function f (x) is one-to-one in a domain D if f (a) ̸= f (b), whenever a ̸= b. There is an easy method to check if a function is one-to-one: draw a horizontal line across the graph. If the line intersects at only one point on the graph, then the function is one-to-one; otherwise, it is not. Notice in the ﬁgure below that the graph of y = x2 is not one-to-one since the horizontal line intersects the graph more than once. But the function y = x3 is a one-to-one function because the graph meets the horizontal line only once.

Example 1: Determine whether the functions are one-to-one: (a) f (x) = |x| (b) h(x) = x1/2 . Solution: It is best to graph both functions and draw on each a horizontal line. As you can see from the graphs, f (x) = |x| is not one-to-one since the horizontal line intersects it at two points. The function h(x) = x1/2 , however, is indeed one-to-one since only one point is intersected www.ck12.org

308

by the horizontal line.

The Inverse of a Function We discussed above the condition for a one-to-one function: for each output, there is only one input. A one-to-one function can be reversed in such a way that the input of the function becomes the output and the output becomes an input. This reverse of the original function is called the inverse of the function. If f −1 is an inverse of a function f, then f −1 ◦ f = x. For example, the two functions f (x) = 2x + 3 and h(x) = x−3 are inverses of each other 2 since

309

www.ck12.org

[

] x−3 f ◦ h = f (h(x)) = 2 + 3 = x − 3 + 3 = x, 2 (2x + 3) − 3 2x h ◦ f = h(f (x)) = = = x. 2 x Thus

f ◦ h = h ◦ f = x, and f and h are inverses of each other. Note: In general, f −1 ̸= f1 . When is a function invertable? It is interesting to note that if a function f (x) is always increasing or always decreasing over its domain, then a horizontal line will cut through this graph at one point only. Then f in this case is a one-to-one function and thus has an inverse. So if we can ﬁnd a way to prove that a function is constantly increasing or decreasing, then it is invertable or monotonic. From previous chapters, you have learned that if f ′ (x) > 0 then f must be increasing and if f ′ (x) < 0 then f must be decreasing. To summarize, a function has an inverse if it is one-to-one in its domain or if its derivative is either f ′ (x) > 0 or f ′ (x) < 0. Example 2: Given the polynomial function f (x) = 3x5 +2x+1, show that it is invertable (has an inverse). Solution: Taking the derivative, we ﬁnd that f ′ (x) = 15x4 + 2 > 0 for all x. We conclude that f (x) is one-to-one and invertable. Keep in mind that it may not be easy to ﬁnd the inverse of f (x) = 3x5 + 2x + 1 (try it!), but we still know that it is indeed invertable. How to ﬁnd the inverse of a one-to-one function: To ﬁnd the inverse of a one-to-one function, simply solve for x in terms of y and then interchange x and y. The resulting formula is the inverse y = f −1 (x). Example 3: Find the inverse of f (x) =

√

4x + 1.

Solution: From the discussion above, we can ﬁnd the inverse by ﬁrst solving for x in y = www.ck12.org

310

√

4x + 1.

√ y = 4x + 1, y 2 = 4x + 1, y2 − 1 x= . 4

Interchanging x ←→ y,

y=

x2 − 1 . 4

Replacing y = f −1 (x),

f −1 (x) =

x2 − 1 4

which is the inverse of the original function f (x) =

√

4x + 1.

Graphs of Inverse Functions What is the relationship between the graphs of f and f −1 ? If the point (a, b) is on the graph of f (x), then from the deﬁnition of the inverse, the point (b, a) is on the graph of f −1 (x). In other words, when we reverse the coordinates of a point on the graph of f (x) we automatically get a point on the graph of f −1 (x). We conclude that f (x) and f −1 (x) are reﬂections of one another about the line y = x. That is, each is a mirror image of the other 2 about the line y = x. The ﬁgure √ below shows an example of y = x and, when the domain is restricted, its inverse y = x and how they are reﬂected about y = x.

311

www.ck12.org

It is important to note that for the function f (x) = x2 to have an inverse, we must restrict its domain to 0 ≤ x < ∞, since that is the domain in which the function is increasing.

Continuity and Diﬀerentiability of Inverse Functions Since the graph of a one-to-one function and its inverse are reﬂections of one another about the line y = x, it would be safe to say that if the function f has no breaks (no discontinuities) then f −1 will not have breaks either. This implies that if f is continuous on the√domain D, then its inverse f −1 is continuous on the range R of f. For example, if f (x) = x, then its domain is x ≥ 0 and its range is y ≥ 0. This means that f (x) is continuous for all x ≥ 0. The inverse of f (x) is f −1 (x) = x2 , where its domain is all x > 0 and its range is y ≥ 0. We conclude that if f is a function with domain D and range R and it is continuous and one-to-one on D, then its inverse f −1 is continuous and one-to-one on the range R of f. Suppose that f has a domain D and a range R. If f is diﬀerentiable and one-to-one on D, then its inverse f −1 is diﬀerentiable at any value x in R for which f ′ (f −1 (x)) ̸= 0 and

d −1 1 [f (x)] = ′ −1 . dx f (f (x)) The formula above can be written in a form that is easier to remember:

1 dy = . dx dx/dy www.ck12.org

312

In addition, if f on its domain is either f ′ (x) > 0 or f ′ (x) < 0, then f has an inverse function f −1 and f −1 is diﬀerentiable at all values of x in the range of f. In this case, f −1 is given by the formula above. The example below illustrate this important theorem. Example 4: In Example 3, we were given the polynomial function f (x) = 3x5 + 2x + 1 and we showed that it is invertable. Show that it is diﬀerentiable and ﬁnd the derivative of its inverse. Solution: Since f ′ (x) = 15x4 + 2 > 0 for all x ∈ R, f −1 (x) is diﬀerentiable at all values of x. To ﬁnd the derivative of f −1 , if we let x = f (y), then

x = f (y) = 3y 5 + 2y + 1. So

dx = 15y 4 + 2 dy and

1 1 dy = = . dx dx/dy 15y 4 + 2 Since we are unable to solve for y in terms of x, we leave the answer above in terms of y. Another way of solving the problem is to use Implicit Diﬀerentiation: Since

x = 3y 5 + 2y + 1, diﬀerentiating implicitly,

d d [x] = [3y 5 + 2y + 1], dx dx dy 1 = (15y 4 + 2) . dx

313

www.ck12.org

Solving for

dy dx

we ﬁnally obtain

dy 1 = , 4 dx 15y + 2 which is the same result.

Review Questions In problems #1 - 3, ﬁnd the inverse function of f and verify that f ◦ f −1 = f −1 ◦ f = x. 1. f√(x) = 3x + 1 2. 3 x 3. f (x) = x−1 3 In problems #4 - 6, use the horizontal line test to verify whether the following functions have inverse. 4. h(x) = 4−x 6 5. g(x) = |x + √ 4| − |x − 4| 6. f (x) = −2x 16 − x2 In problems #7 - 8, use the functions f (x) = x + 4 and g(x) = 2x − 5 to ﬁnd the speciﬁed functions. 7. g −1 ◦ f −1 8. (f ◦ g)−1 In problems #9 - 10, show that f is monotonic (invertable) on the given interval (and therefore has an inverse.) 2 9. f (x) = (x − 5) ] ∞) [ , [5, 10. f (x) = cos x, 0, π2

Review Answers 1. f −1 (x) = 2. x3 www.ck12.org

x−1 3

314

3. 4. 5. 6. 7. 8. 9. 10.

6.2

3x + 1 Function has an inverse. Function does not have an inverse. Function does not have an inverse. x+1 2 x+1 2 ′

f (x) = 2(x − 5) > 0 on x > 5. f ′ (x) = − sin x, which is negative on the interval in question, so f (x) is monotonically decreasing.

Exponential and Logarithmic Functions

Learning Objectives A student will be able to:

• Understand and use the basic deﬁnitions of exponential and logarithmic functions and how they are related algebraically.

• Distinguish between an exponential and logarithmic functions graphically.

A Quick Algebraic Review of Exponential and Logarithmic Functions Exponential Functions Recall from algebra that an exponential function is a function that has a constant base and a variable exponent. A function of the form f (x) = bx where b is a constant and b > 0 and b ̸=( 1) is called an exponential function with base b. Some examples are f (x) = 2x , x f (x) = 21 , and f (x) = ex . All exponential functions are continuous and their graph is one of the two basic shapes, depending on whether 0 < b < 1 or b > 1. The graph below shows the two basic shapes:

315

www.ck12.org

Logarithmic Functions Recall from your previous courses in algebra that a logarithm is an exponent. If the base b > 0 and b ̸= 1, then for any value of x > 0, the logarithm to the base b of the value of x is denoted by

y = logb x. This is equivalent to the exponential form

by = x. For example, the following table shows the logarithmic forms in the ﬁrst row and the corresponding exponential forms in the second row. 1 = −2 25 1 = 25

Logarithmic Form →

log2 16 = 4

log5

log10 100 = 2

loge e = 1

Exponential Form →

24 = 16

5−2

102 = 100

e1 = e

Historically, logarithms with base of 10 were very popular. They are called the common logarithms. Recently the base 2 has been gaining popularity due to its considerable role in the ﬁeld of computer science and the associated binary number system. However, the most widely used base in applications is the natural logarithm, which has an irrational base denoted by e, in honor of the famous mathematician Leonhard Euler. This irrational constant is e ≈ 2.718281. Formally, it is deﬁned as the limit of (1 + x)1/x as x approaches zero. That is, www.ck12.org

316

lim (1 + x)1/x = e.

x→0

We denote the natural logarithm of x by ln x rather than loge x. So keep in mind, that ln x is the power to which e must be raised to produce x. That is, the following two expressions are equivalent:

y = ln x ⇐⇒ x = ey The table below shows this operation.

Natural Logarithm ln Equivalent Exponential Form

ln 2 = 0.693 e0.693 = 2

ln 1 = 0 e0 = 1

ln e = 1 e1 = e

ln e3 = 3 e 3 = e3

A Comparison between Logarithmic Functions and Exponential Functions Looking at the two graphs of exponential functions above, we notice that both pass the horizontal line test. This means that an exponential function is a one-to-one function and thus has an inverse. To ﬁnd a formula for this inverse, we start with the exponential function

y = bx . Interchanging x and y,

x = by . Projecting the logarithm to the base b on both sides,

logb x = logb by = y logb b = y(1) = y. Thus y = f −1 (x) = logb x is the inverse of y = f (x) = bx .

317

www.ck12.org

This implies that the graphs of f and f −1 are reﬂections of one another about the line y = x. The ﬁgure below shows this relationship.

Similarly, in the special case when the base b = e, the two equations above take the forms

y = f (x) = ex

and

f −1 (x) = ln x.

The graph below shows this relationship: www.ck12.org

318

Before we move to the calculus of exponential and logarithmic functions, here is a summary of the two important relationships that we have just discussed: • The function y = bx is equivalent to x = logb y if y > 0 and x ∈ R. • The function y = ex is equivalent to x = ln y if y > 0 and x ∈ R. You should also recall the following important properties about logarithms: • • • • •

logb vw = logb v + logb w logb wv = logb v − logb w logb wn = n logb w To express a logarithm with base in terms of the natural logarithm: logb w = aw To express a logarithm with base b in terms of another base a: logb w = log log b

ln w ln b

a

Review Questions Solve for x. 1. 2. 3. 4. 5. 6.

1 6x = 216 x e =3 log2 z = 3 ln x2 = 5 3e−5x = 132 e2x − 7ex + 10 = 0

319

www.ck12.org

7. 8. 9. 10.

−4(3)x = −36 ln x − ln 3 = ( 22 ) y = 5 log10 2−x y = 3e−2x/3

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

6.3

x = −3 ln 3 z=8 e5/2 x = −0.757 ln 2 and ln 5 x=2 x = 3e(2 ) x = 2 1 − 101y/5 ( ) x = − 32 ln y3

Diﬀerentiation and Integration of Logarithmic and Exponential Functions

Learning Objectives A student will be able to: • Understand and use the rules of diﬀerentiation of logarithmic and exponential functions. • Understand and use the rules of integration of logarithmic and exponential functions. In this section we will explore the derivatives of logarithmic and exponential functions. We will also see how the derivative of a one-to-one function is related to its inverse.

The Derivative of a Logarithmic Function Our goal at this point to ﬁnd an expression for the derivative of the logarithmic function y = logb x. Recall that the exponential number e is deﬁned as e = lim (1 + a)1/a a→ 0

www.ck12.org

320

(where we have substituted a for x for convenience). From the deﬁnition of the derivative of f (x) that you already studied in Chapter 2, f (w) − f (x) f (x + h) − f (x) = lim . w→ x h→0 h w−x

f ′ (x) = lim

We want to apply this deﬁnition to get the derivative to our logarithmic function y = logb x. Using the deﬁnition of the derivative and the rules of logarithms from the Lesson on Exponential and Logarithmic Functions, d logb w − logb x [logb x] = lim w→ x dx w−x logb (w/x) = lim w→ x [ w−x ( w )] 1 = lim logb w→ x w − x x [ ( )] 1 x + (w − x) = lim logb w→ x w − x x ( )] [ 1 w−x = lim logb 1 + w→ x w − x x [ ( )] 1 w−x = lim logb 1 + . w→ x x(w − x) x At this stage, let a = (w − x)/(x), the limit of w → x then becomes a → 0. Substituting, we get [

] 11 = lim logb (1 + a) a→ 0 x a ] [ 1 1 = lim logb (1 + a) x a→ 0 a [ ] 1 = lim logb (1 + a)1/a . x a→ 0 Inserting the limit,

=

[ ] 1 logb lim (1 + a)1/a . a→ 0 x

321

www.ck12.org

But by the deﬁnition e = lima→ 0 (1 + a)1/a ,

1 d [logb x] = logb e. dx x From the box above, we can express logb e in terms of natural logarithm by the using the formula logb w = ln w/ ln b. Then

logb e =

ln e 1 = . ln b ln b

Thus we conclude

d 1 [logb x] = > 0, dx x ln b and in the special case where b = e,

d 1 [lnx] = > 0. dx x To generalize, if u is a diﬀerentiable function of x and if u(x) > 0, then the above two equations, after the Chain Rule is applied, will produce the generalized derivative rule for logarithmic functions. Derivatives of Logarithmic Functions

d 1 du [logb u] = dx u ln b dx d 1 du u′ [ln u] = = dx u dx u Remark: Students often wonder why the constant e is deﬁned the way it is. The answer is in the derivative of f (x) = ln x. With any other base the derivative of f (x) = logb x would 1 , a more complicated expression than 1/x. Thinking back to another be equal f ′ (x) = x ln b unexpected unit, radians, the derivative of f (x) = sin(x) is the simple expression f ′ (x) = π cos(x) only if x is in radians. In degrees, f ′ (x) = 180 cos(x), which is more cumbersome and harder to remember. www.ck12.org

322

Example 1: Find the derivative of y = ln(2x2 − 4x + 3). Solution: Since

d dx

[lnu] =

1 du u dx

, for u = 2x2 − 4x + 3, ] dy 1 d [ 2 = 2 2x − 4x + 3 dx 2x − 4x + 3 dx 1 = 2 (4x − 4) 2x − 4x + 3 4(x − 1) . = 2 2x − 4x + 3

Example 2: Find

d dx

[ln(sin x)].

Solution:

d 1 [ln(sin x)] = · [cos x] dx sin x cos x = sin x = cot x. Example 3: Find

d dx

[ln(cos 5x)3 ] .

Solution: Here we use the Chain Rule: ] [ ] d [ 1 2 ln(cos 5x)3 = · 3(cos 5x) · (− sin 5x) · (5) dx cos3 5x [ ] 1 2 = · −15 cos 5x · sin 5x cos3 5x −15 sin 5x = cos 5x = −15 tan 5x. Example 4:

323

www.ck12.org

Find the derivative of y = x3 log5 2x. Solution: Here we use the Product Rule along with

d [logb dx

u] =

1 du u ln b dx

:

d d d 3 [x log5 2x] = x3 · [log5 2x] + [x3 ] · log5 2x dx dx dx 1 3 2 =x · + 3x · log5 2x 2x ln 5 x2 + 3x2 log5 2x. = 2 ln 5 Example 5: x Find the derivative of y = ln x+1 ·

Solution: We use the Quotient Rule and the natural logarithm rule: (x + 1)(1) − (1)(x) (x + 1)2 x+1 1 = · x (x + 1)2 1 = . x(x + 1)

y′ =

1

x x+1

·

Integrals Involving Natural Logarithmic Function In the last section, we have learned that the derivative of y = ln u(x) is y ′ = antiderivative is ∫

u′ (x) dx = ln |u(x)| + C. u(x)

If the argument of the natural logarithm is x, then ∫

www.ck12.org

d [ln x] dx

1 dx = ln |x| + C. x

324

= 1/x, thus

1 .u′ (x). u(x)

The

Example 6: ∫ 1 Evaluate x+1 dx. Solution: In general, whenever you encounter an integral with an integrand as a rational function, it might be possible that it can be integrated with the rule of natural logarithm. To do so, determine the derivative of the denominator. If it is the numerator itself, then the integration is simply the ln of the absolute value of the denominator. Let’s test this technique. ∫

1 dx. x+1

Notice that the derivative of the denominator is 1, which is equal to the numerator. Thus the solution is simply the natural logarithm of the absolute value of the denominator: ∫

1 dx = ln |x + 1| + C. x+1

The formal way of solving such integrals is to use u−substitution by letting u equal the denominator. Here, let u = x + 1, and du = dx. Substituting, ∫

∫ 1 1 dx = du x+1 u = ln |u| + C = ln |x + 1| + C.

Remark: The integral must use the absolute value symbol because although x may have negative values, the domain of ln(x) is restricted to x ≥ 0. Example 7: ∫ Evaluate 4x24x+1 dx. +2x+1 Solution: As you can see here, the derivative of the denominator is 8x + 2. Our numerator is 4x + 1. However, when we multiply the numerator by 2, we get the derivative of the denominator. Hence

325

www.ck12.org

∫

∫ 1 4x + 1 2(4x + 1) dx = dx 2 4x + 2x + 1 2 4x2 + 2x + 1 ∫ 1 8x + 2 = dx 2 2 4x + 2x + 1 1 = ln |4x2 + 2x + 1| + C. 2

Again, we could have used u−substitution. Example 8: ∫ Evaluate tan xdx. Solution: To solve, we rewrite the integrand as ∫

∫ tan xdx =

sin x dx. cos x

Looking at the denominator, its derivative is − sin x. So we need to insert a minus sign in the numerator: ∫

− sin x dx cos x = − ln | cos x| + C. =−

Derivatives of Exponential Functions We have discussed above that the exponential function is simply the inverse function of the logarithmic function. To obtain a derivative formula for the exponential function with base b, we rewrite y = bx as

x = logb y. Diﬀerentiating implicitly,

1= www.ck12.org

1 dy · . y ln b dx

326

Solving for

dy dx

and replacing y with bx ,

dy = y ln b = bx ln b. dx Thus the derivative of an exponential function is

d x [b ] = bx ln b. dx In the special case where the base is bx = ex , since ln e = 1 the derivative rule becomes

d x [e ] = ex . dx To generalize, if u is a diﬀerentiable function of x, with the use of the Chain Rule the above derivatives take the general form

d u du [b ] = bu · ln b · . dx dx And if b = e,

d u du [e ] = eu · . dx dx Derivatives of Exponential Functions

d u du [b ] = bu · ln b · = u′ bu ln b dx dx du d u [e ] = eu · = u′ eu dx dx Example 9: 2

Find the derivative of y = 2x . Solution:

327

www.ck12.org

Applying the rule for diﬀerentiating an exponential function,

y ′ = (2x)2x ln 2 2

= 2x

2 +1

· x · ln 2.

Example 10: 2

Find the derivative of y = ex . Solution: Since

d u [e ] = u′ eu , dx 2 y ′ = 2xex . Example 11: Find f ′ (x) if

1 2 f (x) = √ e−αk(x−x0 ) . πσ where σ, α, x0 , and k are constants and σ ̸= 0. Solution: We apply the exponential derivative and the Chain Rule:

1 2 f ′ (x) = √ (−2αk(x − x0 ))e−αk(x−x0 ) πσ 2αk(x − x0 ) −αk(x−x0 )2 √ =− e . πσ

Integrals Involving Exponential Functions Associated with the exponential derivatives in the box above are the two corresponding integration formulas: www.ck12.org

328

∫ bu du = ∫

1 u b + C, ln b

eu du = eu + C. The following examples illustrate how they can be used. Example 12: ∫ Evaluate 5x dx. Solution: ∫

1 x 5 +C ln 5 5x = + C. ln 5

5x dx =

Example 13: ∫ ex dx. Solution: ∫ ex dx = ex + C. In next chapter, we will learn how to integrate more complicated integrals, such as ∫ the 3 x2 ex dx, with the use of u−substitution and integration by parts along with the logarithmic and exponential integration formulas.

Review Questions 1. 2. 3. 4. 5.

Find Find Find Find Find

dy/dx dy/dx dy/dx dy/dx dy/dx

of of of of of

y y y y y

= e6x 3 2 = e3x −2x (+6 ) = ex2 · ln x1 x −x = eex −e +e−x = cos(ex )

329

www.ck12.org

6. 7. 8. 9. 10. 11.

Find dy/dx ∫ 1of y = ln(sin(ln x)) Evaluate ∫ ex dx √ Evaluate ∫ ex dx Evaluate 4x24x−3 dx ∫ ex +e−6x+7 −x Evaluate ex −e−x dx ∫ e dx Evaluate 0 x+e ∫ ln 3 x 12. Evaluate − ln 3 exe+4 dx

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

6.4

y ′ = 6e6x 3 2 y ′ = (9x2 − 4x)e3x −2x +6 2 y ′ = −xex [2 ln(x) + 1] y ′ = (ex +e4−x )2 y ′ = −ex sin ex x) y ′ = tan(ln x −e−x + C 2ex/2 + C 1 ln |4x2 − 6x + 7| + C 2 ln |ex − e−x | + C ln 2 ln 21 13

Exponential Growth and Decay

Learning Objectives A student will be able to: • Apply the laws of exponential and logarithmic functions to a variety of applications. • Model situations of growth and decay in a variety of problems. When the rate of change in a substance or population is proportional to the amount present at any time t, we say that this substance or population is going through either a decay or a growth, depending on the sign of the constant of proportionality. This kind of growth is called exponential growth and is characterized by rapid growth or decay. For example, a population of bacteria may increase exponentially with time because the rate of change of its population is proportional to its population at a given instant of time (more bacteria make more bacteria and fewer bacteria make fewer bacteria). The www.ck12.org

330

decomposition of a radioactive substance is another example in which the rate of decay is proportional to the amount of the substance at a given time instant. In the business world, the interest added to an investment each day, month, or year is proportional to the amount present, so this is also an example of exponential growth. Mathematically, the relationship between amount y and time t is a diﬀerential equation:

dy = ky. dt Separating variables,

dy = kdt, y and integrating both sides, ∫

dy = y

∫ kdt,

gives us

ln y = kt + C, y = ekt eC = Cekt . So the solution to the equation (dy/dt) = ky has the form y = Cekt . The box below summarizes the details of this function. The Law of Exponential Growth and Decay The function y = Cekt is a model for exponential growth or decay, depending on the value of k. • If k > 0: The function represents exponential growth (increase). • If k < 0: The function represents exponential decay (decrease). Where t is the time, C is the initial population at t = 0, and y is the population after time t.

331

www.ck12.org

Applications of Growth and Decay Radioactive Decay In physics, radioactive decay is a process in which an unstable atomic nucleus loses energy by emitting radiation in the form of electromagnetic radiation (like gamma rays) or particles (such as beta and alpha particles). During this process, the nucleus will continue to decay, in a chain of decays, until a new stable nucleus is reached (called an isotope). Physicists measure the rate of decay by the time it takes a sample to lose half of its nuclei due to radioactive decay. Initially, as the nuclei begins to decay, the rate starts very fast and furious, but it slows down over time as more and more of the available nuclei have decayed. The ﬁgure below shows a typical radioactive decay of a nucleus. As you can see, the graph has the shape of an exponential function with k < 0.

The equation that is used for radioactive decay is y = Cekt . We want to ﬁnd an expression for the half-life of an isotope. Since half-life is deﬁned as the time it takes for a sample to lose half of its nuclei, then if we starting with an initial mass C (measured in grams), then after some time t, y will become half the amount that we started with, C/2. Substituting this into the exponential decay model,

y = Cekt C = Cekt . 2 Canceling C from both sides, www.ck12.org

332

1 = ekt . 2 Solving for t, which is the half-life, by taking the natural logarithm on both sides,

1 = ln ekt 2 − ln 2 = kt. ln

Solving for t, and denoting it with new notation t1/2 for half-life (a standard notation in physics),

t1/2 =

− ln 2 −0.693 = k k

This is a famous expression in physics for measuring the half-life of a substance if the decay constant k is known. It can also be used to compute k if the half-life t1/2 is known. Example 1: A radioactive sample contains 2 grams of nobelium. If you know that the half-life of nobelium is 25 seconds, how much will remain after 3 minutes? Solution: Before we compute the mass of nobelium after 3 minutes, we need to ﬁrst know its decay rate k. Using the half-life formula, − ln 2 k − ln 2 k= t1/2 − ln 2 = 25 = −0.028 sec−1

t1/2 =

So the decay rate is k = −0.028/sec. The common unit for the decay rate is the Becquerel (Bq): 1 Bq is equivalent to 1 decay per sec. Since we found k, we are now ready to calculate the mass after 3 minutes. We use the radioactive decay formula. Remember, C represents the initial mass, C = 2 grams, and t = 3 minutes = 180 seconds. Thus

333

www.ck12.org

y = Cekt = 2e(−0.028)(180) = 0.013 grams. So after 3 minutes, the mass of the isotope is approximately 0.013 grams. Population Growth The same formula y = Cekt can be used for population growth, except that k > 0, since it is an increasing function. Example 2: A certain population of bacteria increases continuously at a rate that is proportional to its present number. The initial population of the bacterial culture is 140 and jumped to 720 bacteria in 4 hours. 1. How many will be there in 10 hours? 2. How long will it take the population to double? Solution: From reading the ﬁrst sentence in the problem, we learn that the bacteria is increasing exponentially. Therefore, the exponential growth formula is the correct model to use. 1. Just like we did in the previous example, we need to ﬁrst ﬁnd k, the growth rate. Notice that C = 140, t = 4, and y = 720. Substituting and solving for k.

y = Cekt 720 = 140ek(4) . Dividing both sides by 140 and then projecting the natural logarithm on both sides,

720 = ln e4k 140 ln 5.143 = 4k k = 0.409. ln

Now that we have found k, we want to know how many will be there after 10 hours. Substituting, www.ck12.org

334

y = Cekt = 140e(0.409)(10) = 8364 bacteria. 2. We are looking for the time required for the population to double. This means that we are looking for the time at which y = 2C. Substituting,

y = Cekt 2C = Cekt 2 = ekt . Solving for t requires taking the natural logarithm of both sides:

ln 2 = ln ekt ln 2 = kt. Solving for t,

ln 2 k ln 2 = 0.409 t = 1.7 hours. t=

This tells us that after about 1.7 hours (around 100 minutes) the population of the bacteria will double in number. Compound Interest Investors and bankers depend on compound interest to increase their investment. Traditionally, banks added interest after certain periods of time, such as a month or a year, and the phrase was “the interest is being compounded monthly or yearly.” With the advent of computers, the compunding could be done daily or even more often. Our exponential model represents continuous, or instantaneous, compounding, and it is a good model of current banking practices. Our model states that

335

www.ck12.org

A = P ert ,

where P is the initial investment (present value) and A is the future value of the investment after time t at an interest rate of r. The interest rate r is usually given in percentage per year. The rate must be converted to a decimal number, and t must be expressed in years. The example below illustrates this model. Example 3: An investor invests an amount of $10, 000 and discovers that its value has doubled in 5 years. What is the annual interest rate that this investment is earning? Solution: We use the exponential growth model for continuously compounded interest,

A = P ert 20, 000 = 10, 000er(5) 2 = e5r ln 2 = 5r.

Thus

ln 2 5 = 0.139 r = 13.9% r=

The investment has grown at a rate of 13.9% per year. Example 4: Going back to the previous example, how long will it take the invested money to triple? Solution: www.ck12.org

336

A = P ert 30, 000 = 10, 000e(0.139)(t) 3 = e0.139t ln 3 = 0.139t ln 3 t= 0.139 = 7.9 years. Other Exponential Models and Examples Not all exponential growths and decays are modeled in the natural base e or by y = Cekt . Actually, in everyday life most are constructed from empirical data and regression techniques. For example, in the business world the demand function for a product may be described by the formula

p = 12, 400 −

11, 000 , 2.2 + e−0.003x

where p is the price per unit and x is the number of units produced. So if the business is interested in basing the price of its unit on the number that it is projecting to sell, this formula becomes very helpful. If a motorcycle factory is projecting to sell 7000 units in one month, what price should the factory set on each motorcycle? 11, 000 2.2 + e0.003x 11, 000 = 12, 400 − 2.2 + e0.0003(7000) 11, 000 = 12, 400 − 2.2 + 0.122 = 7, 663.

p = 12, 400 −

Thus the factory’s base price for each motorcycle should be set at $7663. As another example, let’s say a medical researcher is studying the spread of the ﬂu virus through a certain campus during the winter months. Let’s assume that the model for the spread is described by

P =

4500 , x ≥ 0, 1 + 4490e−0.8x

337

www.ck12.org

where P represents the total number of infected students and x is the time, measured in days. Suppose the researcher is interested in the number of students who will be infected in the next week (7 days). Substituting x = 7 into the model,

4500 1 + 4490e−0.8x 4500 = 1 + 4499e−0.8(7) 4500 = 1 + 4499(0.004) = 255.

P =

According to the model, 255 students will become infected with the ﬂu virus. Assume further that the researcher wants to know how long it will take until 1000 students become infected with the ﬂu virus. Solving for x,

P =

4500 . 1 + 4490e−0.8x

Cross-multiplying,

P (1 + 4499e−0.8x) = 4500 4500 1 + 4499e−0.8x = P 4500 4499e−0.8x = −1 P 4500 − P = P 4500 − P −0.8x e = . 4499P Projecting ln on both sides, [

] 4500 − P −0.8x = ln 4499P ] [ 4500 − P ÷ (−0.8). x = ln 4499P www.ck12.org

338

Substituting for P = 1000,

x = 9 days. So the ﬂu virus will spread to 1000 students in 9 days. Other applications are introduced in the exercises.

Review Questions 1. In 1990, the population of the USA was 249 million. Assume that the annual growth rate is 1.8%. (a) According to this model, what was the population in the year 2000? (b) According to this model, in which year the population will reach 1 billion? 2. Prove that if a quantity A is exponentially growing and if A1 is the value at t1 and A2 at time t2 , then the growth rate will be given by ( ) 1 A1 k= ln . t1 − t2 A2 3. Newton’s Law of Cooling states that the rate of cooling is proportional to the diﬀerence in temperature between the object and the surroundings. The law is expressed by the formula T (t) = (T0 − Tr )e−kt + Tr where T0 is the initial temperature of the object at t = 0, Tr is the room temperature (the surroundings), and k is a constant that is unique for the measuring instrument (the thermometer) called the time constant. Suppose a liter of juice at 23◦ C is placed in the refrigerator to cool. If the temperature of the refrigerator is kept at 11◦ C and k = 0.417, what is the temperature of the juice after 3 minutes? 4. Referring back to problem 3, if it takes an object 320 seconds to cool from 40◦ C above room temperature to 22◦ C above room temperature, how long will it take to cool another 10◦ C? 5. Polonium−210 is a radioactive isotope with half-life of 140 days. If a sample has a mass of 10 grams, how much will remain after 10 weeks? 6. In the physics of acoustics, there is a relationship between the subjective sensation of loudness and the physically measured intensity of sound. This relationship is called the sound level β. It is speciﬁed on a logarithmic scale and measured with units of decibels (dB). The sound level β of any sound is deﬁned in terms of its intensity I (in the SI-mks unit system, it is measured in watts per meter squared, W/m2 ) as β = 10 log

339

I . 10−12 www.ck12.org

For example, the average decibel level of a busy street traﬀic is 70 dB, normal conversation at a dinner table is 55 dB, the sound of leaves rustling is 10 dB, the siren of a ﬁre truck at 30 meters is 100 dB, and a loud rock concert is 120 dB. The sound level 120 dB is considered the threshold of pain for the human ear and 0 dB is the threshold of hearing (the minimum sound that can be heard by humans.) (a) If at a heavy metal rock concert a dB meter registered 130 dB, what is the intensity I of this sound level? (b) What is the sound level (in dB) of a sound whose intensity is 2.0 × 10−6 W/m2 ? 7. Referring to problem #6, a single mosquito 10 meters away from a person makes a sound that is barely heard by the person (threshold 0 dB). What will be the sound level of 1000 mosquitoes at the same distance? 8. Referring back to problem #6, a noisy machine at a factory produces a sound level of 90 dB. If an identical machine is placed beside it, what is the combined sound level of the two machines?

Review Answers 1. (a) 298.1 million (b) 2077 2. Hint: use A1 = Cekt1 3. 14.4◦ C 4. 923 seconds, about 15 minutes 5. 7 grams 6. (a) 10 W/m2 (b) 63 dB 7. 30 dB 8. 93 dB

6.5

Derivatives and Integrals Involving Inverse Trigonometric Functions

Learning Objectives A student will be able to: • Learn the basic properties inverse trigonometric functions. • Learn how to use the derivative formula to use them to ﬁnd derivatives of inverse trigonometric functions. • Learn to solve certain integrals involving inverse trigonometric functions. www.ck12.org

340

A Quick Algebraic Review of Inverse Trigonometric Functions You already know what a trigonometric function is, but what is an inverse trigonometric function? If we ask what is sin(π/6) equal to, the answer is (1/2). That is simple enough. But what if we ask what angle has a sine of (1/2)? That is an inverse trigonometric function. So we say sin(π/6) = (1/2), but sin−1 (1/2) = (π/6). The “sin−1 ” is the notation for the inverse of the sine function. For every one of the six trigonometric functions there is an associated inverse function. They are denoted by

sin−1 x, cos−1 x, tan−1 x, sec−1 x, csc−1 x, cot−1 x Alternatively, you may see the following notations for the above inverses, respectively,

arcsin x, arccos x, arctan x, arcsec x, arccsc x, arccot x Since all trigonometric functions are periodic functions, they do not pass the horizontal line test. Therefore they are not one-to-one functions. The table below provides a brief summary of their deﬁnitions and basic properties. We will restrict our study to the ﬁrst four functions; the remaining two, csc−1 and cot−1 , are of lesser importance (in most applications) and will be left for the exercises. Table 6.1: Inverse Function

Domain

Range

sin cos−1

−1 ≤ x ≤ 1 −1 ≤ x ≤ 1

−π 2

≤y≤ 0≤y≤π

tan−1

all R

( −π

sec−1

(−∞, −1] ∪ [1, +∞)

[ −π ) ( π ] 0, 2 ∪ 2 , π

−1

2

, π2

Basic Properties π 2

)

sin(sin−1 (x)) = x cos−1 (cos x) = −1 cos(cos (x)) = x tan−1 (tan x) = −1 tan(tan (x)) = x sec−1 (sec x) = sec(sec−1 (x)) = x

The range is based on limiting the domain of the original function so that it is a one-to-one function. Example 1:

√ What is the exact value of sin−1 ( 3/2)?

341

www.ck12.org

Solution:

√ √ This is equivalent to sin x = 23 . Thus sin−1 ( 3/2) = π/3. You can easily conﬁrm this result by using your scientiﬁc calculator.

Example 2: Most calculators do not provide a way to calculate the inverse of the secant function, sec−1 x. A practical trick however is to use the identity

sec−1 x = cos−1

(Recall that sec θ =

1 x

1 .) cos θ

For practice, use your calculator to ﬁnd sec−1 (3.24). Solution: Since

1 1 = = 0.3086, x 3.24

sec−1 3.24 = cos−1 0.3086 = 72◦ . Here are two other identities that you may need to enter into your calculator:

1 csc−1 x = sin−1 , x −1 −1 1 cot x = tan . x

The Derivative Formulas of the Inverse Trigonometric Functions If u is a diﬀerentiable function of x then the generalized derivative formulas for the inverse trigonometric functions are (we introduce them here without a proof): www.ck12.org

342

d [ −1 ] sin u = dx d [ −1 ] cos u = dx d [ −1 ] tan u = dx d [ −1 ] sec u = dx d [ −1 ] csc u = dx d [ −1 ] cot u = dx

1 du −1 < u < 1 1 − u2 dx −1 du √ −1 < u < 1 1 − u2 dx 1 du −∞ < x < ∞ 1 + u2 dx 1 du √ u >1 x u2 − 1 dx −1 du √ u >1 u u2 − 1 dx √

−1 du −∞ < x < ∞ 1 + u2 dx

Example 3: Diﬀerentiate y = sin−1 (2x4 ) Solution: Let u = 2x4 , so

dy 1 · (8x2 ) =√ 4 2 dx 1 − (2x ) 8x3 =√ . 1 − 4x8 Example 4: Diﬀerentiate tan−1 (e3x ). Solution: Let u = e3x , so

1 dy = · 3e3x dx 1 + (e3x )2 3e3x = . 1 + e6x Example 5:

343

www.ck12.org

Find dy/dx if y = sec−1 (sin x). Solution: Let u = sin x.

dy 1 √ 2 · cos x = dx sin x sin x − 1 cos x = sin x cos x 1 . = sin x

The Integration Formulas of the Inverse Trigonometric Functions The derivative formulas in the box above yield the following integrations formulas for inverse trigonometric functions:

∫

du √ = sin−1 u + c 2 ∫ 1−u du = tan−1 u + c 1 + u2 ∫ du √ = sec−1 u + c u u2 − 1

Example 6: ∫ dx Evaluate 1+4x 2. Solution: Before we integrate, we use u−substitution. Let u = 2x (the square root of 4x2 ). Then du = 2dx. Substituting, www.ck12.org

344

∫

dx = 1 + 4x2

∫

1/2 du 1 + u2 ∫ 1 1 = du 2 1 + u2 1 = tan−1 u + c 2 1 = tan−1 (2x) + c. 2

Example 7: ∫ ex Evaluate √1−e 2x dx. Solution: We use u−substitution. Let u = ex , so du = ex dx. Substituting, ∫

∫ du ex ex √ √ dx = 1 − e2x 1 − u2 e x ∫ 1 √ = du 1 − u2 = sin−1 u + c = sin−1 (ex ) + c.

Example 8: Evaluate the deﬁnite integral

∫ ln(2/√3) ln2

−x √ e dx. 1−e−2x

Solution: Substituting u = e−x , du = −e−x dx. To change the limits,

1 x = ln 2 → u = e−x = e− ln 2 = eln 1/2 = , 2√ ( ) √ 2 3 . x = ln √ → u = e−x = e− ln 2/ 3 = 2 3 Thus our integral becomes

345

www.ck12.org

∫

√ ln(2/ 3)

ln2

e−x √ dx = 1 − e−2x

∫

√ 3/2

1/2

∫

√ 3/2

=− [

√

u −du 1 − u2 u 1 du 1 − u2 ]√3/2

√

1/2

= − sin−1 u 1/2

( √ ) ( )] 3 1 = − sin − − sin−1 2 2 [π π ] =− − 3 6 π = . 3 [

Review Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Find dy/dx of y = sec−1 x2 . Find dy/dx of y = tan1−1 x . Find dy/dx of y = ln(cos−1 x). Find dy/dx of y = sin−1 e−4x . −1 2 Find dy/dx ∫ ofdxy = sin (x ln x). Evaluate √9−x2 . ∫ 3 dx Evaluate 1 √x(x+1) . ∫ x−3 Evaluate x2 +1 dx. ∫0 x Evaluate −√3 1+x 2 dx. Given the points A(2, 1) and B(5, 4), ﬁnd a point Q in the interval [2, 5] on the x−axis that maximizes angle ]AQB.

Review Answers 1. x√x24 −1 −1 2. (1+x2 )(tan −1 x)2 1√ 3. − (cos−1 x) 1−x2 −4x

4. √−4e 1−e−8x √ 5. x+2x4 ln x 2 1−x (ln x) ( ) 1 6. 3 sin−1 x3 + C www.ck12.org

346

7. π/6 8. 12 ln(x2 + 1) − 3 tan−1 x + C 9. − ln 2√ 10. 1 + 2 2

6.6

L’Hospital’s Rule

Learning Objectives A student will be able to: • Learn how to ﬁnd the limit of indeterminate form (0/0) by L’Hospital’s rule. If the two functions f (x) and g(x) are both equal to zero at x = a, then the limit

lim

x→ a

f (x) g(x)

cannot be found by directly substituting x = a. The reason is because when we substitute x = a, the substitution will produce (0/0), known as an indeterminate form, which is a meaningless expression. To work around this problem, we use L’Hospital’s rule, which enables us to evaluate limits of indeterminate forms. L’Hospital’s Rule If limx→a f (x) = limx→a g(x) = 0, and f ′ (a) and g ′ (a) exist, where g ′ (a) ̸= 0, then f (x) f ′ (x) = lim ′ . a g(x) x→ a g (x)

lim

x→

The essence of L’Hospital’s rule is to be able to replace one limit problem with a simpler one. In each of the examples below, we will employ the following three-step process: (x) is an indeterminate form 0/0. To do so, directly substitute x = a 1. Check that limx→a fg(x) into f (x) and g(x). If you get f (a) = g(a) = 0, then you can use L’Hospital’s rule. Otherwise, it cannot be used. 2. Diﬀerentiate f (x) and g(x) separately. ′ (x) (x) 3. Find limx→a fg′ (x) . If the limit is ﬁnite, then it is equal to the original limit limx→a fg(x) .

347

www.ck12.org

Example 1: Find limx→0

√

√ 2+x− 2 . x

Solution: When x = 0 is substituted, you will get 0/0. Therefore L’Hospital’s rule applies: √ lim

x→0

2+x− x

√

2

[

√ ] 2 + x − 2) = lim d x→0 (x) dx √ ] [ 1/(2 2 + x) = 1 x=0 1 = √ 2 2 √ 2 = . 4 √

d ( dx

Example 2: Find limx→0

1−cos 2x x2 +2x

Solution: We can see that the limit is 0/0 when x = 0 is substituted. Using L’Hospital’s rule, [ ] 2 sin 2x 1 − cos 2x lim = x→0 x2 + 2x 2x + 2 x=0 = 0/2 = 0. Example 3: Use L’Hospital’s rule to evaluate limx→3

x2 −9 x−3

.

Solution:

lim =

x→3

www.ck12.org

x2 − 9 2x = lim = 6. x→3 1 x−3

348

Example 4: Evaluate limx→0

sin 3x . x

Solution: sin 3x 3 cos 3x = lim = 3. x→0 x→0 x 1 lim

Example 5: Evaluate limx→π/2

5−5 sin x cos x

.

Solution: We can use L’Hospital’s rule since the limit produces the 0/0 once x = π/2 is substituted. Hence 5 − 5 sin x 0 − 5 cos x 0 = lim = = 0. x→π/2 x→π/2 cos x − sin x −1 lim

A broader application of L’Hospital’s rule is when x = a is substituted into the derivatives of the numerator and the denominator but both still equal zero. In this case, a second diﬀerentiation is necessary. Example 6: Evaluate limx→0

ex −cos x . x2

Solution: ex − cos x ex + sin x = lim . x→0 x→0 x2 2x lim

As you can see, if we apply the limit at this stage the limit is still indeterminate. So we apply L’Hospital’s rule again: ex − cos x = lim x→0 2 0 1−1 = = 0. = 2 2

Review Questions Find the limits.

349

www.ck12.org

1. 2. 3. 4. 5. 6.

limx→0 tanθ θ ln x limx→1 tan πx 10x 6x limx→0 e x−e x limx→π sin x−π x xe limx→0 1−e x If k is a nonzero constant and x > 0. ∫x 1 k (a) Show that 1 t1−k dt = x k−1 . (b) Use L’Hospital’s rule to ﬁnd limk→0

xk −1 . k

7. Cauchy’s Mean Value Theorem states that if the functions f and g are continuous on the interval (a, b) and g ′ ̸= 0, then there exists a number c such that f ′ (c) f (b) − f (a) = . ′ g (c) g(b) − g(a) Find all possible values of c in the interval (a, b) that satisfy this property for f (x) = cos x g(x) = sin x on the interval [ π] [a, b] = 0, . 2

Review Answers 1. 2. 3. 4. 5. 6. 7.

1 1/π 4 −1 −1 ln x π 4

Image Sources

www.ck12.org

350

Chapter 7 Integration Techniques 7.1

Integration by Substitution

Each basic rule of integration that you have studied so far was derived from a corresponding diﬀerentiation rule. Even though you have learned all the necessary tools for diﬀerentiating exponential, logarithmic, trigonometric, and algebraic functions, your set of tools for integrating these functions is not yet complete. In this chapter we will explore diﬀerent ways of integrating functions and develop several integration techniques that will greatly expand the set of integrals to which the basic integration formulas can be applied. Before we do that, let us review the basic integration formulas that you are already familiar with from previous chapters. 1. The Power Rule (n ̸= −1): ∫ xn dx =

xn + 1 + C. n+1

2. The General Power Rule (n ̸= −1): ∫ u

n du

dx

∫ dx =

un du =

un+1 + C. n+1

3. The Simple Exponential Rule: ∫ ex dx = ex + C.

351

www.ck12.org

4. The General Exponential Rule: ∫

u du

e

dx

∫ eu du = eu + C.

dx =

5. The Simple Log Rule: ∫

1 dx = ln |x| + C. x

6. The General Log Rule: ∫

du/dx dx = u

∫

1 du = ln |u| + C. u

It is important that you remember the above rules because we will be using them extensively to solve more complicated integration problems. The skill that you need to develop is to determine which of these basic rules is needed to solve an integration problem.

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using the technique of u−substitution. • Apply the u−substitution technique to deﬁnite integrals. • Apply the u−substitution technique to trig functions. Probably one of the most powerful techniques of integration is integration by substitution. In this technique, you choose part of the integrand to be equal to a variable we will call u and then write the entire integrand in terms of u. The diﬀiculty of the technique is deciding which term in the integrand will be best for substitution by u. However, with practice, you will develop a skill for choosing the right term. Recall from Chapter 2 that if u is a diﬀerentiable function of x and if n is a real number and n ̸= −1, then the Chain Rule tells us that d n du [u ] = nun−1 . dx dx www.ck12.org

352

The reverse of this formula is the integration formula, ∫ un du =

un+1 + C, n ̸= −1. n+1

Sometimes it is not easy to integrate directly. For example, look at this integral: ∫ (5x − 2)2 dx. One way to integrate is to ﬁrst expand the integrand and then integrate term by term. ∫

∫ (5x − 2) dx =

(25x2 − 20x + 4)dx ∫ ∫ ∫ 2 = 25 x dx − 20 xdx + 4dx

2

=

25 3 x − 10x2 + 4x + C. 3

That is easy enough. However, what if the integral was ∫ (5x − 2)15 dx? Would you still expand the integrand and then integrate term by term? That would be impractical and time-consuming. A better way of doing this is to change the variables. Changing variables can often turn a diﬀicult integral, such as the one above, into one that is easy to integrate. The method of doing this is called integration by substitution, or for short, the u-substitution method. The examples below will show you how the method is used. Example 1: ∫ Evaluate (x + 1)5 dx. Solution: Let u = x + 1. Then du = d(x + 1) = 1dx = dx. Substituting for u and du we get ∫

∫ 5

(x + 1) dx =

353

u5 du. www.ck12.org

Integrating using the power rule, u6 = + C. 6 Since u = x + 1, substituting back,

=

(x + 1)6 + C. 6

Example 2: ∫√ Evaluate 4x + 3dx. Solution: Let u = 4x + 3. Then du = 4dx. Solving for dx,

dx = du/4. Substituting, ∫

1 u1/2 · dx 4 ∫ 1 u1/2 dx = 4 1 u3/2 = + C. 4 3/2 =

Simplifying, 1 = u3/2 + C 6 1 = (4x + 3)3/2 + C. 6

Trigonometric Integrands We can apply the change of variable technique to trigonometric functions as long as u is a diﬀerentiable function of x. Before we show how, recall the basic trigonometric integrals: www.ck12.org

354

∫ cos udu = sin u + C, ∫ ∫

sin udu = − cos u + C, sec2 udu = tan u + C,

∫ csc2 udu = − cot u + C,

∫

(sec u)(tan u)du = sec u + C, ∫ (csc u)(cot u)du = − csc u + C. Example 3: ∫ Evaluate cos(3x + 2)dx. Solution: The argument of the cosine function is 3x + 2. So we let u = 3x + 2. Then du = 3dx, or dx = du/3. Substituting, ∫

∫

1 cos u · dx 3 ∫ 1 cos udx. = 3

cos(3x + 2)dx =

Integrating, 1 sin u + C 3 1 = sin(3x + 2) + C. 3 =

Example 4: This example requires us to use trigonometric identities before we substitute. Evaluate ∫

1 dx. cos2 3x

355

www.ck12.org

Solution: Since sec 3x =

1 , cos 3x

the integral becomes ∫

1 dx = cos2 3x

∫ sec2 3xdx.

Substituting for the argument of the secant, u = 3x, then du = 3dx, or dx = du/3. Thus our integral becomes, ∫

∫ 1 1 sec u. du = sec2 udu 3 3 1 = tan u + C 3 1 = tan(3x) + C. 3 2

Some integrations of trigonometric functions involve the logarithmic functions as a solution, as shown in the following example. Example 5: ∫ Evaluate tan xdx. Solution: As you may have guessed, this is not a straightforward integration. We need to make use of trigonometric identities to simplify it. Since tan x = sin x/ cos x, ∫

∫ tan xdx =

sin x dx. cos x

Now make a change of variable x. Choose u = cos x. Then du = − sin xdx, or dx = −du/ sin x. Substituting, ∫

( ) sin x −du u sin x ∫ du . =− u

sin x dx = cos x

∫

This integral should look obvious to you. The integrand is the derivative of the natural logarithm ln u. www.ck12.org

356

= − ln |u| + C = − ln | cos x| + C.

1 Another way of writing it, since − ln |u| = ln |u| , is

1 +C = ln cos x = ln | sec x| + C.

Using Substitution on Deﬁnite Integrals Example 6: ∫3 x Evaluate 1 √2x−1 dx. Solution: Let u = 2x − 1. Then du = 2dx, or dx = du/2. Before we substitute, we need to determine the new limits of integration in terms of the u variable. To do so, we simply substitute the limits of integration into u = 2x − 1: Lower limit: For x = 1, u = 2(1) − 1 = 1. Upper limit: For x = 3, u = 2(3) − 1 = 5. We now substitute u and the associated limits into the integral:

∫ 1

5

x du √ . u 2

As you may notice, the variable x is still hanging there. To write it in terms of u, since u = 2x − 1, solving for x, we get, x = (u + 1)/(2). Substituting back into the integral,

357

www.ck12.org

∫ = = = = =

5

u + 1 du √ 1 2 u 2 ∫ 1 5 u+1 √ du 4 1 u ∫ 5 1 (u + 1)u−1/2 du 4 1 ∫ 1 5 1/2 (u + u−1/2 )du 4 1 [ ]5 1 2u3/2 2u1/2 + . 4 3 1 1

Applying the Fundamental Theorem of Calculus by inserting the limits of integration and calculating,

1 = 4

([

] [ ]) 2(5)3/2 2(5)1/2 2(1)3/2 2(1)1/2 + − + . 3 1 3 1

Calculating and simplifying, we get √ 4 5−2 . = 3 √ We could have chosen u = 2x − 1 instead. You may want to try to solve the integral with this substitution. It might be easier and less tedious. Example 7: Let’s try the substitution method of deﬁnite integrals with a trigonometric integrand. ∫ π/4 Evaluate 0 tan x sec2 xdx. Solution: Try u = tan x. Then du = sec2 xdx, dx = du/ sec2 x. Lower limit: For x = 0, u = tan 0 = 0. Upper limit: For x = π/4, u = tan π4 = 1. Thus www.ck12.org

358

∫

∫

π/4 2

1

tan x sec xdx = 0

udu [ 2 ]1 u = 2 0 1 1 = −0= . 2 2 0

Review Questions In the following exercises, evaluate the integrals. ∫ 3 1. (x−8) 2 dx ∫√ 2. ∫ 2 + xdx 1 3. √2+x dx ∫ x2 4. x+1 dx ∫ e−x 5. e−x dx +2 ∫ 3√t+5 6. ∫ t dt 2 7. √3x−1 dx ∫ cos xdx 8. ∫ sin x √ 9. ∫ cos x 1 − cos2 xdx 10. ∫ sin5 x cos xdx 11. ∫ x3 cos 4x4 dx 12. sec2 (2x + 4)dx ∫2 2 13. 0√xex dx ∫ π 14. 0 x sin x2 dx ∫1 15. 0 x(x + 5)4 dx

Review Answers 1. 2. 3. 4. 5. 6. 7. 8.

−3 x−8 2 (2 3√

+C + x)3/2 + C 2 2+x+C x2 − x − 32 + ln |x + 1| + C 2 −√ln |ex + 2| + C 6 √t + 5 ln |t| + C 4 3x − 1 + C 3 1 sin2 x + C 2

359

www.ck12.org

cos2 x + C 9. −1 2 10. 16 sin6 x + C 1 11. 16 sin 4x4 + C 1 12. 2 tan(2x + 4) + C 13. 12 (e4 − 1) 14. 1 15. 520 56

7.2

Integration By Parts

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using technique of Integration by Parts. • Combine this technique with the u−substitution method to solve integrals. • Learn to tabulate the technique when it is repeated. In this section we will study a technique of integration that involves the product of algebraic and exponential or logarithmic functions, such as ∫ x ln xdx and ∫ xex dx. Integration by parts is based on the product rule of diﬀerentiation that you have already studied:

d dv du [uv] = u + v . dx dx dx If we integrate each side, www.ck12.org

360

∫

∫ dv du uv = u dx + v dx dx dx ∫ ∫ = udv + vdu. Solving for

∫

udv, ∫

∫ udv = uv −

vdu.

This is the formula for integration by parts. With the proper choice of u and dv, the second integral may be easier to integrate. The following examples will show you how to properly choose u and dv. Example 1: ∫ Evaluate x sin xdx. Solution: We use the formula

∫

udv = uv −

∫

vdu.

Choose

u=x and

dv = sin xdx. To complete the formula, we take the diﬀerential of u and the simplest antiderivative of dv = sin xdx.

du = dx v = − cos x. The formula becomes

361

www.ck12.org

∫

∫ x sin xdx = −x cos x −

(− cos x)dx ∫

= −x cos x +

cos xdx

= −x cos x + sin x + C. A Guide to Integration by Parts Which choices of u and dv lead to a successful evaluation of the original integral? In general, choose u to be something that simpliﬁes when diﬀerentiated, and dv to be something that remains manageable when integrated. Looking at the example that we have just done, we chose u = x and dv = sin xdx. That led to a successful evaluation of our integral. However, let’s assume that we made the following choice,

u = sin x dv = xdx. Then

du = cos xdx v = x2 /2. Substituting back into the formula to integrate, we get ∫

∫ udv = uv −

vdu ∫ 2 x2 x = sin x − cos xdx 2 2

As you can see, this integral is worse than what we started with! This tells us that we have made the wrong choice and we must change (in this case switch) our choices of u and dv. ∫ Remember, the goal of the integration by parts is to start with an∫integral in the form udv that is hard to integrate directly and change it to an integral vdu that looks easier to evaluate. However, here is a general guide that you may ﬁnd helpful: 1. Choose dv to be the more complicated portion of the integrand that ﬁts a basic integration formula. Choose u to be the remaining term in the integrand. www.ck12.org

362

2. Choose u to be the portion of the integrand whose derivative is simpler than u. Choose dv to be the remaining term. Example 2: ∫ Evaluate xex dx. Solution: Again, we use the formula

∫

udv = uv −

∫

vdu.

Let us choose

u=x and

dv = ex dx. We take the diﬀerential of u and the simplest antiderivative of dv = ex dx:

du = dx v = ex . Substituting back into the formula, ∫

∫ udv = uv − = xe − x

vdu ∫ ex dx.

∫ ∫ We have made the right choice because, as you can see, the new integral vdu = ex dx is deﬁnitely simpler than our original integral. Integrating, we ﬁnally obtain our solution ∫ xex dx = xex − ex + C. Example 3:

363

www.ck12.org

Evaluate

∫

ln xdx.

Solution: Here, we only have one term, ln x. We can always assume that this term is multiplied by 1: ∫ ln x1dx.

So let u = ln x, and dv = 1dx. Thus du = 1/xdx and v = x. Substituting, ∫

∫ udv = uv −

vdu ∫ ∫ 1 ln xdx = x ln x − x dx x ∫ = x ln x − 1dx = x ln x − x + C.

Repeated Use of Integration by Parts Oftentimes we use integration by parts more than once to evaluate the integral, as the example below shows. Example 4: ∫ Evaluate x2 ex dx. Solution: With u = x2 , dv = ex dx, du = 2xdx, and v = ex , our integral becomes ∫

∫ x e dx = x e − 2 2 x

2 x

xex dx.

2 x x As you can see, the integral has become less complicated than the original, ∫ x x e → xe . This tells us that we have made the right choice. However, to evaluate xe dx we still need to integrate by parts with u = x and dv = ex dx. Then du = dx and v = ex , and

www.ck12.org

364

∫

∫ x e dx = x e − 2 2 x

2 x

xex dx [ ] ∫ 2 x = x e − 2 uv − udv [ ] ∫ 2 x x x = x e − 2 xe − e dx

= x2 ex − 2xex + 2ex + C. Actually, the method that we have just used works for any integral that has the form ∫ n x x e dx, where n is a positive integer. The following section illustrates a systematic way of solving repeated integrations by parts.

Tabular Integration by Parts Sometimes, we need to integrate by parts several times. This leads to cumbersome calculations. In situations like these it is best to organize our calculations to save us a great deal of tedious work and to avoid making unpredictable mistakes. The example below illustrates the method of tabular integration. Example 5: ∫ Evaluate x2 sin 3xdx. Solution: Begin as usual by letting u = x2 and dv = sin 3xdx. Next, create a table that consists of three columns, as shown below: Table 7.1: Alternate signs

u and its derivatives

dv and its antiderivatives

+ − + −

x2 ↘ 2x ↘ 2↘ 0

sin 3x −1 cos 3x 3 −1 sin 3x 9 1 cos 3x 27

To ﬁnd the solution for the integral, pick the sign from the ﬁrst row (+), multiply it by u of the ﬁrst row (x2 ) and then multiply by the dv of the second row, −1/3 cos 3x (watch the direction of the arrows.) This is the ﬁrst term in the solution. Do the same thing to obtain the second term: Pick the sign from the second row, multiply it by the u of the same row and then follow the arrow to multiply the product by the dv in the third row. Eventually

365

www.ck12.org

we obtain the solution ∫ x2 sin 3xdx =

−1 2 2 2 x cos 3x + x sin 3x + cos 3x + C. 3 9 27

Solving for an Unknown Integral There are some integrals that require us to evaluate two integrations by parts, followed by solving for the unknown integral. These kinds of integrals crop up often in electrical engineering and other disciplines. Example 6: ∫ Evaluate ex cos xdx. Solution: Let u = ex , and dv = cos xdx. Then du = ex dx, v = sin x, and ∫

∫ e cos xdx = e sin x − x

x

ex sin xdx.

Notice that the second integral looks the same as our original integral in form, except that it has a sin x instead of cos x. To evaluate it, we again apply integration by parts to the second term with u = ex , dv = sin xdx, v = − cos x, and du = ex dx. Then ∫

[ ] ∫ x x e cos xdx = e sin x − −e cos x − (− cos x)(e dx) ∫ x x = e sin x − e cos x − ex cos xdx. x

x

Notice that the unknown integral now appears on both sides of the equation. We can simply move the unknown integral on the right to the left side of the equation, thus adding it to our original integral: ∫ 2

ex cos xdx = ex sin x + ex cos x + C.

Dividing both sides by 2, we obtain www.ck12.org

366

∫

1 1 1 ex cos xdx = ex sin x + ex cos x + C. 2 2 2

Since the constant of integration is just a “dummy” constant, let

C 2

→ C.

Finally, our solution is ∫

1 1 ex cos xdx = ex sin x + ex cos x + C. 2 2

Multimedia Link To see this same ”classic” example worked out with narration, see Khan Academy Indeﬁnite Integration Series Part 7 (9:39) .

Figure 7.1: Another example using integration by parts. http://www.youtube.com/watch? v=F-OsMq7QKEQ&feature=youtube_gdata

Review Questions Evaluate the following integrals. (Remark: Integration by parts is not necessarily a requirement to solve the integrals. In some, you may need to use u−substitution along with integration by parts.) 1. 2. 3. 4. 5. 6.

∫ ∫ ∫ ∫ ∫ ∫

3xex dx x2 e−x dx ln(3x + 2)dx sin−1 xdx sec3 xdx 2x ln(3x)dx

367

www.ck12.org

∫ 2 7. (lnx) dx x 8. Use both the method of u−substitution and the method of integration by∫ parts √ to integrate the integral below. Both methods will produce equivalent answers. x 5x − 2dx ∫ 9. Use the method of tabular integration by parts to solve x2 e5x dx. ∫1 10. Evaluate the deﬁnite integral 0 x2 ex dx. ∫3 11. Evaluate the deﬁnite integral 1 ln(x + 1)dx.

Review Answers 1. 2. 3. 4. 5. 6. 7. 8.

3xex − 3ex + C −e−x (x2 + 2x + 2) + C 3x+2 [ln |3x + 2| − 1] + C 3 √ x sin−1 x + 1 − x2 + C 1 (sec x)(tan x) + 12 ln | sec x + tan x| + C 2 x2 ln |3x| − 12 x2 + C 1 (ln x)3 + C 3 2 4 3/2 (5x − 2)5/2 + 75 125 [ ] (5x − 2) + C 2

9. e5x x5 − 10. e − 2 11. 6 ln 2 + 2

7.3

2x 25

+

2 125

+C

Integration by Partial Fractions

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using technique of Integration by Partial Fractions. • Combine the technique of partial fractions with u−substitution to solve various integrals. This is the third technique that we will study. This technique involves decomposing a rational function into a sum of two or more simple rational functions. For example, the rational function

x2 www.ck12.org

x+4 +x−2

368

can be decomposed into

x2

x+4 2 3 = + . +x−2 x+2 x−1

The two partial sums on the right are called partial factions. Suppose that we wish to integrate the rational function above. By decomposing it into two partial fractions, the integral becomes manageable: ∫

) 2 3 + x+2 x−1 ∫ ∫ 1 1 =2 dx + 3 dx x+1 x−1 = 2 ln |x + 1| + 3 ln |x − 1| + C.

x+4 dx = 2 x +x−2

∫ (

To use this method, we must be able to factor the denominator of the original function and then decompose the rational function into two or more partial fractions. The examples below illustrate the method. Example 1: Find the partial fraction decomposition of 2x − 19 . +x−6

x2 Solution:

We begin by factoring the denominator as x2 + x − 6 = (x + 3)(x − 2). Then write the partial fraction decomposition as 2x − 19 A B = + . +x−6 x+3 x−2

x2

Our goal at this point is to ﬁnd the values of A and B. To solve this equation, multiply both sides of the equation by the factored denominator (x + 3)(x − 2). This process will produce the basic equation.

2x − 19 = A(x − 2) + B(x + 3).

369

www.ck12.org

This equation is true for all values of x. The most convenient values are the ones that make a factor equal to zero, namely, x = 2 and x = −3. Substituting x = 2, 2(2) − 19 = A(2 − 2) + B(2 + 3) −15 = 0 + 5B −3 = B Similarly, substituting for x = −3 into the basic equation we get 2(−3) − 19 = A(−3 − 2) + B(−3 + 3) −25 = −5A + 0 5=A We have solved the basic equation by ﬁnding the values of A and B. Therefore, the partial fraction decomposition is 2x − 19 5 3 = − . +x−6 x+3 x−2

x2

General Description of the Method To be able to write a rational function f (x)/g(x) as a sum of partial fractions, must apply two conditions: • The degree of f (x) must be less than the degree of g(x). If so, the rational function is called proper. If it is not, divide f (x) by g(x) (use long division) and work with the remainder term. • The factors of g(x) are known. If not, you need to ﬁnd a way to ﬁnd them. The guide below shows how you can write f (x)/g(x) as a sum of partial fractions if the factors of g(x) are known. A Guide to Finding Partial Fractions Decomposition of a Rational Function 1. To ﬁnd the partial fraction decomposition of a proper rational function, f (x)/g(x), factor the denominator g(x) and write an equation that has the form f (x) = (sum of partial fractions.) g(x) www.ck12.org

370

2. For each distinct factor ax + b, the right side must include a term of the form

A . ax + b 3. For each repeated factor (ax + b)n , the right side must include n terms of the form

A1 A2 A3 An + + + ... + . 2 3 (ax + b) (ax + b) (ax + b) (ax + b)n Example 2: Use the method of partial fractions to evaluate

∫

x+1 dx (x+2)2

.

Solution: According to the guide above (item #3), we must assign the sum of n = 2 partial sums:

x+1 A B = + . 2 (x + 2) (x + 2) (x + 2)2 Multiply both sides by (x + 2)2 :

x + 1 = A(x + 2) + B x + 1 = Ax + (2A + B). Equating the coeﬀicients of like terms from both sides,

1=A 1 = 2A + B. Thus

A = 1. B = −1.

371

www.ck12.org

Therefore the partial fraction decomposition is x+1 1 1 = − . 2 (x + 2) x + 2 (x + 2)2 The integral will become ∫

) 1 1 − x + 2 (x + 2)2 ∫ ∫ 1 1 = dx − dx x+2 (x + 2)2 1 = ln x + 2 + + c, x+1

x+1 dx = (x + 2)2

∫ (

where we have used u−substitution for the second integral. Example 3: ∫ 2 +3x+1 Evaluate x3x3 +2x 2 +x dx . Solution: We begin by factoring the denominator as x(x + 1)2 . Then the partial fraction decomposition is 3x2 + 3x + 1 A B C = + + . x3 + 2x2 + x x x + 1 (x + 1)2 Multiplying each side of the equation by x(x + 1)2 we get the basic equation

3x2 + 3x + 1 = A(x + 1)2 + Bx(x + 1) + Cx. This equation is true for all values of x. The most convenient values are the ones that make a factor equal to zero, namely, x = −1 and x = 0. Substituting x = −1, 3(−1)2 + 3(−1) + 1 = A(−1 + 1)2 + B(−1)(−1 + 1) + C(−1) 1=0+0−C −1 = C. www.ck12.org

372

Substituting x = 0,

3(0)2 + 3(0) + 1 = A(0 + 1)2 + B(0)(0 + 1) + C(0) 1=A+0+0 1 = A. To ﬁnd B we can simply substitute any value of x along with the values of A and C obtained. Choose x = 1:

3(1)2 + 3(1) + 1 = A(1 + 1)2 + B(1)(1 + 1) + C(1) 7 = 4 + 2B − 1 2 = B. Now we have solved for A, B, and C. We use the partial fraction decomposition to integrate. ∫

) 1 2 1 + − dx x x + 1 (x + 1)2 1 + C. = ln x + 2ln x + 1 + x+1

3x2 + 3x + 1 dx. = x3 + 2x2 + x

∫ (

Example 4: This problem is an example of an improper rational function. Evaluate the deﬁnite integral ∫ 1

2

x3 − 4x2 − 3x + 3 dx. x2 − 3x

Solution: This rational function is improper because its numerator has a degree that is higher than its denominator. The ﬁrst step is to divide the denominator into the numerator by long division and obtain

−6x + 3 x3 − 4x2 − 3x + 3 = (x − 1) + 2 . 2 x − 3x x − 3x

373

www.ck12.org

Now apply partial function decomposition only on the remainder, −6x + 3 A B −6x + 3 = = + . 2 x − 3x x(x − 3) x x−3 As we did in the previous examples, multiply both sides by x(x − 3) and then set x = 0 and x = 3 to obtain the basic equation

−6x + 3 = A(x − 3) + Bx For x = 0,

3 = −3A + 0 −1 = A. For x = 3,

−18 + 3 = 0 + 3B −15 = 3B −5 = B. Thus our integral becomes ∫ 1

2

x3 − 4x2 − 3x + 3 dx = x2 − 3x

∫ 1

2

] ] [ ∫ 2[ −6x + 3 1 5 dx = dx. (x − 1) + 2 x−1− − x − 3x x x−3 1

Integrating and substituting the limits, [

] 2 x2 = − x − ln x − 5ln x − 3 2 1 ) ( ) ( 1 4 − 2 − ln2 − 5ln1 − − 1 − ln1 − 5ln2 = 2 2 1 = 4ln2 − . 2 www.ck12.org

374

Review Questions Evaluate the following integrals. ∫ 1. ∫ x21−1 dx x 2. x2 −2x−3 dx ∫ 1 3. x3 +x2 −2x dx ∫ 3 4. x2x+4 dx ∫1 ϕ 5. 0 1+ϕ dϕ ∫ 5 x−1 6. 1 x2 (x+1) dx 7. Evaluate the by making the proper u−substitution to convert to a rational ∫ integral cosθ function: sin2 θ+4 sin θ−5 dθ. 8. Evaluate the by making the proper u−substitution to convert to a rational ∫ 3eintegral θ function: e2θ −1 dθ. 9. Find the area under the curve y = 1/(2 + ex ), over the interval [−ln3, ln4]. (Hint: make a u−substitution the integrand into a rational function.) ∫ 1 to convert 1 a+x 10. Show that a2 −x2 dx = 2a ln a−x + C.

Review Answers C 1. 12 ln x−1 x+1 + 2. 14 ln x + 1 + 34 ln x − 3 + C 3. 16 ln x + 2 + 13 ln x − 1 − 12 ln x + C 4. 12 x2 − 2ln(x2 + 4) + C 5. 1 − ln2 6. −4 + 2ln 53 ) 5 ( θ−1 7. 16 ln sin +C 5+sin θ 3 θ 8. 2 ln e − 1 + 32 ln eθ + 1 + C 9. 12 ln 14 3 10. Hint: Decompose the integrand into partial fractions.

7.4

Trigonometric Integrals

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using the Trigonometric Integrals. • Combine this technique with u−substitution.

375

www.ck12.org

Integrating Powers of Sines and Cosines In this section we will study methods of integrating functions of the form ∫ sinm xcosn xdx, where m and n are nonnegative integers. The method that we will describe uses the famous trigonometric identities sin2 x = 12 (1 − cos 2x) and cos2 x = 12 (1 + cos 2x). Example 1: ∫ ∫ Evaluate sin2 xdx and cos2 xdx. Solution: Using the identities above, the ﬁrst integral can be written as ∫

∫

2

1 (1 − cos 2x)dx 2 ∫ 1 − (1 − cos 2x)dx = 2 1 1 = (x − sin 2x) + C 2 2 x 1 = − sin 2x + C. 2 4

sin xdx =

Similarly, the second integral can be written as ∫

∫

2

1 (1 + cos 2x)dx 2 ∫ 1 (1 + cos 2x)dx = 2 ) ( 1 1 = x + sin 2x + C 2 2 x 1 = + sin 2x + C. 2 4

cos xdx =

Example 2: www.ck12.org

376

Evaluate

∫

cos4 xdx.

Solution: ∫

∫ (

∫ 4

2

cos xdx =

2

)2 1 (1 + cos 2x) dx 2

(cos x) dx = ∫ 1 = (1 + 2 cos 2x + cos2 2x)dx 4 ) ∫ ( 1 1 1 = 1 + 2 cos 2x + + cos 4x dx 4 2 2 ) ∫ ( 3 1 1 + 2 cos 2x + cos 4x dx. = 4 2 2

Integrating term by term, [ ] 1 3 1 = x + sin 2x + sin 4x + C 4 2 8 1 1 3 sin 4x + C. = x + sin 2x + 8 4 32 Example 3: ∫ Evaluate sin3 xdx. Solution: ∫

∫ 3

sin xdx =

sin2 x sin xdx

Recall that sin2 x + cos2 x = 1, so by substitution, ∫ (1 − cos2 x) sin xdx ∫ ∫ = sin xdx − cos2 x sin xdx. =

The ﬁrst integral should be straightforward. The second can be done by the method of u−substitution by letting u = cos x, so du = − sin xdx. The integral becomes

377

www.ck12.org

] ∫ [ du 2 = − cos x − −u sin x sin x ∫ = − cos x + u2 du u3 +C 3 1 = − cos x + cos3 x + C. 3 = − cos x +

If m and n are both positive integers, then an integral of the form ∫ sinm x cosn xdx can be evaluated by one of the procedures shown in the table below, depending on whether m and n are odd or even. Table 7.2: ∫

sinm x cosn xdx

n odd m odd n and m even

Procedure

Identities

Let u = sin x Let u = cos x Use identities to reduce powers

cos2 x = 1 − sin2 x sin2 x = 1 − cos2 x sin2 x = (1/2)(1 − cos 2x) cos2 x = (1/2)(1 + cos 2x)

Example 4: Evaluate sin3 x cos4 xdx. Solution: Here, m is odd. So according to the second procedure in the table above, let u = cos x, so du = − sin x. Substituting, ∫

∫

3

4

−1 du u4 sin3 x sin x ∫ = − u4 sin2 xdu.

sin x cos xdx =

www.ck12.org

378

Referring to the table again, we can now substitute sin2 x = 1 − cos2 x in the integral: ∫ =−

∫

u4 (1 − cos2 x)du

= − u4 (1 − u2 )du ∫ = (−u4 + u6 )du −1 5 1 7 u + u +C 5 7 1 1 = − cos5 x + cos7 x + C. 5 7 =

Example 5: ∫ Evaluate sin4 x cos4 xdx. Solution: Here, m = n = 4. We follow the third procedure in the table above: ∫

∫ 4

4

(sin2 x)2 (cos2 x)2 dx ]2 [ ]2 ∫ [ 1 1 (1 − cos 2x) (1 + cos 2x) dx = 2 2 ∫ 1 (1 − cos2 2x)2 dx = 16 ∫ 1 = sin4 2xdx. 16

sin x cos xdx =

At this stage, it is best to use u−substitution to integrate. Let u = 2x, so du = 2dx. ∫ 4

4

sin x cos xdx = = = =

∫ 1 sin4 udu 32 ]2 ∫ ∫ [ 1 1 1 2 2 (sin u) du = (1 − cos 2u) du 32 32 2 ) ( 1 3 1 1 u − sin 2u + sin 4u + C 32 8 4 32 3 1 1 x− sin 4x + sin 8x + C. 256 128 1024

379

www.ck12.org

Integrating Powers of Secants and Tangents In this section we will study methods of integrating functions of the form ∫ tanm x secn xdx, where m and n are nonnegative integers. However, we will begin with the integrals ∫ tan xdx and ∫ sec xdx. The ﬁrst integral can be evaluated by writing ∫

∫ tan xdx =

sin x dx. cos x

Using u−substitution, let u = cos x, so du = − sin xdx. The integral becomes ∫

∫

sin x −1 u sin x ∫ 1 du = − du = − ln |u| + C u = − ln | cos x| + C = ln(1/| cos x|) + C = ln | sec x| + C.

tan xdx =

The second integral

∫

sec xdx, however, is not straightforward—it requires a trick. Let ∫

∫

sec x + tan x dx sec x + tan x ∫ sec2 x + sec x tan x dx. = sec x + tan x

sec xdx =

www.ck12.org

sec x

380

Use u−substitution. Let u = sec x + tan x, then du = (sec2 x + sec x tan x)dx, the integral becomes, ∫

∫

du u = ln |u| + C = ln | sec x + tan x| + C.

sec xdx =

There are two reduction formulas that help evaluate higher powers of tangent and secant: ∫

∫ secn−2 x tan x n − 2 sec xdx = + secn−2 xdx, n−1 n−1 ∫ ∫ tanm−1 x m tan xdx = − tanm−2 xdx. m−1 n

Example 6: ∫ Evaluate sec3 xdx. Solution: We use the formula above by substituting for n = 3. ∫

∫ sec x tan x 3 − 2 sec xdx + sec xdx = 3−1 3−1 ∫ 1 1 sec xdx = sec x tan x + 2 2 1 1 = sec x tan x + ln | sec x + tan x| + C. 2 2 3

Example 7: ∫ Evaluate tan5 xdx. Solution: We use the formula above by substituting for m = 5. ∫

tan4 x tan xdx = − 4 5

381

∫ tan3 xdx. www.ck12.org

We need to use the formula again to solve the integral ∫

∫

tan3 xdx:

∫ tan4 x tan xdx = − tan3 xdx 4 [ ] ∫ tan4 x tan2 x = − − tan xdx 4 2 1 1 = tan4 x − tan2 x − ln | cos x| + C. 4 2 5

If m and n are both positive integers, then an integral of the form ∫ tanm x secn xdx can be evaluated by one of the procedures shown in the table below, depending on whether m and n are odd or even. Table 7.3: ∫

tanm x secn xdx

n even m odd m even n odd

Procedure

Identities

Let u = tan x Let u = sec x Reduce powers of sec x

sec2 x = tan2 x + 1 tan2 x = sec2 x − 1 tan2 x = sec2 x − 1

Example 8: ∫ Evaluate tan2 x sec4 xdx. Solution: Here n = 4 is even, and so we will follow the ﬁrst procedure in the table above. Let u = tan x, so du = sec2 xdx. Before we substitute, split oﬀ a factor of sec2 x. ∫

∫ 2

4

tan x sec xdx =

tan2 x sec2 x sec2 xdx.

Since sec2 x = tan2 x + 1, www.ck12.org

382

∫ tan2 x(tan2 x + 1) sec2 xdx.

= Now we make the u−substitution:

∫ =

u2 (u2 + 1)du

1 1 = u5 + u3 + C 5 3 1 1 = tan5 x + tan3 x + C. 5 3 Example 9: ∫ Evaluate tan3 x sec3 xdx. Solution: Here m = 3 is odd. We follow the third procedure in the table. Make the substitution, u = sec x and du = sec x tan xdx. Our integral becomes ∫

∫ 3

3

tan2 x sec2 x(sec x tan x)dx

tan x sec xdx = ∫ = ∫ =

(sec2 x − 1) sec2 x(sec x tan x)dx (u2 − 1)u2 du

1 1 = u5 − u3 + C 5 3 1 1 = sec5 x − sec3 x + C. 5 3

Review Questions Evaluate the integrals. ∫ 1. ∫ cos4 x sin xdx 2. ∫ sin2 5ϕdϕ 3. ∫ sin2 2z cos3 2zdz 4. sin x cos(x/2)dx

383

www.ck12.org

∫ 4 3 ∫ sec 4 x tan xdx √ x sec xdx ∫ tan tan x sec4 xdx ∫ π/2 tan5 x2 dx 0 Graph and then ﬁnd the volume of the solid that results when the region enclosed by y = sin x, y = cos∫x, x = 0, and x = π/4 is revolved around the x−axis. 10. (a) Prove that csc xdx = − ln | csc x + cot x| + C ∫ (b) Show that it can also be written in the following two forms: csc xdx = ln | tan 12 x|+ C = ln | csc x − cot x| + C. 5. 6. 7. 8. 9.

Review Answers 1. 15 cos5 x + C 1 2. 12 ϕ − 20 sin 10ϕ + C 1 1 3 3. 6 sin 2z − 10 sin5 2z + C 4. −1 cos( 3x ) − cos( x2 ) + C 3 2 5. 17 sec7 x − 15 sec5 x + C 6. 14 sec3 x tan x − 58 sec x tan x + 38 ln | sec x + tan x| + C 7. 23 tan3/2 x + 27 tan7/2 x + C 8. −1 + ln 2 2 9. V = π/2 10. .

7.5

Trigonometric Substitutions

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using technique of Trigonometric Substitution. • Combine this technique with other integration techniques to integrate. √ √ 2 − x2 , a x2 − a2 , and When we are faced with integrals that involve radicals of the forms √ 2 2 x + a , we may make substitutions that involve trigonometric functions to eliminate the radical. For example, to eliminate the radical in the expression √

www.ck12.org

a2 − x 2

384

we can make the substitution

x = a sin θ,

−π/2 ≤ θ ≤ π/2, (Note: θ must be limited to the range of the inverse sine function.) which yields, √

a2

−

x2

=

√

a2

−

√ a2

2

sin θ =

√ = a cos2 θ = a cos θ.

a2 (1 − sin2 θ)

The reason for the restriction −π/2 ≤ θ ≤ π/2 is to guarantee that sin θ is a one-to-one function on this interval and thus has an inverse. The table below lists the proper trigonometric substitutions that will enable us to integrate functions with radical expressions in the forms above. Table 7.4: Expression in Integrand √ 2 2 √a − x 2 2 √a + x x2 − a2

Substitution

Identity Needed

x = a sin θ x = a tan θ x = a sec θ

1 − sin2 θ = cos2 θ 1 + tan2 θ = sec2 x sec2 θ − 1 = tan2 θ

In the second column are listed the most common substitutions. They come from the reference right triangles, as shown in the ﬁgure below. We want any of the substitutions we use in the integration to be reversible so we can change back to the original variable afterward. The right triangles in the ﬁgure below will help us reverse our substitutions.

385

www.ck12.org

Description: 3 triangles. Example 1: ∫ . Evaluate x2 √dx 4−x2 Solution: Our goal ﬁrst is to eliminate the radical. To do so, look up the table above and make the substitution

x = 2 sin θ, −π/2 ≤ θ ≤ π/2, so that

dx = 2 cos θ dθ Our integral becomes ∫

∫

2 cos θdθ √ (2 sin θ)2 4 − 4 sin2 θ ∫ 2 cos θdθ = (2 sin θ)2 (2 cos θ) ∫ dθ 1 = 2 4 ∫ sin θ 1 csc2 θdθ = 4 1 = − cot θ + C. 4

dx √ = x2 4 − x2

Up to this stage, we are done integrating. To complete the solution however, we need to express cot θ in terms of x. Looking at the ﬁgure of triangles above, we can √ see that the second triangle represents our case, with a = 2. So x = 2 sin θ and 2 cos θ = 4 − x2 , thus √ cot θ =

4 − x2 , x

since www.ck12.org

386

cot θ =

cos θ . sin θ

so that ∫

dx 1 √ = − cot θ + C 4 x 2 4 − x2 √ 1 4 − x2 =− + C. 4 x

Example 2: ∫ √x2 −3 Evaluate dx. x Solution: Again, √ we want to ﬁrst to √ eliminate the radical. Consult the table above and substitute x = 3 sec θ. Then dx = 3 sec θ tan θdθ. Substituting back into the integral, ∫ √

x2 − 3 dx = x =

∫ √ √

3 sec2 θ − 3 √ √ 3 sec θ tan θdθ 3 sec θ ∫

3

tan2 θdθ.

Using the integral identity from the section on Trigonometric Integrals, ∫

tanm−1 x tan xdx = − m−1 m

∫ tanm−2 xdx.

and letting m = 2 we obtain ∫ √

√ x2 − 3 dx = 3 tan θ − θ + C. x

√ Looking at the triangles above, the third triangle represents our case, with a = 3. So √ √ √ √ 2 x = 3 sec θ and thus cos x = 3/x, which gives tan θ = x − 3/ 3. Substituting,

387

www.ck12.org

∫ √

√ x2 − 3 dx = 3 tan θ − θ + C x (√ 2 ) √ x −3 −1 2 √ = x − 3 − tan + C. 3

Example 3: ∫ Evaluate x2 √dx . x2 +1 Solution: From the table above, let x = tan θ then dx = sec2 θdθ. Substituting into the integral, ∫

dx √ = x2 x2 + 1

∫

sec2 θdθ √ . tan2 θ tan2 θ + 1

But since tan2 θ + 1 = sec2 θ, ∫

sec2 θdθ tan2 θ sec θ ∫ sec θ = dθ tan2 θ ∫ 1 cos2 θ = dθ cos θ sin2 θ ∫ = cot θ csc θdθ. =

Since

d (csc θ) dθ

= − cot θ csc θ, ∫

∫ dx √ = cot θ csc θdθ x2 x2 + 1 = − csc θ + C.

Looking at the triangles above, the ﬁrst triangle represents our case, with a = 1. So x = tan θ √ x 1+x2 √ and thus sin x = 1+x2 , which gives csc θ = x . Substituting, ∫ x2

www.ck12.org

dx √ = − csc θ + C x2 + 1 √ 1 + x2 =− + C. x

388

Review Questions Evaluate the integrals. ∫√ 1. ∫ 4 − x2 dx 1 2. √9+x dx ∫ x3 2 3. √1−x2 dx ∫ 1 4. √1−9x dx ∫ 3√ 2 5. ∫ x 4 − x2 dx 6. x2 √x12 −36 dx ∫ 1 7. (x2 +25) 2 dx ∫ 4 3√ 8. 0 x 16 − x2 dx √ ∫0 9. −π ex 1 − e2x dx (Hint: First use u−substitution, letting u = ex ) 10. Graph and then ﬁnd the area of the surface generated by the curve y = x2 from x = 1 to x = 0 and revolved about the x−axis.

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

7.6

√ ( ) 2 sin−1 x2 + 21 x 4 − x2 + C √ ln( 13√| 9 + x2 + x|) + C √ −x2 1−x2 − 23 1 − x2 + C 3 1 sin−1 3x + C 3 − 43√(4 − x2 )3/2 + 15 (4 − x2 )5/2 + C 1 x2 −36 +C 36 x x 1 1 + 250 tan−1 (x/5) + C 50 x2 +25 2048 15 [ 1 π 2 2

√ ] − sin−1 (e−π ) − e−π 1 − e−2π √ ] [ √ Surface area is π3 18 5 − ln(2 + 5)

Improper Integrals

Learning Objectives A student will be able to: • Compute by hand the integrals of a wide variety of functions by using the technique of Improper Integration. • Combine this technique with other integration techniques to integrate.

389

www.ck12.org

• Distinguish between proper and improper integrals. The concept of improper integrals is an extension to the concept of deﬁnite integrals. The reason for the term improper is because those integrals either • include integration over inﬁnite limits or • the integrand may become inﬁnite within the limits of integration. ∫b We will take each case separately. Recall that in the deﬁnition of deﬁnite integral a f (x)dx we assume that the interval of integration [a, b] is ﬁnite and the function f is continuous on this interval.

Integration Over Inﬁnite Limits If the integrand f is continuous over the interval [a, ∞), then the improper integral in this case is deﬁned as ∫

∫

∞

f (x)dx = lim a

l→∞

l

f (x)dx. a

If the integration of the improper integral exists, then we say that it converges. But if the limit of integration fails to exist, then the improper integral is said to diverge. The integral above has an important geometric interpretation that you need to keep in mind. Recall that, ∫b geometrically, the deﬁnite integral a f (x)dx represents the area under the curve. Similarly, ∫l the integral a f (x)dx is a deﬁnite integral that represents the area under the curve f (x) over the interval [a, l], as the ﬁgure below shows. However, as l approaches ∞, this area will expand to the area under ∫ ∞ the curve of f (x) and over the entire interval [a, ∞). Therefore, the improper integral a f (x)dx can be thought of as the area under the function f (x) over the interval [a, ∞).

www.ck12.org

390

Example 1: ∫∞ Evaluate 1 dx . x Solution: We notice immediately that the integral is an improper integral because the upper limit of integration approaches inﬁnity. First, replace the inﬁnite upper limit by the ﬁnite limit l and take the limit of l to approach inﬁnity: ∫

∞

1

∫ l dx dx = lim l→∞ 1 x x = lim ln x]l1 l→∞

= lim (ln l − ln 1) l→∞

= lim ln l l→∞

= ∞. Thus the integral diverges. Example 2: ∫∞ . Evaluate 2 dx x2 Solution: ∫

∞

2

dx = lim l→∞ x2

∫

l

dx 2 2 x [ ]l −1 = lim l→∞ x 2 ( ) −1 1 = lim + l→∞ l 2 1 = . 2

Thus the integration converges to 12 . Example 3: ∫ −∞ Evaluate +∞

dx . 1+x2

Solution: What we need to do ﬁrst is to split the integral into two intervals (−∞, 0] and [0, +∞). So the integral becomes

391

www.ck12.org

∫

+∞

−∞

dx = 1 + x2

∫

0

−∞

dx + 1 + x2

∫

+∞

0

dx . 1 + x2

Next, evaluate each improper integral separately. Evaluating the ﬁrst integral on the right, ∫

0 −∞

∫

0

dx 2 l 1+x [ −1 ]0 = lim tan x l l→−∞ [ ] = lim tan−1 0 − tan−1 l l→−∞ [ ( π )] π = lim 0 − − = . l→−∞ 2 2

dx = lim 1 + x2 l→−∞

Evaluating the second integral on the right, ∫

∞ 0

dx = lim 1 + x2 l→∞

∫

l

dx 2 0 1+x [ −1 ]l = lim tan x 0 l→∞ π π = −0= . 2 2

Adding the two results, ∫

+∞

−∞

dx π π = + = π. 2 1+x 2 2

Remark: In the previous example, we split the integral at x = 0. However, we could have split the integral at any value of x = c without aﬀecting the convergence or divergence of the integral. The choice is completely arbitrary. This is a famous thoerem that we will not prove here. That is, ∫

∫

+∞

∫

c

f (x)dx = −∞

www.ck12.org

f (x)dx + −∞

392

+∞

f (x)dx. c

Integrands with Inﬁnite Discontinuities This is another type of integral that arises when the integrand has a vertical asymptote (an inﬁnite discontinuity) at the limit of integration or at some point in the interval of integration. Recall from Chapter 5 in the Lesson on Deﬁnite Integrals that in order for the function f to be integrable, it must be bounded on the interval [a, b]. Otherwise, the function is not integrable and thus does not exist. For example, the integral ∫

4

0

dx x−1

develops an inﬁnite discontinuity at x = 1 because the integrand approaches inﬁnity at this point. However, it is continuous on the two intervals [0, 1) and (1, 4]. Looking at the integral more carefully, we may split the interval [0, 4] → [0, 1) ∪ (1, 4] and integrate between those two intervals to see if the integral converges. ∫

4

0

dx = x−1

∫

1 0

dx + x−1

∫ 1

4

dx . x−1

We next evaluate each improper integral. Integrating the ﬁrst integral on the right hand side, ∫ 0

1

∫ l dx dx = lim− x − 1 l→1 0 x − 1 = lim− [ln |x − 1|]l0 l→1

= lim− [ln |l − 1| − ln | − 1|] l→1

= −∞. The integral diverges because ln(−1) and ln(0) are not deﬁned, and thus there is no reason to evaluate the second integral. We conclude that the original integral diverges and has no ﬁnite value. Example 4: ∫3 . Evaluate 1 √dx x−1 Solution:

393

www.ck12.org

∫

3

1

∫

3

dx x−1 l [ √ ]3 = lim+ 2 x − 1 l l→1 [ √ ] √ = lim+ 2 2 − 2 l − 1 l→1 √ = 2 2.

dx √ = lim x − 1 l→1+

√

√ So the integral converges to 2 2. Example 5: In Chapter 5 you learned to ﬁnd the volume of a solid by revolving a curve. Let the curve be y = xe−x , 0 ≤ x ≤ ∞ and revolving about the x−axis. What is the volume of revolution? Solution:

From the ﬁgure above, the area of the region to be revolved is given by A = πy 2 = πx2 e−2x . Thus the volume of the solid is ∫ V =π

∞

2 −2x

xe

∫ l→∞

0

As you can see, we need to integrate by parts twice: www.ck12.org

l

dx = π lim

394

0

x2 e−2x dx.

∫

2 −2x

xe

∫ x2 −2x dx = − e + xe−2x dx 2 x 1 x2 = − e−2x − e−2x − e−2x + C. 2 2 4

Thus [ 2 ]l x −2x x −2x 1 −2x V = π lim − e − e − e l→∞ 2 2 4 0 [ 2 ]l 2x + 2x + 1 = π lim l→∞ −4e2x 0 [ 2 ] 2l + 2l + 1 1 = π lim − l→∞ −4e2l −4e0 [ 2 ] 2l + 2l + 1 1 = π lim + . l→∞ 4e2l 4 At this stage, we take the limit as l approaches inﬁnity. Notice that the when you substitute 2 +2l+1 inﬁnity into the function, the denominator of the expression 2l−4e 2l , being an exponential function, will approach inﬁnity at a much faster rate than will the numerator. Thus this expression will approach zero at inﬁnity. Hence [ ] 1 π V =π 0+ = , 4 4 So the volume of the solid is π/4. Example 6: ∫ +∞ Evaluate −∞

dx . ex +e−x

Solution: This can be a tough integral! To simplify, rewrite the integrand as

1 ex ex 1 = = = . ex + e−x e−x (e2x + 1) e2x + 1 1 + (ex )2 Substitute into the integral:

395

www.ck12.org

∫

dx = x e + e−x

∫

ex dx. 1 + (ex )2

Using u−substitution, let u = ex , du = ex dx. ∫

∫ dx du = x −x e +e 1 + u2 = tan−1 u + C = tan−1 ex + C.

Returning to our integral with inﬁnite limits, we split it into two regions. Choose as the split point the convenient x = 0. ∫

+∞

−∞

dx = ex + e−x

∫

0

−∞

dx + ex + e−x

∫ 0

+∞

dx . ex + e−x

Taking each integral separately, ∫

0

−∞

∫

0

dx + e−x l [ ]0 = lim tan−1 ex l l→−∞ [ ] = lim tan−1 e0 − tan−1 el

dx = lim x e + e−x l→−∞

ex

l→−∞

π −0 4 π = . 4 =

Similarly, ∫ 0

www.ck12.org

+∞

dx = lim x e + e−x l→∞

∫

1

dx + e−x 0 [ −1 x ]l = lim tan e 0 l→∞ [ ] = lim tan−1 el − tan−1 1 l→∞ π π π = − = . 2 4 4

396

ex

Thus the integral converges to ∫

+∞

−∞

ex

dx π π π = + = . −x +e 4 4 2

Review Questions 1. Determine whether the following integrals are improper. If so, explain why. ∫7 (a) 1 x+2 dx x−3 ∫ 7 x+2 (b) 1 x+3 dx ∫1 (c) ∫0 lnxdx ∞ 1 (d) 0 √x−2 dx ∫ π/4 (e) 0 tan xdx Evaluate the integral or state that it diverges. ∫∞ 1 2. 1 x2.001 dx ] ∫ −2 [ 1 1 3. −∞ x−1 − x+1 dx ∫ 0 5x 4. −∞ e dx ∫5 1 5. 3 (x−3) 4 dx ∫ π/2 6. −π/2 tan xdx ∫1 1 7. 0 √1−x 2 dx 8. The region between the x−axis and the curve y = e−x for x ≥ 0 is revolved about the x−axis. (a) Find the volume of revolution, V. (b) Find the surface area of the volume generated, S.

Review Answers 1. (a) (b) (c) (d) (e) 1 2. 1.001 3. ln 3 4. 15

Improper; inﬁnite discontinuity at x = 3. Not improper. Improper; inﬁnite discontinuity at x = 0. Improper; inﬁnite interval of integration. Not improper.

397

www.ck12.org

5. divergent 6. divergent 7. π2 . 8. (a) V = π/2 √ ] [√ (b) S = π 2 + ln(1 + 2) .

Homework Evaluate the following integrals. ∫√ 1. sin xcos xdx ∫ 2. x tan2 (x2 ) sec2 (x2 )dx ∫ ln3 √ 3. 0 e2x − 1dx ∫∞ 4. 0 x12 dx ∫8 1 5. −1 √ 3 x dx ∫ x2 +x−16 6. (x+1)(x−3) 2 dx 7. Graph and ﬁnd the volume of the region enclosed by the x−axis, the y−axis, x = 2 and y = x2 /(9 − x2 ) when revolved about the x−axis. 8. The Gamma Function, Γ(x), is an improper integral that appears frequently in quantum physics. It is deﬁned as ∫

∞

Γ(x) =

tx−1 e−t dt.

0

The integral converges for all x ≥ 0. a. Find Γ(1). b. Prove that Γ(x + 1) = xΓ(x), for all x ≥ 0. ( ) √ c. Prove that Γ 21 = π. 9. Refer to the Gamma Function deﬁned in the previous exercise to prove that ( ) ∫∞ n (a) 0 e−x dx = Γ n+1 , n ≥ 0 [Hint: Let t = xn ] n ∫1 (b) 0 (lnx)n dx = (−1)n Γ(n + 1), n ≥ 0 [Hint: Let t = − ln x] 10. In wave mechanics, a sawtooth wave is described by the integral www.ck12.org

398

∫

+π/ω

t sin(kωt)dt, −π/ω

where k is called the wave number, ω is the frequency, and t is the time variable. Evaluate the integral.

Answers 1.

3 2

sin2/3 x + C

2.

1 6

tan3 (x2 ) + C

3. ln 3 4. divergent 5.

9 8 2

1 6. ln ˛˛(x−3) ˛˛ + x−3 +C ˛x + 1˛ ( ) 9 7. π 19 − ln5 5 4

8. a. Γ(1) = 1 9. a. Hint: Let t = xn . b. Hint: t = − ln x 10.

2 (kω)2

7.7

sin(kπ)

Ordinary Diﬀerential Equations

General and Particular Solutions Diﬀerential equations appear in almost every area of daily life including science, business, and many others. We will only consider ordinary diﬀerential equations (ODE). An ODE is a relation on a function y of one independent variable x and the derivatives of y with respect to x, i.e. y (n) = F (x, y, y ′ , . . . ., y (n−1) ). For example, y ′′ + (y ′ )2 + y = x. An ODE ∑ is linear if F can be written as a linear combination of the derivatives of y, i.e. (n) y = ai (x)y(i) + r(x). A linear ODE is homogeneous if r(x) = 0. A general solution to a linear ODE is a solution containing a number (the order of the ODE) of arbitrary variables corresponding to the constants of integration. A particular

399

www.ck12.org

solution is derived from the general solution by setting the constants to particular values. For example, for linear ODE of second degree y ′′ + y = 0, a general solution have the form yg = A cos x + B sin x where A, B are real numbers. By setting A = 1 and B = 0, yp = cos x It is generally hard to ﬁnd the solution of diﬀerential equations. Graphically and numerical methods are often used. In some cases, analytical method works, and in the best case, y has an explicit formula in x.

Slope Fields and Isoclines dy We now only consider linear ODE of the ﬁrst degree, i.e. dx = F (x, y). In general, the solutions of a diﬀerential equation could be visualized before trying an analytic method. A solution curve is the curve that represents a solution (in the xy− plane).

The slope ﬁeld of the diﬀerential |eq|uation is the set of all short line segments through each point (x, y) and with slope F (x, y).

An isocline (for constant k) is the line along which the solution curves have the same gradient (k). By calculating this gradient for each isocline, the slope ﬁeld can be visualized; making dy = xy . The isoclines it relatively easy to sketch approxi- mate solution curves. For example, dx x are y = k . Example 1 Consider

dy dx

The solutions are y 3 =

3 2

www.ck12.org

=

x . y2

We brieﬂy sketch the slope ﬁeld as above.

x2 + C.

400

Exercise

dy = 1 − y. Sketch the solution curves 1. Sketch the slope ﬁeld of the diﬀerential equation dx based on it. dy 2. Sketch the slope ﬁeld of the diﬀerential equation dx = y − x. Find the isoclines and sketch a solution curve that passes through (1, 0).

401

www.ck12.org

Diﬀerential Equations and Integration We begin the analytic solutions of diﬀerential equations with a simple type where F (x, y) is dy a function of x only. dx = f (x) is a function of x. Then any antiderivative of f is a solution by the Fundamental Theorem of Calculus: ∫x d f (t) dt = f (x). dx a dy Example 1 Solve the diﬀerential equation dx = x with y(0) = 1. ∫ 2 Solution. y = x dx = x2 + C. Then y(0) = 1 gives 1 = 0 + C, i.e. C = 1 Therefore 2 y = x2 + 1. dy 1 = √1+x Example 2 Solve the diﬀerential equation dx 2. ∫ 1 ∫ Solution. We have y = √1+x2 dx and a substitution u = sinh−1 x gives y = ∫ cos h u du ∫ = du = u = sin h−1 x + C. cos h u

√cos h u du 1+sin h2 u

=

Exercise 1. Solve the diﬀerential equation 2. Solve the diﬀerential equation

dy dx dy dx

= =

√

9 − x2 with y(0) = 3. √ 1 . 1+x+x2

Hint: Let u = x + 21 .

Solving Separable First-Order Diﬀerential Equations The next type of diﬀerential equation where analytic solution are rela- tively easy is when the dy = F (x, y) where F (x, y) = f (x)g(y) is dependence of F (x, y) on x and y are separable: dx the product of a functions of x and y respectively. The solution is in the form P (x) = Q(y). Here g(y) is never 0 or the values of y in the solutions will be restricted by where g(y) = 0. Example 1 Solve the diﬀerential equation y ′ = xy with the initial condition y(0) = 1. Solution. Separating x and y turns the equation in diﬀerential form both sides, we have ln |y| = 21 x2 + C.

dy y

= xdx. Integrating

Then y(0) = 1 gives ln |1| = 12 (0)2 + C, i.e. C = 0 and ln |y| = 12 x2 . 1 2

So |y| = e 2 x . 1 2

Therefore, the solutions are y = ±e 2 x . Here Q(y) = y is 0 when y = 0 and the values of y in the solutions satisfy y > 0 or y < 0. Example 2. Solve the diﬀerential equation 2xy ′ = 1 − y 2 . Solution. Separating x and y turns the equation in diﬀerential form www.ck12.org

402

2 dy 1−y 2

=

dx . x

2 A B Resolving the partial fraction 1−y 2 = 1−y + 1+y gives linear equations A + B = 2 and A − B = 0. ( ) 1 1 So 1−y + 1+y dy = dx . Integrating both sides, we have − ln |1 − y| + ln |1 + y| = ln |x| + C x 1+y 1+y C C or ln 1−y = ln(e |x|) = ln D|x| with D = e > 0. Then = ±Dx 1−y = D|x|, i.e. 1+y 1−y where D > 0.

Therefore, the solution has form y = ± Dx+1 where D > 0. Dx−1 Exercise 1. Solve the diﬀerential equation 2. Solve the diﬀerential equation 3. Solve the diﬀerential equation

dy dx dy dx dy dx

= e1y which satisﬁes the condition y(e) = 0. = x(y 2 + 1). = √x . 1−y 2

Exponential and Logistic Growth In some model, the population grows at a rate proportional to the current population without restrictions. The population is given by the diﬀerential equation dP = kP , where k )> 0 are dt ( P growth. In a reﬁned model, the rate of growth is adjusted by another factor 1 − K where K is the carrier capacity. This is close to 1 when P is small compared with K but close to 0 when P is close to K. Both diﬀerential equations are separable and could be solved as in last section. The solutions are respectively: P (t) = P (0)ekt and P (t) =

P0 1+Aekt

with A =

K−P0 . P0

Example 1 (Exponential Growth) The population of a group of immigrant increased from 10000 to 20000 from the end of ﬁrst year to the end of second year they came to an island. Assuming an exponential growth model on the population, estimate the size of the group of initial immigrants. Solution. The population of the group is given by P = P0 ekt where the initial population and relative growth rate are to be determined. At t = 1 (year), P = 10000, so 10000 = P0 ek·1 = P0ek . At t = 2 (year), P = 20000, so 20000 = P0 ek·2 = P0 e2k . Dividing both sides of the second equation by the ﬁrst, we have 2 = ek . Then back in the ﬁrst equation, 10000 = P0 (2). So P0 = 5000. There are 5000 initial immigrants. Example 2 (Logistic Growth) The population on an island is given by the equation dP = dt ) ( P0 0.05P 1 − 5000 , P0 = 1000. Find the population sizes P (20), P (30). At what time will the

403

www.ck12.org

population ﬁrst exceed 4000? Solution. The solution is given by P =

P0 1−Ae0.05t

where A =

5000−1000 1000

= 4.

5000 5000 = = 2023 −0.05(20) 1 + 4e 1 + 4e−1 5000 5000 P (30) = = = 3785. 1 + 4e−0.05(30) 1 + 4e−1.5 P (20) =

(−0.05t) Solve for time, 4000 = 1+4e5000 = −0.05(t) gives e th ﬁrst exceed 4000 in the 56 year.

5000 −1 4000

4

= 0.0625. So t = 56. The population

Exercise 1. (Exponential Growth) The population of a suburban city increased from 10000 in 2005 to 30000 in 2007. Assuming an exponential growth model on the population, by which year will the population ﬁrst exceeds 100000? ( ) P0 2. (Logistic Growth) The population of a city is given by the equation dP = 0.06P 1 − , P0 = dt 100000 25000. Find the population sizes P (10), P (25). At what time will the population ﬁrst exceed 90000?

Numerical Methods (Euler’s, Improved Euler, Runge-Kutta) The Euler’s method is a numerical approximation to a solution curve starting from the point (a, b) through the algorithm: yn+1 = yn + hF (xn , yn ) where x0 = a, y0 = b and h is the step size. The shorter step size, the better is the approximation to the solution curve. Improved Euler (Heun) method adapts on Euler’s method by using both end point values: yn+1 = yn + h2 [F (xn , yn ) + F (xn+1 , yn+1 )]. Since yn+1 also appears on the right side, we replace it by Euler’s formula, h yn+1 = yn + [F (xn , yn ) + F (xn+1 , yn + hF (xn , yn ))]. 2 The Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of our ODE. On them, apply Simpson’s rule: ∫ yn+1 − yn =

xn+1

′

xn +h

f (x)dx = xn

www.ck12.org

∫

xn

404

f ′ (x)dx

h ≈ 6

{ ( ) } h ′ ′ ′ y (xn ) + 4y xn + + y (xn + 1) . 2

Exercise 1. Apply the Euler’s, improved Euler’s and the Runge-Kutta methods on the ODE dy dx

= y to approximate the solution that satisfy y(0) = 1 from x = 0 to x = 1 with h = 0.2.

We know the exact solution is y = ex . Compare their relative accuracy against the exact solution.

405

www.ck12.org

www.ck12.org

406

Chapter 8 Inﬁnite Series This chapter introduces the study of sequences and inﬁnite series. In calculus, we are interested in the behavior of sequences and series, including ﬁnding whether a sequence approaches a number or whether an inﬁnite series adds up to a number. The tests and properties in this chapter will help you describe the behavior of a sequence or series.

8.1

Sequences

Learning Objectives • Demonstrate an understanding of sequences and their terms • Determine if the limit of a sequence exists and, if it exists, ﬁnd the limit • Apply rules, theorems, and Picard’s method to compute the limits of sequences

Sequences (rules, terms, indices) The alphabet, the names in a phone book, the numbered instructions of a model airplane kit, and the schedule in the local television guide are examples of sequences people may use. These examples are all sets of ordered items. In mathematics, a sequence is a list of numbers. You can make ﬁnite sequences, such as 2, 4, 6, 8. These sequences end. You can also make inﬁnite sequences, such as 3, 5, 7, 9, . . . , which do not end but continue on as indicated by the three dots. In this chapter the word sequence refers to an inﬁnite sequence. Each term in a sequence is deﬁned by its place of order in the list. Consider the sequence 3, 5, 7, 9, . . . The ﬁrst term is 3 because it belongs to place 1 of the sequence. The second term is 5 because it belongs to the second place of the sequence. Likewise, The third term is 7 because it is in the third place. Notice that there is a natural relationship between the

407

www.ck12.org

counting numbers, or the positive integers, and the terms of the sequence. This leads us to the deﬁnition of a sequence. Sequence A sequence is a function from the domain of the set of counting numbers, or positive integers, to the range which consists of the members of a sequence. A sequence can be denoted by {an } or by a1 , a2 , a3 , a4 , . . . , an , . . . The numbers a1 , a2 , a3 , a4 , . . . , an , . . . that belong to a sequence are called terms of the sequence. Each subscript of 1, 2, 3, . . . on the terms a1 , a2 , a3 , a4 , . . . refers to the place of the terms in the sequence, or the index. The subscripts are called the indices of the terms. We assume that n = 1, 2, 3, . . . , unless otherwise noted. Instead of listing the elements of a sequence, we can deﬁne a sequence by a rule, or formula, in terms of the indices. Example 1 The formula an =

1 n

is a rule for a sequence.

We can generate the terms for this rule as follows:

n an =

1 1 =1 1

1 n

2 1 2

3 1 3

4 1 3

... ...

Example 2 Consider the sequence rule an =

n2 . n+1

The terms of the sequence are:

n

1 2

an =

n n+1

2 2

1 1 = 1+1 2

2

4 2 = 2+1 3

You can also ﬁnd rule for a sequence. Example 3 Find the rule for the sequence below. www.ck12.org

408

3 9 4

4 16 5

... ...

n an =?

1 1 2

2 −

3 3 4

2 3

4 −

... 4 5

...

Look at each term in terms of its index. The numerator of each term matches the index. The n denominator is one more than the index. So far, we can write the formula an as n+1 . However, we are not done. Notice that each even-indexed term has a negative sign. This means that all of terms of the sequence have a power of −1. The powers of −1 alternate between odd and even. Usually, alternating powers of −1 can be denote by (−1)n or (−1)n+1 . Since the terms nn are negative for even indices, we use (−1)n+1 . Thus, the rule for the sequence is an = (−1) . n+1 (−1)n n You can check the rule by ﬁnding the ﬁrst few terms of the sequence an = n+1 .

Limit of a Sequence We are interested in the behavior of the sequence as the value of n gets very large. Many times a sequence will get closer to a certain number, or limit, as n gets large. Finding the limit of a sequence is very similar to ﬁnding the limit of a function. Let’s look at some graphs of sequences. Example 4 Find the limit of the sequence

{

1 2n+1

}

as n goes to inﬁnity.

Solution 1 for n = 1, 2, 3, . . . . The graph of is similar We can graph the corresponding function y = 2n+1 1 to the continuous function y = 2x+1 for the domain of x ≥ 1.

409

www.ck12.org

To determine the limit, we look at the trend or behavior of the graph of sequence as n gets larger or travels out to positive inﬁnity. This means we look at the points of sequence that correspond to the far right end of the horizontal axis in the ﬁgure on the right. We see that the points of the are getting closer to the horizontal axis, y = 0. Thus, the limit of { 1sequence } 1 the sequence 2n+1 is 0 as n tends to inﬁnity. We write: limn→+∞ 2n+1 = 0. Here is the precise deﬁnition of the limit of a sequence. Limit of a Sequence The limit of a sequence an is the number L if for each ε > 0, there exists an integer N such that | an − L | < ε for all n > N. Recall that | an − L | < ε means the values of an such that L − ε < an < L + ε. What does the deﬁnition of the limit of a sequence mean? Here is another example. Example 5 Look at Figure 3.

www.ck12.org

410

{ } Figure 3 shows the graph of the sequence ln n(n) . Notice that from N on, the terms of lnn n are between L − ε and L + ε. In other words, for this value of ε, there is a value N such that all terms of an are in the interval from L − ε and L + ε. Thus, limn→+∞ ln n(n) = 0. Not every sequence has a limit. Example 6 Here is a graph of the sequence {n + 1}.

Consider the sequence {n + 1} in Figure 4. As n gets larger and goes to inﬁnity, the terms of an = n + 1 become larger and larger. The sequence {n + 1} does not have a limit. We

411

www.ck12.org

write limn→+∞ (n + 1) = +∞.

Convergence and Divergence We say that a sequence {an } converges to a limit L if sequence has a ﬁnite limit L. The sequence has convergence. We describe the sequence as convergent. Likewise, a sequence {an } diverges to a limit L if sequence does not have a ﬁnite limit. The sequence has divergence and we describe the sequence as divergent. Example 7 The sequence {ln (n)} grows without bound as n approaches inﬁnity. Note that the related function y = ln(x) grows without bound. The sequence is divergent because it does not have a ﬁnite limit. We write limn→+∞ ln (n) = +∞. Example 8

{ } The sequence 4 − n8 converges to the limit L = 4 and hence is convergent. If you graph the function y = 4 − n8 for n = 1, 2, 3, . . . , you will see that the graph approaches 4 as n gets 8 larger. Algebraically, as n goes to inﬁnity, ( ) the term − n gets smaller and tends to 0 while 4 8 stays constant. We write limn→+∞ 4 − n = 4. Example 9 Does the sequence sn with terms 1, −1, 1, −1, 1, −1, . . . . have a limit? Solution This sequence oscillates, or goes back and forth, between the values 1 and −1. The sequence does not get closer to 1 or −1 as n gets larger. We say that the sequence does not have a limit, or limn→+∞ sn does not exist. Note: Each sequence’s limit falls under only one of the four possible cases: 1. A limit exists and the limit is L: limn→+∞ sn = L. 2. There is no limit: limn→+∞ sn does not exist. 3. The limit grows without bound in the positive direction and is divergent: limn→+∞ sn = +∞. 4. The limit grows without bound in the negative direction and is divergent: limn→+∞ sn = −∞. If a sequence has a ﬁnite limit, then it only has one value for that limit. Theorem If a sequence is convergent, then its limit is unique. Keep in mind that being divergent is not the same as not having a limit. www.ck12.org

412

L’Hôpital’s Rule Realistically, we cannot graph every sequence to determine if it has a ﬁnite limit and the value of that limit. Nor can we make an algebraic argument for the limit for every possible sequence. Just as there are indeterminate forms when ﬁnding limits of functions, there are indeterminate forms of sequences, such as 00 , ∞ , 0 + ∞. To ﬁnd the limit of such sequences, ∞ we can apply L’Hôpital’s rule. Example 10 Find limn→+∞

ln (n) . n

Solution We solved this limit by using a graph in Example 5. Let’s solve this problem using L’Hôpital’s rule. The numerator is ln (n) and the denominator { } is n. Both functions y = ln (n) and ln (n) ∞ y = n do not have limits. So, the sequence is of the indeterminate form ∞ . Since n the functions y = ln (n) and y = n are not diﬀerentiable, we apply L’Hôpital’s rule to the corresponding problem, limx→+∞ ln x(x) , ﬁrst. Taking the ﬁrst derivative of the numerator and 1

denominator of y = ln x(x) , we ﬁnd limx→+∞ ln x(x) = limx→+∞ 1x = 0. Thus, limn→+∞ ln n(n) = 0 because the points of y = ln n(n) are a subset of the points of the function y = ln x(x) as x approaches inﬁnity. We also conﬁrmed the limit of the sequence with its graph in Example 5.

Rules, Sandwich/Squeeze Properties of function limits are also used with limits of sequences. Theorem (Rules) Let {an } and {bn } be sequences such that limn→+∞ an = L1 and limn→+∞ bn = L2 . Let c be any constant. Then the following statements are true: 1. limn→+∞ c = c The limit of a constant is the same constant. 2. limn→+∞ c × an = c × limn→+∞ an = cL1 The limit of a constant times a sequence is the same as the constant times the limit of the sequence. 3. limn→+∞ (an + bn ) = limn→+∞ an + limn→+∞ = L1 + L2 The limit of a sum of sequences is the same as the sum of the limits of the sequences. 4. limn→+∞ (an × bn ) = limn→+∞ an × limn→+∞ bn = L1 L2

413

www.ck12.org

The limit of the product of sequences is the same as the product of the limits of the sequences. ( ) n→+∞ an 5. If L2 ̸= 0, then limn→+∞ abnn = lim = LL12 . limn→+∞ bn The limit of the quotient of two sequences is the same as the quotient of the limits of the sequences. Let’s apply these rules to help us ﬁnd limits. Example 11 Find limn→+∞

7n . 9n+5

Solution We could use L’Hôpital’s rule or we could use some of the rules in the preceding theorem. Let’s use the rules in the theorem. Divide both the numerator and denominator by the highest power of n in the expression and using rules from the theorem, we ﬁnd the limit:

7n lim = lim n→+∞ 9n + 5 n→+∞

7n n

Dividing both numerator and denominator by n + n5 7 ) Simplifying = lim ( n→+∞ 9 + 5 n limn→+∞ 7 ( ) Applying the division rule for limits. = limn→+∞ 9 + n5 limn→+∞ 7 = Applying the rule for the limit of a sum to the denominator limn→+∞ 9 + limn→+∞ n5 7 7 = = Evaluating the limits 9+0 9

Example 12 Find limn→+∞

( 11 n

−

9n n

8 n2

)

.

Solution ( lim

n→+∞

11 8 − 2 n n

)

11 8 − lim 2 Applying the rule for the diﬀerence of two limits n→+∞ n n→+∞ n 1 1 − 8 lim 2 Applying the rule for the limit of c times a limit = 11 lim n→+∞ n n→+∞ n = 11 × 0 − 8 × 0 = 0 Evaluating the limits

= lim

As with limits of functions, there is a Sandwich/Squeeze Theorem for the limits of sequences. www.ck12.org

414

Sandwich/Squeeze Theorem Let {an } , {bn } and {cn } be sequences. Let N be a positive integer. Suppose cn is a sequence such that an ≤ cn ≤ bn for all n ≥ N . Suppose also that lim an = lim bn = L. Then lim cn = L. You can see how the name of the theorem makes sense from the statement. After a certain point in the sequences, the terms of a sequence cn are sandwiched or squeezed between the terms of two convergent sequences with the same limit. Then the limit of the sequence cn is squeezed to become the same as the limit of the two convergent sequences. Let’s look at an example. Example 13 Prove limn→+∞

8n n!

= 0.

Solution Recall that n! is read as “n factorial” and is written as n! = n × (n − 1) × (n − 2) × . . . × 1. We want to apply the Sandwich theorem by squeezing the sequence that converge to the same limit.

8n n!

between two sequences

n

First, we know that 0 ≤ 8n! . Now we want to ﬁnd a sequence whose terms greater than or n equal to the terms of the sequence 8n! for some n. We can write 8n 8 × 8 × 8... × 8 = n! n × (n − 1) × (n − 2) × . . . 1 8 8 8 8 = × × ... × × n )n(− 1 2 1 )( ( ) 8 8 8 8 8 8 8 8 = × ... × × × × × ... × n n−1 9 8 7 6 5 1 8 Since each factor in the product n−1 × . . . × 89 × 88 is less than or equal to 1, then the product 8 × . . . × 89 × 88 ≤ 1. Then we make an inequality: n−1

( )( )( ) ( ) ( ) 8 8 8 8 8 8 8 8 8 8 8 8 8 × ... × × × × × ... × × × × ... × ≤ (1) n n−1 9 8 7 6 5 1 7 7 6 5 1 ( )( ) 8 8 8 8 8 × × × ... × = 7 7 6 5 1 ( )( 7) 8 8 = n 7!

415

www.ck12.org

Thus, limn→+∞ 0 ≤ limn→+∞

8n n!

have limn→+∞ 0 = 0 and limn→+∞ 0 ≤ limn→+∞

8n n!

( 8 ) ( 87 )

≤ limn→+∞ ( 8 ) ( 87 ) n

n

8 7!

=

7!

( 77!)

. By using the Rules Theorem, we ( 7) 8 limn→+∞ n = 87! × 0 = 0. Thus,

≤ 0. By the Sandwich/Squeeze Theorem, limn→+∞

8n n!

= 0.

Picard’s Method The following method appeared in 1891 by Emile Picard, a famous French mathematician. It is a method for solving initial value problems in diﬀerential equations that produces a sequence of functions which converge to the solution. Start with the initial value problem: y ′ = f (x, y) with y(x0 ) = y0 If f (x, y) and fx (x, y) are both continuous then a unique solution to the initial value problem exists by Picard’s theory. Now if y(x) is the solution to the given problem, then a reformulation of the diﬀerential equation is possible: ∫

x

∫

′

x

y (t)dt =

f (t, y(t))dt

x0

x0

Now the Fundamental Theorem of Calculus is utilized to integrate the left hand side of the problem and upon isolating , the following result is obtained: ∫

x

y(x) = y0 +

f (t, y(t))dt x0

The equation above is the starting point for the Picard iteration because it will be used to build the sequence of functions which will describe the actual solution to the initial value problem. The Picard sequence of functions is calculated as follows: Step 1 - Deﬁne Y0 (x) = y0 Step 2 - Substitute Y0 (t) = y0 for y(t) in f (t, y(t)): ∫

x

Y1 (x) = y0 +

f (t, Y0 (t))dt ∫x0x

Y1 (x) = y0 +

f (t, Y0 )dt x0

Step 3 - Repeat step 2 with Y1 (t) for y(t) : www.ck12.org

416

∫

x

Y2 (x) = y0 +

f (t, Y1 (t))dt x0

The substitution process is repeated n times and generates a sequence of functions {Yn (x)} which converges to the initial value problem. To summarize this procedure mathematically, Picard’s Method Let {Yn (x)} be sequence deﬁned successively by, ∫x Yn (x) = y0 + x0 f (t, Yn−1 (t))dt for n ≥ 0 The sequence of approximations converges to the solution y(x), i.e.

lim Yn (x) = y(x).

n→∞

Now that we have deﬁned Picard’s method, let’s calculate a sequence of functions for an initial value problem. Example 1 Find the ﬁrst four functions {Yn (x)}3n=0 deﬁned by Picard’s method for the solution to the initial value problem y ′ (x) = xy(x) with y(−1) = 1. Solution We want to apply the Fundamental Theorem of Calculus to the diﬀerential equations so that it is reformulated for use in the Picard method. Thus, ∫

x

∫

′

x

y (t)dt = −1

ty(t)dt ∫−1x

y(x) − y(−1) =

ty(t)dt ∫ x y(x) = 1 + ty(t)dt −1

−1

Now that the diﬀerential equation has been rewritten for Picard’s method, we begin the calculations for the sequence of functions. In all cases the ﬁrst function Y0 (x) is given by the initial condition: Step 1 - Deﬁne Y0 (x) = 1

417

www.ck12.org

Step 2 - Substitute Y0 (x) = 1 for y(t) in the integrand of y(x) = 1 + ∫ Y1 (x) = 1 +

∫x −1

ty(t)dt:

x

tdt −1 x 2

t 2

Y1 (x) = 1 +

−1

2

Y1 (x) = Step 3 - Substitute Y1 (x) =

1 2

+

x2 2

1 x + 2 2

for y(t) in the integrand as above: ∫

x

(

1 t2 Y2 (x) = 1 + t + 2 2 −1 x ( 2 ) t t4 Y2 (x) = 1 + + 4 8

) dt

−1

2

Y2 (x) = Step 4 - Substitute Y2 (x) =

5 8

+

x2 4

+

x4 8

4

5 x x + + 8 4 8

for y(t) in the integrand as done previously:

) 5 t2 t4 Y3 (x) = 1 + + + dt t 8 4 8 −1 ( 2 ) x 5t t4 t6 Y3 (x) = 1 + + + 16 16 48 ∫

x

(

−1

2

Y3 (x) =

4

6

29 5x x x + + + 48 16 16 48

Thus, the initial four functions in the sequence deﬁned by Picard’s method are: {

1 x2 1, + , 2 2

5 x2 x4 + + , 8 4 8

29 5x2 x4 x6 + + + 48 16 16 48

}

The method also states that this sequence will converge to the solution y(x) of the initial value problem, i.e. www.ck12.org

418

lim Yn (x) = y(x)

n→+∞

A pattern of the functions in the sequence Yn (x) is emerging but it is not an obvious one. We do know Yn (x) will converge to the solution for this problem by Picard’s method. The exact solution for this problem can be calculated and is given by:

y(x) = e

x2 −1 2

Clearly this solution satisﬁes y(x) = xy(x) and y(−1) = 1.

Review Questions 1. Find the rule for the sequence an . n an =?

1 −2

2 2

3 −2

4 2

... ...

Tell if each sequence is convergent, is divergent, or has no limit. If the sequence is convergent, ﬁnd its limit. { } 2. {n4 + n32 } 3. 6 − √7n 4. −5, { 65, −5, } 5, −5, 5, . . . 4n −7 5. { 3n n } 6. (−1) 5n2 n 7. {(−1) n} { } 3n4 −2 n 8. (−1) 2n4 +6n2 −4n { 2} 9. 6n en 10. Let {an } be a sequence such that limn→+∞ |an | = 0. Show that limn→+∞ an = 0. (|an | is the absolute value of an .) 11. Find the ﬁrst four functions {Yn (x)}3n=0 deﬁned by Picard’s method for the solution to the initial value problem y ′ (x) = 1 + y with y(0) = 0. 12. Find the ﬁrst four functions {Yn (x)}3n=0 deﬁned by Picard’s method for the solution to the initial value problem y ′ (x) = 1 + y 2 with y(0) = 0. 13. Find the ﬁrst three functions {Yn (x)}2n=0 deﬁned by Picard’s method for the solution to the initial value problem y ′ (x) = y 1/3 with y(0) = 18 .

419

www.ck12.org

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

an = (−1)n 2 convergent; Limit is 0 convergent; Limit is 6 No limit exists. divergent convergent; Limit is 0 No limit exists. No limit exists. convergent; Limit is 0 By deﬁnition of absolute value, −|an | ≤ an ≤ |an | . Then take limits of all three terms: lim (−1|an |) ≤ lim an ≤ lim |an |

n→+∞

n→+∞

n→+∞

− lim (|an |) ≤ lim an ≤ lim |an | n→+∞

n→+∞

n→+∞

0 ≤ lim an ≤ 0 n→+∞

By the Sandwich/Squeeze Theorem, limn→+∞ an = 0 also. { } 2 2 3 11. {Yn (x)}3n=0 = 0, x, x + x2 , x + x2 + x6 { } 3 3 5 x7 12. {Yn (x)}3n=0 = 0, x, x + x3 , x + x3 + 2x + 15 { }63 ( ) 4/3 2 1 13. {Yn (x)}n=0 = 18 , 18 + x2 , 32 + 32 18 + x2

Keywords 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

sequence rules terms index, indices limit convergence divergence L’Hôpital’s Rule Sandwich/Squeeze Theorem Picard’s Method

www.ck12.org

420

8.2

Inﬁnite Series

Learning Objectives • • • •

Demonstrate an understanding of series and the sequence of partial sums Recognize geometric series and determine when they converge or diverge Compute the sum of a convergent geometric series Determine convergence or divergence of series using the nth-Term Test

Inﬁnite Series (series, sequence of partial sums, convergence, divergence) Series Another topic that involves an inﬁnite number of terms is the topic of inﬁnite series. We can represent certain functions and numbers with an inﬁnite series. For example, any real number that can be written as a non-terminating decimal can be represented as an inﬁnite series. Example 1 The rational number an inﬁnite series:

4 9

can be written as 0.44444…. We can expand the decimal notation as

4 = 0.4 + 0.04 + 0.004 + 0.0004 + . . . 9 4 4 4 4 + + + + ... = 10 100 1000 10, 000 4 4 4 4 = + 2 + 3 + 4 + ... 10 10 10 10 On the other hand, the number we get a ﬁnite series:

1 4

can be written as 0.25. If we expand the decimal notation,

1 = 0.2 + 0.05 4 2 5 = + 10 100 2 5 = + 2 10 10

421

www.ck12.org

Do you see the diﬀerence between an inﬁnite series and a ﬁnite series? Let’s deﬁne what we mean by an inﬁnite series. Inﬁnite Series An inﬁnite series is the sum of an inﬁnite number of terms, u1 , u2 , u3 , u4 , . . . , usually written as.

u1 + u2 + u3 + u4 + . . . . A shorthand notation for an inﬁnite series is to use sigma notation: ∑∞ k=1 uk , which can be read as “the sum of the terms uk ’s for k equal to 1 to inﬁnity.” We can make ﬁnite sums from the terms of the inﬁnite series:

s 1 = u1 s 2 = u1 + u2 s 3 = u1 + u2 + u3 The ﬁrst sum is the ﬁrst term of the sequence. The second sum is the sum of the ﬁrst two terms. The third term is the sum of the ﬁrst three terms. Thus, the nth ﬁnite sum, sn is the sum of the ﬁrst n terms of the inﬁnite series: sn = u1 + u2 + u3 + . . . + un .

Sequence of Partial Sums As you can see, the sums sn = u1 + u2 + u3 + . . . + un form a sequence. The sequence is very important for the study of the related inﬁnite series for it tells a lot about the inﬁnite series. Partial Sums

∑ For an inﬁnite series∑ ∞ k=1 uk , the nth partial sum, sn is the sum of the ﬁrst n terms of the n inﬁnite series: sn = k=1 uk . The sequence {sn } formed from these sums is called the sequence of partial sums. Example 2 Find the ﬁrst ﬁve partial sums of the inﬁnite series 1 + 0.1 + 0.01 + 0.001 + …. Solution www.ck12.org

422

s1 s2 s3 s4 s5

= u1 = 1 = u1 + u2 = 1 + 0.1 = 1.1 = 1 + 0.1 + 0.01 = 1.11 = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1.1111 = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1.1111

Multimedia Link To further explore series, try experimenting with this applet. The applet shows the terms of a series as well as selected partial sums of the series. Series Applet. As you see from this applet, for some series the partial sums appear to approach a ﬁxed number, while for other series the partial sums do not. Exploring this phenomenon is the topic of the next sections.

Convergence and Divergence Just as with sequences, we can talk about convergence and divergence of inﬁnite series. It turns out that the convergence or divergence of an inﬁnite series depends on the convergence or divergence of the sequence of partial sums. Convergence/Divergence of Series ∑ Let ∞ k=1 uk be an inﬁnite series and let {sn } be the sequence of partial ∑∞ sums for the series. If {sn } has a ﬁnite limit l , then the inﬁnite series converges and k=1 uk = l. If {sn } does not have a ﬁnite limit, then the inﬁnite series diverges. The inﬁnite series does not have a sum. Example 3 Does the inﬁnite series 1 + 0.1 + 0.01 + 0.001 + . . . converge or diverge? Solution To make our work easier, write the inﬁnite series 1 + 0.1 + 0.01 + 0.001 + . . . as an inﬁnite series of fractions:

1+

1 1 1 + 2 + 3 + ... 10 10 10

To solve for convergence or divergence of the inﬁnite series, write the formula for the nth ∑n 1 1 partial sum sn = k=1 uk : sn = 1 + 10 + 1012 + 1013 + . . . + 10n−1 . Note that the nth partial sum ends with a power of n − 1 in the denominator because 1 is the ﬁrst term of the inﬁnite series.

423

www.ck12.org

1 1 + 1012 + 1013 + . . . + 10n−1 as it is It is rather diﬀicult to ﬁnd limn→+∞ sn = limn→+∞ 1 + 10 written. We will “work” the sum into a diﬀerent form so that we can ﬁnd the limit of the sequence of partial sums.

First, multiply both sides of the equation sn = 1 +

1 10

+

1 102

+

1 103

+ ... +

1 10n−1

by

1 : 10

) ( 1 1 1 1 1 1 sn = + + + . . . + n−1 1+ 10 10 10 102 103 10 1 1 1 1 1 1 sn = + 2 + 3 + 4 + ... + n 10 10 10 10 10 10 Now we have two equations:

1 1 1 1 + 2 + 3 + . . . + n−1 10 10 10 10 1 1 1 1 1 1 sn = + 2 + 3 + 4 + ... + n 10 10 10 10 10 10 sn = 1 +

Subtract the bottom equation from the top equation to cancel terms and simplifying:

1 1 1 1 1 + 2 + 3 + 4 + . . . + n−1 10 ( 10 10 10 10 ) 1 1 1 1 1 1 − sn = − + + + + ... + n 10 10 102 103 104 10

sn = 1 +

1 1 sn = 1 − n 10 10 9 1 sn = 1 − n 10 10

sn −

Solve for sn by multiplying both sides of the last equation by

sn =

10 1 (1 − n ) 9 10

Now we ﬁnd the limit of both sides: www.ck12.org

424

10 9

:

( ) 1 10 1− n lim sn = lim n→+∞ n→∞ 9 10 ( ) ( ) 10 1 10 10 1 lim 1 − n = lim − lim n→+∞ 9 n→∞ 9 n→+∞ 9 10 10n 10 10 = −0= 9 9 The sum of the inﬁnite series is

10 9

and so the series converges.

Geometric Series The geometric series is a special kind of inﬁnite series whose convergence or divergence is based on a certain number associated with the series. Geometric Series A geometric series is an inﬁnite series written as

a + ar + ar2 + ar3 + . . . + ar{i−1} + . . . . In sigma notation, a geometric series is written as

∑∞ k=1

ark−1 .

The number r is the ratio of the series. Example 4 Here are some examples of geometric series. Table 8.1: Geometric Series

a

r

1 1+ 14 + 412 + 413 +. . .+ 4k−1 +. . . (−1)k 5 5 5 5 − 6 + 62 + 63 +. . .+ 6k +. . . 1+3+32 +33 +. . .+3k−1 +. . .

1 5 1

1 4

− 61 3

The convergence or divergence of a geometric series depends on r. Theorem Suppose that the geometric series

∑∞ k=1

ark−1 has ratio r.

425

www.ck12.org

1. The geometric series converges if |r| < 1 and the sum of the series is. 2. The geometric series diverges if |r| ≥ 1. Example 5 7 Determine if the series 7 + 78 + 872 + 873 + . . . + 8i−1 + . . . converges or diverges. If it converges, ﬁnd the sum of the series.

Solution

∑∞ ( 1 )k−1 . Then a = 7 and the The series is a geometric series that can be written as k=1 7 8 a ratio r = 18 . Because 18 < 1 , the series converges. The sum of the series is 1−r = 1−7 1 = 7 7 8

8

= 8.

Example 6 Determine if the series series.

∑+∞ k=1

9k−1 converges or diverges. If it converges, ﬁnd the sum of the

Solution The series is a geometric series with a = 1 and the ratio r = 9. Because |9| > 1, the series diverges. Example 7 k

Determine if − 43 + 432 + 433 + . . . + 3(−1) + . . . converges or diverges. If it converges, ﬁnd the 4k sum of the series. Solution If we rewrite the series in terms of powers of k, the series looks like this: ( )1 ( )2 ( )k 3(−1)1 3(−1)2 3(−1)3 3(−1)k 1 1 1 + + + ... + + ... = 3 − +3 − + ... + 3 − + .... 1 2 3 k 4 4 4 4 4 4 4 It looks like a geometric series with a = 3 and r = − 14 .Since − 41 = 14 < 1, the series converges. However, if we write the deﬁnition of a geometric series for a = 3 and r = − 14 , the series looks like this: ) ( )0 ( )1 ( )2 +∞ ( ∑ 1 1 1 1 k−1 =3 − +3 − +3 − + ... 3 − 4 4 4 4 k=1 ( )1 ( )2 1 1 =3+3 − +3 − + ... 4 4 www.ck12.org

426

1

2

k

3

The original problem, 3(−1) + 3(−1) + 3(−1) + . . . + 3(−1) + . . ., does not have the leading 41 42 43 4k term of 3. This does not aﬀect the convergence but will aﬀect ( the ) sum of the series. We ( 1 )1 1 2 need to subtract 3 from the sum of the series 3 + 3 − 4 + 3 − 4 + . . . to get the sum of 2 3 k 3(−1)1 + 3(−1) + 3(−1) + . . . + 3(−1) + . . .. 41 42 43 4k The sum of the series is:

a 1−r

−3=

3 1−(− 14 )

−3=

3 5 4

−3=

12 5

−3=

12 5

−

15 5

= − 53 .

Other Convergent Series There are other inﬁnite series that will converge. Example 8 Determine if

∑+∞ ( 2 n=1

4

−

2 k+1

)

converges or diverges. If it converges, ﬁnd the sum.

Solution The nth partial sum sn is: n ( ∑ 2

) 2 Sn = − k k+1 k=1 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 2 − + − + − + ... + − = 1 1+1 2 2+1 3 3+1 n n+1 ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 = − + − + − + ... + − 1 2 2 3 3 4 n n+1 We can simplify sn further. Notice that the ﬁrst parentheses has − 22 while the second parentheses has 22 . These will add up to 0 and cancel out. Likewise, the − 23 and 23 will cancel out. Continue in this way to cancel opposite terms. This sum is a telescoping sum, which is a sum of terms that cancel each other out so that the sum will fold neatly like a folding telescope. Thus, we can write the partial sum as

sn = ( Then limn→+∞ sn = limn→+∞ 2 −

2 2 2 − =2− . 1 n+1 n+1 2 n+1

)

= 2 and

∑+∞ ( 2 k=1

k

−

2 k+1

)

= 2.

Other Divergent Series (nth-Term Test) Determining convergence by using the limit of the sequence of partial sums is not always feasible or practical. For other series, it is more useful to apply tests to determine if an inﬁnite

427

www.ck12.org

series converges or diverges. Here are two theorems that help us determine convergence or divergence. Theorem (The nth-Term Test) ∑ If the inﬁnite series ∞ k=1 uk converges, then limk→+∞ uk=0 Theorem ∑ ∑ ∑ If k→+∞ uk ̸= 0 or if k→+∞ uk does not exist, then the inﬁnite series ∞ k=1 uk diverges. The ﬁrst theorem tells us that if an inﬁnite series converges, then the limit of the sequence of terms is 0. The converse is not true: If the limit of the sequence of terms is 0, then the series converges. So, we cannot use this theorem as a test of convergence. The second theorem tells us that if limit of the sequence of terms is not zero, then the inﬁnites series diverges. This gives us the ﬁrst test of divergence: the nth-Term Test or Divergence Test. Note that if the test is applied and the limit of the sequence of terms is 0, we cannot conclude anything and must use another test. Example 9 Determine if

∑∞

k k=1 k+5

converges or diverges.

Solution We can use the nth-Term Test to determine if the series diverges. Then we do not have to check for convergence.

lim

k→+∞

Because limk→+∞ Example 10 Determine if

∑∞

k = lim k + 5 k→+∞

k k+5

8 k=1 k−3

k k k+5 k

̸= 0, the series

= lim

k→+∞

∑∞

k k=1 k+5

1 1+

5 k

=

limk→+∞ 1 limk→+∞ 1 +

5 k

=1

diverges.

converges or diverges.

Solution Using the nth-Term Test, limk→+∞ about convergence or divergence. www.ck12.org

8 k−3

. Since the limit is 0, we cannot make a conclusion

428

Rules for Convergent Series, Reindexing Rules As with sequences, there are rules for convergent inﬁnite series that help make it easier to determine convergence. Theorem (Rules for Convergent Series) ∑ ∑∞ ∑∞ ∑∞ 1. Suppose ∞ k=1 uk and n=1 vk are convergent series with k=1 uk = S1 and k=1 vk = S2 . ∑∞ ∑∞ Then k=1 (uk + vk ) and k=1 (uk − vk ) are also convergent where ∑∞ ∑∞ ∑∞ ∑∞ ∑∞ (u + v ) = (u ) + (v ) = S + S and (u − v ) = k k k k 1 2 k k k=1 k=1 k=1 k=1 k=1 (uk ) − ∑∞ k=1 (vk ) = S1 − S2 (The sum or diﬀerence of convergent series is also convergent.) 2. Let c ̸= 0 be a constant. ∑ ∑∞ ∑∞ Suppose ∞ k=1 uk converges and k=1 uk = S Then k=1 cuk also converges where. ∞ ∑ k=1

If

∑∞ k=1

uk diverges, then

∑∞ k=1

cui = c

∞ ∑

uk = cS

k=1

cuk also diverges.

(Multiplying by a nonzero constant does not aﬀect convergence or divergence.) Example 10 Find the sum of

∑∞ (

2

k=1

3k−1

+

1 8k−1

)

.

Solution

) ∑∞ 2 ∑ ( 2 ∑∞ 1 1 Using the Rules Theorem, ∞ k=1 3k−1 + 8k−1 = k=1 3k−1 + k=1 8k−1 . ∑∞ 2 1 k=1 3k−1 is a convergent geometric series with a = 2 and r = 3 . Its sum is ∑∞ 1 1 k=1 8k−1 is a convergent geometric series with a = 2 and r = 8 . Its sum is ) ∑∞ 2 ∑ ( 2 ∑∞ 1 1 8 29 Then ∞ k=1 3k−1 + 8k−1 = k=1 3k−1 + i=1 8k−1 = 3 + 7 = 7 Example 11 Find the sum of

∑∞ k=1

2

(

5 6k−1

)

2 1− 13

=

2

2 1− 18

=

1

2 3 7 8

=3. =

8 7

.

.

Solution

( 5 ) ∑ ∑∞ 5 ∑∞ 2 By the Rules Theorem, ∞ = ∑ k=1 k=1 6k−1 . The series 2 k=1 6k−1 5 5 5 series with a = 5 and r = 16 . The sum of is ∞ k=1 6k−1 1− 1 = 5 = 6. 6

429

5 6k−1

is a geometric

6

www.ck12.org

Then

∑∞ k=1

2

(

5 6k−1

)

= 2 × 6 = 12.

Adding or subtracting a ﬁnite number of terms from an inﬁnite series does not aﬀect convergence or divergence. Theorem ∑ ∑∞ If ∞ k=1 uk converges, then k=1 uk + (u1 + u2 + . . . + um ) is also convergent. ∑∞ ∑∞ If k=1 uk converges, then k=1 uk − (u1 + u2 + . . . + um ) is also convergent. ∑ ∑∞ ∑∞ Likewise, if ∞ k=1 uk diverges, then k=1 uk + (u1 + u2 + . . . + um ) and k=1 uk − (u1 + u2 + . . . + um ) are also divergent. For a convergent series, adding or removing a ﬁnite number of terms will not aﬀect convergence, but it will aﬀect the sum. Example 12 Find the sum of

) ( − 3 + 35 .

∑∞

3 k=1 5k−1

Solution ∑∞ 3

k=1 5k−1 is a geometric series with ) 15 18 ( ∑ 3 3 Then ∞ k=1 5k−1 − 3 + 5 = 4 − 5

a = 3 and r = 15 . Its sum is

15 4

= − 35

Reindexing Another property of convergent series is that we can reindex a series without changing its convergence. This means we can start the indices of the series with another number other than 1. Keep the terms in order though for reindexing. Example 13 ∑∞ 4 k=1 3k−1 is a convergent geometric series. It can be reindexed by changing the starting position of i and the power of i. The new series is still convergent. ∞ ∞ ∑ ∑ 4 4 = k−1 k−6 3 3 k=1 k=6

You can check that the series on the right is the same series as the one of the left by writing out the ﬁrst few terms for each series. Notice that the terms are still in order.

Review Questions 1. Express the number www.ck12.org

1 11

as an inﬁnite series.

430

∑ (−1)k 2. Find s1 , s2 , s3 and for ∞ . k=1 2k 3 3. Determine if the inﬁnite series 3 + 10 + 1032 + 1033 converges or diverges. 4. What are the values of a and r for the geometric series 3 + 3(2)1 + 3(2)2 + 3(2)3 + . . . .? Determine if each inﬁnite series converges or diverges. If a series converges, ﬁnd its sum. ∑∞ ( 3 )k 5. 5 ∑k=1 +∞ ( 2 )k−1 6. −3 ∑k=1 ∞ k3 7. k=1 k3 −5 ∑∞ 4k+2 8. k=1 9k−1

) ∑ (( 2 )k−1 1 9. Find the sum of ∞ + . k=2 3 5k−1 ∑∞ ∑∞ 10. Suppose U is a convergent series and k k=1 k=1 Vk is a divergent series. Explain why ∑∞ ∑∞ (U + V ) and (U − V ) both diverge. k k k k k=1 k=1 11. Give an example of a geometric series whose sum is −3. 12. Give an example of a telescoping sum whose sum is 4.

Review Answers 1 ¯ = 92 + 94 + 96 + . . . . . . + 92k + . . . 1. 11 = 0.09 10 10 10 10 2. s1 = − 12 s2 = − 12 + 14 = − 14 s3 = − 12 + 14 − 81 = − 83 1 s4 = − 12 + 14 − 81 + 16 = − 58 3 3 3 3 3. sn = 3 + 10 ( + 1032 + 1033 + . 3. . . . . + 10n−13 ) 1 1 s = 10 3 + 10 + 102 + 103 + . . . + 10n−1 10 n 3 3 s = 10 + 1032 + 1033 + 1034 + . . . . . . + 103n 10 n

Then sn = 3 +

3 10

+

3 102

+

3 103

+ ...... +

3 − sn = − 10

(

3 10n−1

3 3 3 3 3 + 2 + 3 + 4 + ...... + n 10 10 10 10 10

)

1 3 sn = 3 − n 10 10 7 1 sn = 3 − n 10 ( 10 ) 10 1 sn = 3− n 7 10 sn −

431

www.ck12.org

Then limn→∞ sn = limn→∞

10 (3 7

−

1 ) 10n

lim sn =

n→∞

10 30 (3) − 7 7

The series converges. 4. a = 3, r = 2 ∑ ( 3 )k−1 is a geometric series with a = 1 and r = 35 .|r| = 53 < 1 and so, 5. The series ∞ k=1 5 the geometric series converges. The sum is 1−1 3 = 52 . 5 ∑ ( 2 )k−1 is a geometric series with a = 1 and r = − 32 .|r| 6. The series +∞ 1 )and − k=1 3 ( ) = 123 <( 23 1 4 8 so, the geometric series converges. The sum is 1− − 2 − 1 − 9 + 27 = 5 − 27 = ( 3) 3 3 23 34 − = − . 5 27 27 ∑ 3 ∞ k3 7. diverges by the nth-Term Test: limk→∞ k3k−3 = limk→∞ 1−1 3 = 1 ̸= 0. k=1 k3 −3 k3 ( 4 )k−1 ∑∞ 4k+2 ∑∞ 3 ∑∞ 4k−1 ∑∞ 3 8. . This is a geometric series with k=1 9k−1 = 4 k=1 4 k=1 9k−1 = k=1 64 9 576 a = 64 and r = 94 . The series converges. The sum is 1−644 = 64 . 5 = 5 ) ( 9 9 ( ) ∑∞ ( 2 )k−1 ∑ ( 2 )k−1 ∑∞ 1 1 9. −3 + 5k−1 = ∞ + k=2 5k−1 k=2 k=2 − 3 ∑∞ ( k=2

Then

− 23

)k−1

∑∞ (( k=2

=

1 1−(− 23 )

)k−1 − 23

+

− (1) = 1

)

5k−1

=

3 5

− 1 = − 52 and

∑∞ (( k=2

)k−1 − 23

∑∞

1 k=2 5k−1

+

∑∞

=

1

k=2 5k−1

)

−1=

1 1− 15

=

2 5

+

1 4

5 4

− 1 = 14 .

3 = − 20

∑ ∑ ∑∞ ∑ 10. Since ∞ is divergent, then ∞ Since ∞ k = +∞ or k = −∞ . ∑ k=1 Vk ∑ k=1 V∑ k=1 V k=1 Uk ∑ ∞ ∞ ∞ is convergent, ∞ U is ﬁnite. Then (U + V ) = U + V k k ∑ k=1 k k=1 k=1 k k=1 k is the ∞ sum of something ﬁnite Thus, k=1 (Uk + Vk ) is inﬁnite and ∑∞ and something ∑∞inﬁnite. ∑ diverges. Likewise, k=1 (Uk − Vk ) = k=1 Uk − ∞ k=1 Vk is the diﬀerence of something ﬁnite and something inﬁnite and hence is still inﬁnite and divergent. ( 5 )k−1 ∑ 11. Sample answer: ∑∞ k=1 2 3 ( ) 4 4 − 12. Sample answer: ∞ k=1 k k+1

Keywords 1. 2. 3. 4. 5.

inﬁnite series sequence of partial sums convergence divergence geometric series

www.ck12.org

432

6. ratio of geometric series 7. nth-Term Test 8. reindexing

8.3

Series Without Negative Terms

Learning Objectives • Demonstrate an understanding of nondecreasing sequences • Recognize harmonic series, geometric series, and p− series and determine convergence or divergence • Apply the Comparison Test, the Integral Test, and the Limit Comparison Test

Nondecreasing Sequences In order to extend our study on inﬁnite series, we must ﬁrst take a look at a special type of sequence. Nondecreasing Sequence A nondecreasing sequence {Sn } is a sequence of terms that do not decrease: S1 ≤ S2 ≤ S3 ≤ . . . ≤ Sn ≤ . . . . Each term is greater than or equal to the previous term. Example 1 5, 10, 15, 20, . . . is a nondecreasing sequence. Each term is greater than the previous term: 5 < 10 < 15 < 20 < . . . . 10, 000, 1000, 100, . . . is not a nondecreasing sequence. Each term is less than the previous term: 10, 000, > 1000, > 100 . . . . 3, 3, 4, 4, 5, 5, . . . is a nondecreasing sequence. Each term is less than or equal to the previous term: 3 ≤ 3 ≤ 4 ≤ 4 ≤ 5 ≤ 5 ≤ . . . . A discussion about sequences would not be complete without talking about limits. It turns out that certain nondecreasing sequences are convergent. Theorem Let {Sn } be a nondecreasing sequence: S1 ≤ S2 ≤ S3 ≤ . . . ≤ Sn ≤ . . . .

lim Sn

n→∞

1. If there is a constant B such that Sn ≤ B for all n, then limn→∞ Sn exists and limn→∞ Sn = L where L ≤ B.

433

www.ck12.org

2. If the constant B does not exist, then limn→∞ Sn = +∞. The theorem says that a bounded, convergent, nondecreasing sequence has a limit that is less than or equal to the bound. If we cannot ﬁnd a bound, the sequence diverges. { n } Example 2 Determine if the sequence 6n+5 converges or diverges. If it converges, ﬁnd its limit. Solution 1 2 3 4 Write the ﬁrst few terms: 11 , 17 , 23 , 29 , . . .. The sequence is nondecreasing. To determine n convergence, we see if we can ﬁnd a constant B such that 6n+5 ≤ B. If we cannot ﬁnd such a constant, then the sequence diverges.

If two fractions have the same numerator but diﬀerent denominators, the fraction with the n n n smaller denominator is the larger fraction. Thus, 6n+5 ≤ 6n = 16 . Then 6n+5 ≤ 61 and, in n fact, limn→∞ 6n+5 = 16 .

Series Without Negative Terms (harmonic, geometric, pseries) There are several special kinds of series with nonnegative terms, i.e., terms that are either positive or zero. We will study the convergence of such series by studying their corresponding sequences of partial sums. Let’s start with the harmonic series: ∞ ∑ 1 1 1 1 = 1 + + + + .... K 2 3 4 k=1

The sequence of partial sums look like this:

S1 = 1 1 2 1 1 S1 = 1 + + 2 3 1 1 1 S1 = 1 + + + 2 3 4

S1 = 1 +

.. . www.ck12.org

434

In order for the harmonic series to converge, the sequence of partial sums must converge. The sequence of partial sums of the harmonic series is a nondecreasing sequence. By the previous theorem, if we ﬁnd a bound on the sequence of partial sums, we can show that the sequence of partial sums converges and, consequently, that the harmonic series converges. It turns out that the sequence of partial sums cannot be made less than a set constant B. We will omit the proof here, but the main idea is to show that the a selected inﬁnite subset of terms of the sequence of partial sums are greater than a sequence that diverges, which implies that the sequence of partial sums diverge. Hence, the harmonic series is not convergent. We can also work with geometric series whose terms are all non-negative. ( 3 )k−1 ∑ Example 3 The geometric series ∞ has all non-negative terms. The sequence of k=1 2 partial sums looks like this:

S1 = 1 3 2 3 9 S1 = 1 + + 2 4 3 9 27 S1 = 1 + + + 2 4 8 S1 = 1 +

.. . Intuitively, we can see that there is no bound on the sequence of partial sums and so the series diverges. This is conﬁrmed by the fact that the ratio of the series, r = 32 , tells us that the geometric series does not converge. Another example of an important series is the p−series: ∑∞ 1 1 1 1 k=1 K p = 1 + 2p + 3p + 4p +, where p > 0. The p−series may look like a harmonic series, but it will converge for certain values of p. Theorem

∑∞

converges for p > 1 and diverges for 0 < p ≤ 1. ∑ √1 Example 4 Determine if ∞ k=1 K converges or diverges.

The p−series

1 k=1 K p

Solution

∑ ∑+∞ √1 Rewrite √1K as 11 to get ∞ k=1 K = k=1 2 K which tells us that the series diverges.

1 K2

1

. The value of p is 21 . This is less than 1,

435

www.ck12.org

Comparison Test Now that we have studied series without negative terms, we can apply convergence tests made for such series. The ﬁrst test we will consider is the Comparison Test. The name of the test tells us that we will compare series to determine convergence or divergence. Theorem (The Comparison Test) ∑ ∑∞ Let ∞ k=1 uk and k=1 vk be series without negative terms. Suppose that u1 ≤ v1 , u2 ≤ v 2 , . . . , ui ≤ v i , . . . . ∑ ∑∞ 1. If ∑∞ k=1 vk converges, then∑ k=1 uk converges. ∞ 2. If ∞ k=1 uk diverges, then k=1 vk diverges. In order to use this test, we must check that for each index k, every uk is less than ∑ or equal to vk . This is the comparison part of the test. If the∑series with the greater terms, ∞ k=1 vk , converges, than the series with the lesser terms , ∞ u , converges. If the lesser series k=1 k diverges, then the greater series will diverge. You can only use the test in the orders given for convergence or divergence. You cannot use this test to say, for example, that if the greater series diverges, than the lesser series also diverges. ∑ 1 Example 5 Determine whether ∞ k=1 K 3 +3 converges or diverges. Solution ∑∞ 1

∑ 1 looks similar to ∞ Test. Begin by k=1 K 3 , so we will try to apply the Comparison ∑∞ ∑ 1 1 1 1 comparing each term. For each k, K 3 +3 is less than or equal to K 3 , so k=1 K 3 +3 ≤ ∞ k=1 K 3 . ∑∞ ∑∞ 1 1 Since k=1 K 3 +3 also k=1 K 3 is a convergent p−series, then, by the Comparison Test, converges. ∑ √1 Example 6 Determine whether ∞ k=1 4 k−5 converges or diverges. k=1 K 3 +3

Solution

∑ ∑∞ 1 √1 √ √1 √1 The series ∞ k=1 4 k−5 is similar to k=1 4 k . Using the Comparison Test, 4 k−5 ≥ 4 k for all ∑∞ 1 k. The series k=1 √ diverges since it is a p−series with p = 14 . By the Comparison Test, 4 k ∑∞ √1 k=1 4 k−5 also diverges.

The Integral Test Another useful test for convergence or divergence of an inﬁnite series without negative terms is the Integral Test. It involves taking the integral of the function related to the formula in the series. It makes sense to use this kind of test for certain series because the integral is the limit of a certain series. Theorem (The Integral Test) www.ck12.org

436

∑∞ ∑∞ Let k=1 uk = k=1 f (k) be a series without negative terms. If f (x) is a decreasing, continuous, non-negative function for x ≥ 1, then: 1. 2.

∫∞

∑∞

k=1 uk converges if and only if

∑∞ k=1

uk diverges if and only if

f (x) converges.

1 ∫∞

f (x) diverges.

1

In the statement of the Integral Test, we assumed that uk is a function f of k. We then changed that function f to be a continuous function of x in order to evaluate the integral of f . If the integral is ﬁnite, then the inﬁnite series converges. If the integral is inﬁnite, the inﬁnite series diverges. The convergence or divergence of the inﬁnite series depends on the convergence or divergence of the corresponding integral. ∑ 1 Example 7 Determine if ∞ 3 converges or diverges. k=1 (2k+1) 2

Solution We can use the Integral Test to determine convergence. Write the integral form: ∫∞

1 3

(2k + 1) 2

1

dx.

Next, evaluate the integral. ∫∞ 1

1

dx = lim (2k + 1)− 2 dx 3

(2k + 1)

3 2

t→∞

Use the following u−substitution to evaluate the integral:

u = 2x + 1 du = 2dx Then

1 2

∫

u− 2 du = 3

∫t

1 u−1/2 2 (−1/2) − 32

Thus, limt→∞ (2x + 1) 1

=−

1 1

u2

dx =

= − √1u .

1 limt→∞ − √2x+1 ]41

Since the integral is ﬁnite, the series

∑∞ k=1

= limt→∞

1 3

(2k+1) 2

437

(

− √13

+

1 − √2t+1

)

= − √13 .

converges by the Integral Test. www.ck12.org

Limit Comparison Test, Simpliﬁed Limit Comparison Test Another test we can use to determine convergence of series without negative terms is the Limit Comparison Test. It is easier to use than the Comparison Test. Theorem (The Limit Comparison Test) ∑ Suppose ∞ k=1 uk is a series without negative terms. Then one of the following will hold. ∑ 1. If ∞ v is a convergent series without negative terms and limk→∞ uvkk is ﬁnite, then ∑∞ k=1 k k=1 uk converges. ∑ 2. If ∞ w is a divergent series without negative terms and limk→∞ wvkk is positive, then ∑∞ k=1 k k=1 uk diverges. The Limit Comparison Test says to make a ratio of the terms of two series and compute the limit. This test is most useful for series with rational expressions. ∑ k4 +6k3 −1 Example 8 Determine if ∞ k=1 7k5 +K 2 converges or diverges. Solution

∑ k4 +6k3 −1 Just as with rational functions, the behavior of the series ∞ k=1 7k5 +K 2 when k goes to inﬁnity behaves like the series with only the∑highest powers of k in the numerator and ∑∞ ∞ k4 k4 denominator: k=1 7k 5 . We will use the series k=1 7k5 to apply the Limit Comparison Test. ∑ ∑∞ 1 k4 First, when we∑ simplify the ∑ series ∞ k=1 7k5 , we get the series k=1 7k . This is a harmonic ∞ 1 1 1 1 series because ∞ = and the multiplier does not aﬀect the convergence or k=1 k=1 k 7 7 ∑7k ∞ 1 divergence. Thus, k=1 7k diverges. So, we will next check that the limit of the ratio of the terms of the two series is positive: k2 +6k3 −1 5 2 lim 7k 1+k k→∞ 7k

7k 4 + 42k 3 − 7 = 1 > 0. k→∞ 7k 4 + K

= lim

Using the Limit Comparison Test, because ∑ k4 +6k3 −1 then ∞ k=1 7k5 +K 2 diverges.

1 7k

diverges and the limit of the ratio is positive,

Unlike the Comparison Test, you do not have to compare the terms of both series. You may just make a ratio of the terms. There is a Simpliﬁed Limit Comparison Test, which may be easier for you to use. Theorem ( The Simpliﬁed Limit Comparison Test) ∑∞ ∑ negative terms. If limk→∞ uvkk is ﬁnite and Suppose ∞ k=1 uk and ∑ k=1 vk are series without ∑∞ ∑∞ ∑ ∞ ∞ positive, then either k=1 vk both k=1 uk and k=1 vk converge or k=1 uk and both diverge. www.ck12.org

438

Example 9 Determine if

∑∞

2 k=1 8k +5

converges or diverges.

Solution ∑∞ 2

is a series without negative terms. To apply the Simpliﬁed Limit Comparison ∑ ∑∞ 2 2 Test, we can compare ∞ k=1 8k +5 with the series k=1 8k , which is a convergent geometric 2 ∑ k k 2 series. Then limk→∞ 8 2+5 = limk→∞ 8k8+5 = 1 > 0. Thus, since ∞ k=1 8k converges, then 8k ∑∞ 2 also converges. k k=1 8 +5 k=1 8k +5

Review Question 1. Write an example of a nondecreasing sequence. 2. Write an example of a sequence that is not nondecreasing. 3. Suppose {Sn } is a nondecreasing sequence such that for each M > 0, there is an N , such that Sn { > M for } all n > N . Does the sequence converge? Explain. 5n2 4. Determine if 2n2 +7 converges or diverges. If it converges, ﬁnd its limit. ∑ ( 1 )k−1 5. Determine if ∞ converges or diverges. If it converges, ﬁnd its sum. k=3 4 Determine if each series converges or diverges. 6. 7. 8. 9. 10. 11.

∑∞ k=1 ∑∞ ∑k=1 ∞ ∑k=1 ∞ k=1 ∑∞ ∑k=1 ∞ k=1

1 1

(4k+1) 2 2 3k5 −4 5 (k+1)(k+3) 7 √ 5 2 k k3 +4k2 +1 3k6 +2k4 1 5

(3k−1) 2

+∞ ∫ 3 ∑ 3 12. Maria uses the integral test to determine if ∞ converges. She ﬁnds that = k=1 k2 x2 1 ∑ 3 3. She then states that ∞ k=1 k2 converges and the sum is 3. What error did she make?

Review Answers 1. Sample answer: 4, 4, 4, 5, 5, 5, 6, 6, 7, . . . 2. Sample answer: 200, 20, 2, . . . 3. No, the nondecreasing sequence is not bounded. For any number M , there is a point in the sequence such that the rest of the sequence is greater than M . The terms of the sequence { 2 } are not bounded by any value of M . 4. 2n5n2 +7 is bounded and nonincreasing. It converges to 52

439

www.ck12.org

5.

∑∞ ( 1 )k−1 k=3

4

5 3 − 4 = 4 ∑∞ 1

6. 7.

4 3 1

−

is a geometric series with a = 1 and r = 41 . The sum is 5 4

=

16−15 12

1 diverges k=1 2 ∑∞ (4k+1) 2 k=1 3k5 −4 converges

=

1 . 12

by Comparison test (using

∑∞

1 1− 14

( ) − 1 + 14 =

1

, a divergent p−series). ∑∞ 2 by the Limit Comparison Test (using k=1 3k5 , which is a conk=1

1

(4k) 2

vergent p−series). ∑ ∑∞ 5 ∑ ∞ ∞ 5 5 k=1 (k−1)(k−3) = k=1 k2 +4k+4 converges by the Limit Comparison Test (using k=1 k2 , which is a convergent p−series). ∑ ∞ √7 9. k=1 5 2 is a divergent p−series. ∑∞ 1 ∑∞ k3k+4k2 +1 converges by the Limit Comparison Test (using 10. 6 4 k=1 3k3 , which is a k=1 3k +2k convergent p−series). ∑∞ 1 11. 5 converges by the Integral Test. k=1 8.

(3k−1) 2

12. Maria is correct that the series converges. She made an error by saying ∑ that3 the value of the related integral gives the sum of the inﬁnite series. However, ∞ k=1 (k2 ) = 3 3 3 + 2 + 9 + . . . is greater than 3.

Keywords 1. 2. 3. 4. 5. 6. 7. 8.

8.4

nondecreasing sequence harmonic series geometric series p−series Comparison Test Integral Test Limit Comparison Test Simpliﬁed Limit Comparison Test

Series With Odd or Even Negative Terms

Learning Objectives • • • •

Demonstrate an understanding of alternating series Apply the Alternating Series Test to an appropriate series Explain the diﬀerence between absolute and conditional convergence Determine absolute and/or conditional convergence of series

www.ck12.org

440

Alternating Series (harmonic, geometric, p-series) Alternating series are series whose terms alternate between positive and negative signs. Generally, alternating series look like one of these expressions: u1 − u2 + u3 − u4 + . . . or −u1 + u2 − u3 + u4 − . . . . Either the terms with the even indices can have the negative sign or the terms with the odd indices can have the negative sign. The actual numbers represented by the u′i s are positive. There are several types of alternating series. One type is the alternating harmonic series: ∞ ∑ k=1

(−1)k+1

1 1 1 1 = 1 − + − + .... k 2 3 4

This series has terms that look like the harmonic series but the terms with even indices have a negative sign. Another kind is the alternating geometric series. Here is one example: ( )k−1 ( )2 ( )2 2 2 2 2 (−1) = −1 + − + − .... 3 3 3 3 k=1

∞ ∑

k

The odd-indexed terms of this series have the negative sign. The alternating p−series is another type of alternating series. An example could look like this: ∞ ∑

1 1 1 1 (−1)k−1 √ +√ −√ − .... =1− √ 3 3 3 3 2 3 4 k k=1

From all of these examples, we can see that the alternating signs depend on the expression in the power of −1 in the inﬁnite series.

The Alternating Series Test As its name implies, the Alternating Series Test is a test for convergence for series who have alternating signs in its terms. Theorem (The Alternating Series Test)

441

www.ck12.org

The alternating series u1 − u2 + u3 − u4 + . . . or −u1 + −u1 + u2 − u3 + u4 − . . . converge if: 1. u1 ≥ u2 ≥ u3 ≥ . . . ≥ uk ≥ . . . and 2. limk→+∞ uk = 0. Take the terms of the series and drop their signs. Then the theorem tells us that the terms of the series must be nonincreasing and the limit of the terms is 0 in order for the test to work. Here is an example of how to use The Alternating Series Test. Example 1 Determine how if

∑∞

k+1 k+5 k=1 (−1) k3 +k

converges or diverges.

Solution

∑ k+1 k+5 The series ∞ is an alternating series. We must ﬁrst check that the terms of k=1 (−1) k3 +k the series are nonincreasing. Note that in order for uk ≥ uuk + 1, then 1 ≥ uuk+1 , or uuk+1 ≤ 1. k k So we can check that the ratio of the (k + 1)st term to the k th term is less than or equal to one.

uk+1 = uk

(k+1)+5 (k+1)3 +(k+1) k+5 k3 +k

=

(k + 1) + 5 k3 + k × (k + 1)3 + (k + 1) k+5

Expanding the last expression, we get: uk+1 (k + 6)(k 3 + k) k 4 + 6k 3 + k 2 + 6k = 3 = . uk (k + 3k 2 + 4k + 2)(k + 5) k 4 + 8k 3 + 19k 2 + 20k + 10 Since k is positive and all the sum of the numerator are part of the denominator’s sum, the numerator is less than the denominator and so, uuk+1 < 1. Thus, uk ≥ uk + 1 for all k. By k ∑∞ k+5 the Alternating Series Test, the series k=1 (−1)k+1 k3 +k converges. Keep in mind that both conditions have to be satisﬁed for the test to prove convergence. However, if the limit condition is not satisﬁed, the inﬁnite series diverges.

Alternating Series Remainder We ﬁnd the sequence of partial sums for an alternating series. A partial sum can be used to approximate the sum of the series. If the alternating series converges, we can actually ﬁnd a bound on the diﬀerence between the partial sum and the actual sum. This diﬀerence, or remainder, is called the error. www.ck12.org

442

Theorem (Alternating Series Remainder) Suppose an alternating series satisﬁes the conditions of the Alternating Series Test and has the sum S. Let sn be the nth partial sum of the series. Then |S − sn | ≤ un+1 . The main idea of the theorem is that the remainder |S − sn | cannot get larger than the n + 1st term in the series, un+1 . This is the term whose index is one more than the index of the partial sum used in the diﬀerence. Example 2 Compute s3 for the series

∑∞

k+1 k+5 k=1 (−1) k3 +k

and determine the bound on the remainder.

Solution First we compute the third partial sum to approximate the sum S of the series: 1+5 2+5 3+5 + (−1)3 3 + (−1)4 3 3 1 +1 2 +2 3 +3 6 7 8 = − + 4 10 30 90 − 42 + 16 64 = = 60 60

s3 = (−1)2

The theorem tells us to use the next term in the series, u4 , to calculate the bound on the diﬀerence or remainder. Remember that the part (−1)k+1 just gives the sign of the term and, so we just use the part kk+5 3 +k to calculate u4 . 9 64 9 Thus u4 = 44+5 3 +4 = 68 . Then |S − sn | = S − 60 < 68 ≈ 0.13. This tells us that the absolute value of the error or remainder is less than 0.13.

Absolute and Conditional Convergence There are other types of convergence for inﬁnite series: absolute convergence and conditional convergence. Absolute Convergence ∑ Let ∞ u1 + u2 + u3 . . . + uk + . . . be an inﬁnite series. Then the series is absolutely k=1 uk = ∑ convergent if ∞ k=1 |uk | = |u1 | + |u2 | + |u3 | + . . . + |uk | + . . . converges. ∑ The inﬁnite series ∞ 3 | + . . . + |uk | + . . . is the series made by taking k=1 |uk | = |u1 | + |u2 | + |u∑ the absolute values of the terms of the series ∞ k=1 |uk | = |u1 | + |u2 | + |u3 | + . . . + |uk | + . . . . The convergence of the series of absolute values tells us something about the convergence of the series. Theorem

443

www.ck12.org

∑ ∑∞ If ∞ k=1 |uk | = |u1 |+|u2 |+|u3 |+. . .+|uk |+. . ., then k=1 |uk | = |u1 |+|u2 |+|u3 |+. . .+|uk |+. . . also converges. This tells us that if you can show absolute convergence, then the series converges. If the series of absolute value diverges, we cannot conclude anything about the series. Example 3 Determine if the series

∑∞

(−1)k+1 k=1 3k4 −k

converges absolutely.

Solution

∑∞ (−1)k+1 ∑∞ 1 We ﬁnd the series of absolute values: k=1 3k4 −k = k=1 3k4 −k , which behaves like the ∑ ∑∞ (−1)k+1 1 series ∞ k=1 3k4 . This is a p−series with p = 4. The series k=1 3k4 −k converges absolutely and hence converges. Example 4 Determine if the series

∑∞ k=1

(−1)k+1 2k+1

converges absolutely.

Solution

∑ (−1)k+1 1 1 1 The series made up of the absolute values of the terms is ∞ k=1 2k+1 = 3 + 5 + 7 + . . . . ∑∞ 1 ∑∞ ∑ 1 k+1 1 This series behaves like k=1 2k = 12 ∞ k=1 (−1) k=1 k , which diverges. The series 2k+1 does not converge absolutely. It is possible to have a series that is convergent, but not a absolutely convergent. Conditional Convergence An inﬁnite series that converges, but does not converge absolutely, is called a conditionally convergent series. Example 5 Determine if

∑∞ k=1

(−1)k+1 k

converges absolutely, converges conditionally, or diverges.

Solution

∑ 1 The series of absolute values is ∞ k=1 k . This is the harmonic series, which does not converge. ∑∞ (−1)k+1 So, the series does not converge absolutely. The next step is to check the k ∑∞ k=1 (−1)k+1 convergence k=1 k . This will tell us if the series converges conditionally. Applying the Alternating Series Test: The sequence 11 > 12 > 13 > . . . is nondecreasing and limk→∞ k1 = 0. ∑ (−1)k+1 converges. Hence, the series converges conditionally, but not absoThe series ∞ k=1 k lutely.

www.ck12.org

444

Rearrangement Making a rearrangement of terms of a series means writing all of the terms of a series in a diﬀerent order. The following theorem explains how rearrangement aﬀects convergence. Theorem ∑ If ∞ k=1 uk is an absolutely convergent series, then the new series formed by a rearrangement of the terms of the series also converges absolutely. This tells us that rearrangement does not aﬀect absolute convergence.

Review Questions Determine if the series converges or diverges. ∑∞ (−1)k+1 3k2k+k ∑k=1 ∞ k+1 3k k=1 (−1) 2k ∑ k+1 4 . Compute s3∑ for ∞ k=1 (−1) k3 ∞ k 5 to the Alternating Series Test. Let The ∑∞ seriesk 5 k=1 (−1) k2 converges∑according ∞ k 5 k=1 (−1) k2 = S. Compute s3 for k=1 (−1) k2 and determine the bound on |s3 −S|. ∑∞ (−1)k+1 ∑ (−1)k+1 5. The series k=1 k! converges according to the Alternating Series Test. Let ∞ = k=1 k! ∑∞ (−1)k+1 S. Compute s4 for k=1 k! and determine the bound on |s4 − S|. 1. 2. 3. 4.

∑ ∑ (−1)k+1 The series ∞ converges according to the Alternating Series Test. Let ∞ k=1 k=1 k S. Find the least value of n such that: ∑ 6. nk=1 ∑ 7. nk=1 ∑ 8. nk=1

(−1)k+1 k (−1)k+1 k (−1)k+1 k

(−1)k+1 k

=

− S < 0.05 − S < 0.005 − S < 0.0001

Determine if each series converges absolutely, converges conditionally, or diverges. ∑∞ (−1)k+1 3k 9. 2k ∑k=1 ∞ (−1)k+1 k 10. k=1 2k2 +2 ∑∞ (−4) k+1 11. k=1 7k2 ∑∞ (−1)k+1 12. 7 k=1 k2

445

www.ck12.org

Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

∑∞ (−1)k+1 3k2k+k converges by the Alternating Series Test. ∑k=1 ∞ k+1 3k converges by the Alternating Series Test. k=1 (−1) 2k 197 s3 = 54 ; 5 s3 = 151 ; |s3 − S| < 16 36 1 s3 = 15 ; |s3 − S| < 120 24 19 199 9, 999 converges absolutely converges conditionally divergent converges absolutely

Keywords 1. 2. 3. 4. 5. 6. 7. 8. 9.

8.5

alternating series alternating harmonic series alternating geometric series alternating p−series Alternating Series Test Alternating Series Remainder conditional convergence absolute convergence rearrangement

Ratio Test, Root Test, and Summary of Tests

Ratio Test We have seen the Integral Test,(Limit) Comparison Test and Alternating Series Test which impose conditions on the sign of an . We are going to introduce two tests for a stronger version of convergence that do not. Deﬁnition ∑(Absolute convergence) ∑ A series an is absolutely convergent if the series of the absolute values |an | is convergent. To this end, we need to distinguish the other type of convergence. www.ck12.org

446

Deﬁnition ∑ (Conditional convergence) A series an is conditionally convergent if the series is convergent but not absolutely convergent.

Theorem If

∑

an is absolutely convergent, then it is convergent.

The proof is quite straightforward and is left as an exercise. The converse of Theorem 5.1.1 is ∑ (−1)n−1 not true. The series is convergent by the Al- ternating Series Test, but its absolute n ∑1 series, (the harmonic series), is divergent. n ∑ cos nθ ∑ 1 Example 1 is absolutely convergent since cosn2nθ ≤ n12 for any 1 ≤ n, θ, and n2 n2 ∑ is convergent (e.g. by the p− test). Indeed, by the Integral and Comparison tests, cosnpnθ is absolutely convergent for any θ and p > 1. ∑ an +1 The limit of the ratio an gives us a comparison of the tail part (i.e n Large an ) of the ∑ series an with a geometric series. ∑ Theorem (The Ratio Test) Let an be a series of non-zero numbers∗ . an +1 (A) If limn→∞ an = α < 1, then the series is absolutely convergent. an +1 (B) If limn→∞ ana+1 = α > 1 or lim = ∞ then the series is absolutely divergent. n→∞ an n (C) If limn→∞ ana+1 = α = 1 then the test is inconclusive. n ∗: we could ignore the zero-valued an ’s as far as the sum is concerned. Proof. (A) The proof is by comparison with a geometric series. α+1 If α < 1, then α < 2 < 1 It follows from the deﬁnition of limit that there is an integer N , an +1 Then |aN +1 | < β|aN |, |aN +2 | < β|aN +1 | < β 2 |aN |, . . . an < for all n ≥ N . Let β = α+1 2 ∑∞ ∑∞ n−N and recursively we have |an+1 | < β n−N +1 |aN | for n ≥ N , and |a | < |a | n N n=N n=N β ∑∞ which is ﬁnite. Combining with the ﬁnitely many terms, n=0 |an | is still ﬁnite.

(B) A similar argument concludes limn→∞ an ̸= 0. So the series is divergent. ∑ An Example 2 Test the series for absolute convergence where A is a constant. n! n An+1 A Solution. Let an = An! . Then an+1 = (n+1)! · An!n = n+1 → 0 as n → ∞. So by the Ratio an Test, the series is absolutely convergent for any constant A. Indeed, the sum is eA which is very large for large A, but still ﬁnite. an+1 We see the limitation of the Ratio Test is when lim an does not exist (not ∞) or is 1.

447

www.ck12.org

1 n+1 1 n

n limn→∞ n+1

1 (n+1)2 1 n2

Example 3 (Ratio Test inconclusive) limn→∞ = = 1, and limn→∞ ∑ ∑ 1 1 limn→∞ aan+1 and The former (harmonic series) diverges while the = 1 for both n n2 N latter converges (by, say, the p−test). Questions (related to the Ratio Test) What if a2n+1 a2n 1. limit of an+1 exist separately for n odd and n even, i.e lim , lim n→∞ n→∞ an a2 n a2 n−1 exist but are diﬀerent? a2n+1 a2n+2 2. limn→∞ a2n−1 , limn→∞ a2n exist but are diﬀerent? Exercise Determine whether the following series is absolutely convergent, conditionally convergent, or divergent with the Ratio Test and other tests if necessary: ∑ n3 1. ∑∞ n=1 2n −2n 2. ∑∞ n! n=1 e ∞ n−1 √1 3. (−1) n=1 n 2 3 4 5 4. 1.3 − 2.4 + 3.5 − 4.6 + ...

nth - root Test If the general term an resembles an exponential expression, the following test is handy. Theorem (The Root Test) √ (A) If limn→∞ n an = α < 1, then the series is absolutely convergent. √ √ (B) If limn→∞ n an = α > 1 or limn→∞ n an = ∞, then the series is absolutely divergent. √ (C) If limn→∞ | n n| = α = 1, then the test is inconclusive. The proof is similar to that of the Ratio Test and is left as an exercise. ∑ ∑ 1 Example 1 Consider an = where p > 0. We already know it is convergent exactly np √ n when p > 1. To apply the Root Test, we need limn→∞ √n which is 1 after some work. = 1 so limn→∞ n n = 1 by the argument similar Alternatively, we could check limn→∞ an+1 an to the proof of the Ratio Test. The Root Test is also inconclusive. ∑ ( n+1 )n Example 2 Test the series for convergence. 2n+3 √ ( n+1 )n n+1 . Then limn→∞ n |an | = limn→∞ 2n+3 = 12 . So the series is Solution. Let an = 2n+3 absolutely absolutely convergent. What if we have applied the Ratio Test? www.ck12.org

448

( n+2 )n+1 ( n+1 )n ( n+2 ) [ (n+2)(2n+3) ]n = lim / = lim limn→∞ an+1 n→∞ 2n+5 n→∞ 2n+5 . (n+1)(2n+5) an 2n+3 We could still argue the limit is test.

1 2

with some work. So we should learn to apply the right

Exercise Determine whether the following series is absolutely convergent, conditionally convergent, or divergent with the Root Test and other tests if necessary: 1. 2. 3.

∑∞ ( n )n n=1 (n2 +1 ∑ n ∞ n=1 ∑∞ n=1

)n

(−1) (ln n) ( nn n) 53+2n limn→∞ an+1 = an

√ α, then limn→∞ | n an | = α. ( 2 )n ∑ n −1 5. (Hard) Is the series ∞ convergent? n=1 n2 +1 4. If

Hint:

(

n2 −1 n2 +1

)n =

√[ ( n2 −1 )n2 ] n n2 +1

and limh→0 (1 + h)1/h exists and equals e.

Summary of Procedures for Determining Convergence We have seen various test for convergence of is ”recognize the form of an ”.

∑

an in action. To summarize, the key phrase

Some series are hard for any test even if it works. However, if a speciﬁc test works wells for a series, we should develop the insight to pick it. So applying the list of tests in a ﬁx order is not advised. Nonetheless, some tests are relatively simple to apply if we have no idea towards a series at all. The following is a summary: Table 8.2:

1

Test

Form of of an

Comments

No/little test

geometric, harmonic, p−test limn→∞ an = 0 corresp. integral in nice form ∑ closed n bn (or ∑(−1)n−1 (−1) bn )

clear answer

2 3

Test of divergence Integral Test

4

Alternating Series

449

inexpensive test easy integration check conditions on bn www.ck12.org

Table 8.2: (continued) Test

Form of of an

Comments

5

(Limit) Comparison

compare

6

Ratio Test∗

need companion known series recognize good form

Root Test∗ Combination

7 8

an+1 an

an resembles cn n composite of forms

evaluate ratio evaluate nth root combined methods

∗: the inconclusive cases need other tests. Example 1 ∫ 1 ∑ 1 diverges by the Integral Test since dx = ln |ln |x|| + C diverges. n(ln n) x(ln x) ∑ Example 2 sin n1 n diverges by Limit Comparison Test (against the harmonic series) since limn→∞ Example 3

sin

∑

1 n

1 n

1 n

= limh→0

sin

(1) n

sin h h

= 1 and the harmonic series diverges.

converges with Limit Comparison Test (against

∑

1 ) n2

since limn→∞

sin h h

1 n

= 1 and the latter series converges. ∑ Example∑4 √ Convergence of (cn )n is determined√with √ limn→∞ |cn | by the Root √ √ For ∑Test. example ( n 3 − n 2)n , converges since limn→∞ ( n 3 − n 2) = 1 − 1 = 0, but ( n 2 − 1) (no exponent n) diverges by rationalizing the numerator: √ √ ∑ 1 n 1 n2 2 − 1 = n−1 2−1 ≥ 2−1 and applying Limit Comparison test with . n−2 n−1 = 2 n 2n 2 n +2 n +...+1 n.2 n ∑ The Root Test is inconclusive on (1 + n1 )n , but the simpler Test for Divergence conﬁrms its divergence since (1 + n1 )n > 1 always. ∑ (−1)n−1 Example 5 is convergent for q > 0 by the Alternating Series Test since limn→∞ n1q = nq 1 1 0 and nq ≥ (n+1)q . It is absolutely convergent for q > 1 by the p−test. So it is conditionally convergent for 0 < q ≤ 1. ∑ ∑∞ 2.2n−1 ∑∞ 2n n.2n Example 6 ∞ = = 2 n=1 n! n=2 (n−1)! n=1 n! is absolutely convergent by the Ratio limh→0

2n+1 n! . (n+1)! 2n

2 = limn→∞ n+1 = 0. ∑ √n+1 . Notice n2 − 10n + 1 is never 0 and is positive for Example 7 Consider the series n2 −10n+1 n ≥ 10, we could ignore the terms before n = 10. Dropping the lower powers of√ n leads to the ∑ √n ∑ 1 n+1 n2 √ candidate = 3 for applying Limit Comparison Test since limn→∞ n2 −10n+1 . n = 1. n2 n2 So the series is (absolutely) convergent by the p−test. A combination of tests is applied.

Test since limn→∞

Exercises www.ck12.org

450

sin 1 n2

1 n

limn→∞

1. For what values of p is the series

∑∞

1 n=2 n(ln n)p

convergent?

Test the following series for convergence or divergence: ∑ 2 1 2. ∞ n=1 sin n √ ∑ (−1)n−1 n 3. ∞ n=1 n+1 √ ∑∞ √ n 4. n=1 ( 5 − n 3)n √ √ ∑ n n 5. ∞ ( 5 − 3) n=1 ∑ 3n 6. For what values of c is the series ∞ n=1 2n +cn convergent?

8.6

Power Series

Power Series and Convergence Deﬁnition (Power Series) A Power Series is a series of the form ∑ n 2 3 (PS1) ∞ n=0 an x = a0 + a1 x + a2 x + a3 x + . . . where x is a variable and the an ’s are constants (in our case, real numbers) called the coeﬀicients of the series. ∑ The summation sign is a compact and convenient shorthand notation. Reader unfamiliar with the notation might want to write out the detail a few times to get used to it. Power series are a generalization of polynomials, potentially with inﬁnitely many terms. As observed, the indices n of an are non-negative, so no negative integral exponents of x, e.g. 1 appears in a power series. x More generally, a series of the form ∑ n 2 3 (PS2) ∞ n=0 an (x − x0 ) = a0 + a1 (x − x0 ) + a2 (x − x0 ) + a3 (x − x0 ) + . . . is called a power series in (x − x0 )(1 = (x − x0 )0 ) or a power series centered at x0 ((PS1) represents series centered at x = 0). Given any value of x, a power series ((PS1) and (PS2)) is a series of numbers. The ﬁrst question is: Is the power series (as in (PS1) or (PS2)) a function of x? Since the series is always deﬁned at x = 0 (resp. x = x0 ), the question becomes: For what value of x is a power series convergent? The answers are known for some series. Convergence tests could be applied on some others.

451

www.ck12.org

Example 1 Let r ̸= 0 and x0 be real. ∑∞

n=0 (r(x − x0 ))

= 1 + r(x − x0 ) + r2 (x − x0 )2 + r3 (x − x0 )3 + . . . is absolutely convergent 1 and equals 1−r(x−x for |r(x − x0 )| < 1, i.e. |x − x0 | < 1r , and diverges otherwise. 0) n

∑ n 1 Let r = 1; x0 = 0. Then on (−1, 1). Let r = −1; x0 = 2. x is the power series for 1−x ∑ ∑ n 1 1 n n Then (−1) (x − 2) is the power series for 1−(−1)(x−2) = x−1 on (1, 3). So x and ∑ 1 n−1 n (−1) (x − 2) are the power series for the same function 1−x but on diﬀerent intervals. There will be more detailed discussion in §8.7. ∑∞ 2n Example 2 is absolutely convergent for |x| < 1 by Comparison Test (against n=0 x ∑∞ n x ) and diverges for x = ±1 by the Test for Divergence. n=0 Exercise

∑ n 1. Write a power series ∞ n=0 an (x + 2) centered at x = −2 for the same function Example 6.1.1. On what interval does equality hold? Hint: Substitute y = x + 2 in

1 . 1−x

2. Discuss the convergence of the series

∑∞ n=0

1 1−x

in

n

2n x2 .

Hint: Apply a combination of tests in §8.5.

Interval and Radius of Convergence The following theorem characterizes the values of x where a power series is convergent. Theorem (Interval of convergence) ∑ n Given a power series ∞ n=0 an (x − x0 ) . Exactly one of the following three describes all the values where the series is convergent: (A) The series converges exactly at x = x0 only. (B) The series converges for all x. (C) There is a real number Rc > 0 that the series converges if |x − x0 | < Rc and diverges if |x − x0 | > Rc . This Rc is unique for a power series, called the radius of convergence. By convention Rc = 0 for case (A) and Rc = ∞ for case (B). The only two values of x the Theorem cannot conﬁrm are the endpoints x = x0 ± Rc . In any case, the values x where the series converges is an interval, called the interval of convergence. It is the singleton x0 for case (A) and (−∞, ∞) for case (B). For case (C), it is one of the four possible intervals: (x0 − Rc , x0 + Rc ), (x0 − Rc x0 + Rc ], [x0 − Rc x0 + Rc ), and [x0 − Rc x0 + Rc ]. Here, the endpoints must be checked separately for convergence. www.ck12.org

452

∑ xn Example 1 Find the radius of convergence and interval of convergence of the series ∞ n=0 n2 . ( ) bn +1 xn+1 n2 1 2 xn Solution. Let bn = n2 . Then bn = (n+1)2 · xn = 1+ 1 x → |x| as n → ∞. So the n series is absolutely convergent for |x| < 1 (and divergent for |x| > 1) by the Ratio Test. Rc = 1. This leaves the endpoint values to check. If x = ±1, then the series is absolutely convergent by the p−test. Hence the series is absolutely convergent for|x| ≤ 1. The interval of convergence is [−1, 1]. ∑ ∑∞ n n Example 2 If the series ∞ 2 n=0 an 2 converges, then n=0 an x is convergent ∑∞at x = 2, i.e. n is inside the ∑ interval of convergence. So Rc ≥ 2. Conversely, if the series n=0 an (−3) din verges, then ∞ n=0 an x is divergent at x = −3, i.e. −3 is outside the interval of convergence. So Rc ≤ | − 3| = 3. Exercise Find the radius of convergence and interval of convergence of the following series. 1. 2. 3. 4. 5.

∑∞ nxn ∑n=1 ∞ (−1)n xn n ∑n=1 ∞ xn/3 n! ∑n=1 ∞ √ n(x − x0 )n n=1 ∑ n Given ∞ n=0 an x converges at x = 5 and diverges at x = −7. Deduce where possible, the convergence or divergence of these series: ∑ (a) ∑∞ n=0 an n (b) ∑∞ n=0 an 3 ∞ (c) ∑n=0 an (−8)n n (d) ∑∞ n=0 an (9) ∞ n (e) n=0 an (6)

Term-by-Term Diﬀerentiation of Power Series The goal of the next 3 sections is to ﬁnd power series representations of certain classes of functions, namely derivatives, integrals and products. In the study of diﬀerentiation (resp. integration), we have found the derivatives (resp. integrals) of better known functions, many with known power series representations. The power series representations of the derivatives (resp. integrals) can be found by term-by-term diﬀerentiation (resp. integration) by the following theorem. Theorem (Term-by-term diﬀerentiation and Integration) ∑ n Suppose∑ ∞ n=0 an (x − x0 ) has radius of convergence Rc . Then the function f deﬁned by n f (x) = ∞ n=0 an (x − x0 ) is diﬀerentiable on (x0 − Rc , x0 + Rc ) and

453

www.ck12.org

∑ n−1 (A) f ′ (x) = ∞ , n=1 nan (x − x0 ) ∫ ∑ an (x−x0 )n+1 (B) f (x)dx = ∞ +C, and these power series have same radius of convergence n=0 n+1 Rc . (A) means (dropping x0 the derivative of a power series is the same as the term-by-term diﬀerentiation of the power series: ∑∞ d ∑∞ d n n n=0 an x = n=0 dx (an x ) and dx (B) means the integral of a power series is the same as the term-by-term integration of the power series: ∫ ∑ ∞

n

(an x )dx =

n=0

∞ ∫ ∑

(an xn )dx.

n=0

Example 1 Find a power series for g(x) =

1 (1−x)2

and its radius of convergence.

1 whose power series representation is Solution. We recognize g(x) as the derivative of 1−x ∑∞ n ∑∞ n ∑∞ d n−1 x with radius of convergence R = 1. By (A), g(x) = and c n=0 n=0 x = n=1 nx dx has radius of convergence 1.

Exercise Find a power series and the radius of convergence for the following functions: 1. 2. 3.

x 2−x x2 (1−x)3 2x (1−x)3

+

3x2 (1−x)4

Term-by-Term Integration of Power Series Example 1 Find a power series for h(x) = tan−1 x and its radius of convergence. ∑∞ 1 n 2n Solution. We recognize h(x) as the antiderivative of 1+x 2 = n=0 (−1) x . ∫ ∑ ∑∞ (−1)n x2n+1 n 2n By Term-by-Term Theorem (B), h(x) = ∞ + C and has n=0 (−1) x dx = n=0 2n+1 radius of convergence 1. ∑ (−1)n x2n+1 Then C = tan−1 0 = 0 and h(x) = ∞ . n=0 2n+1 Exercise 1. Find a power series for ln(1 + x2 ) and ﬁnd the radius of convergence. ∫ −1 2. Express tan xdx as a power series and ﬁnd the radius of convergence. 3. Find a power series for ln(1+x+x2 ) in x and in x+ 12 , and ﬁnd the radius of convergence. www.ck12.org

454

Series Multiplication of Power Series ∑ n Deﬁnition∑(Series Multiplication) The power series product of two∑ power series ∞ n=0 an x ∑ ∞ ∞ n n n and n=0 bn x is a power series n=0 cn x deﬁned by cn = i=0 ai bn−i = a0 bn + a1 bn−1 + . . . + an b0 (like polynomials). A result is:∑the product of power series ∑∞ is then power series of the product. n If f (x) = ∞ a x and g(x) = bn x converges on a common interval |x| < Ra b, n=0 n n=0 ∑∞ then their product power series n=0 cn xn also converges on Ra b and is the power series for the product function f (x)g(x). 1 . Example 1 Find a power series for (1−x)(1−2x) ∑ 1 n Solution. 1−x = ∞ n=0 x with radius of convergence 1 and of convergence 12 .

1 1−2x

=

∑∞

n n=0 (2x)

with radius

So for |x| < 21 , 1 = (1 + x + x2 + . . .)(1 + 2x + 4x2 + . . .) (1 − x)(1 − 2x) = 1 + 3x + 7x2 + 15x3 + . . . by the above result ∞ ∑ = (2n+1 − 1)xn . n=0

Exercise 1. Find the ﬁrst 4 terms of a power series for √ 2. Find a power series for

1 (1−rx)(1−sx)

1 . (1+x)(1+2x)

where r, s > 0 are real numbers and

(a) r ̸= s (b) r = s.

8.7

Taylor and Maclaurin Series

Taylor and Maclaurin Polynomials We know the linear approximation function L1 (x) to a (smooth) function f (x) at x = x0 is given by the tangent line at the point. L1 (x0 ) = f (x0 ) and L′1 (x0 ) = f ′ (x0 ). Indeed, this is the only linear function with these 2 properties.

455

www.ck12.org

Theorem (nth −degree Taylor polynomial) Given a function f with continuous nth derivative in an open interval containing x∗0 . There exists unique nth −degree polynomial p(x) with p(j) (x0 ) = f (j) (x0 ), for 0 ≤ j ≤ n. ⋆: the functions in this text have continuous derivatives at the center x0 unless otherwise stated. This polynomial (n)

(x−0) 0) (x−x0 )2 +. . .+ f n! (x−x0 )n ) is called the nth −degree Tn (x) = f (x0 )+f ′ (x0 )(x−x0 )+ f (x 2! th Taylor polynomial of f at x0 . If x0 = 0, it is called the n − degree Maclaurin polynomial of f and denoted by Mn (x). Rn (x) = f (x) − Tn (x) is the remainder of the Taylor polynomial.

Example 1 Let f (x) = f (1) = 1. Hence T1 (x) = itself.

x3 , x0 = 1. Then f ′ (x) 6 1 + 12 (x − 1), T2 (x) = 16 6

= x2 and f (x) = x. So f ′ (1) = 12 and + 21 (x − 1) + (x − 1)2 , and T3 (x) = f (x) 2

Example 2 Let f (x) = sin x, x0 = 0 and take n = 3. Then f (x) = cos x, f ′′ (x) = − sin x, f ′′′ (x) = − cos x. So f ′ (0) = 1, f ′′ (0) = 0, f ′′′ (0) = −1. M3 (x) = x − 3!1 x3 = x − 16 x3 is the third-degree Maclaurin polynomial of f . Example 3 Find the second-degree Taylor polynomial of f (x) = tan x at x0 =(π4 .) Solution. f ′ (x) = sec2 x and f ′′ (x) = 2 sec x · sec x tan x = 2 sec2 x tan x. So f ′ π4 = 2 and ( ) ( ) ( )2 ( ) ( )2 f ′′ π4 = 2 · 2 = 4. Then T2 (x) = 1 + 2 x − π4 + 2!4 x − π4 = 1 + 2 x − π4 + 2 x − π4 . Exercise Find the Taylor series of the following functions at the given x0 with given degree n. 1. f (x) = ex at x = 0, n = 3 2. f (x) = ln x at x = 1, n = 4 www.ck12.org

456

3. f (x) = 1 + x + x2 + x3 + x4 at x = −1, n = 4

Taylor and Maclaurin Series Deﬁnition (Taylor Series of f ) The Taylor series of a function f at x = x0 is the power series

T (x) =

∞ ∑ f (n) (x0 ) n=0

n!

(x − x0 )n

= f (x0 ) + f ′ (x0 )(x − x0 ) +

f ′′′ (x0 ) f ′′ (x0 ) (x − x0 )2 + (x − x0 )3 + . . . 2! 3!

taking all the terms of the Taylor polynomials. The Maclaurin series M (x) of f is the Taylor series at x = 0. Example 1 Find the Maclaurin series of f (x) = cos x. Solution. f ′ (x) = − sin x, f ′ (x) = − cos x, f ′′ (x) = sin x, f ′′′ (x) = cos x, f (4) (x) = − sin x, f (5) (x) = − cos x, . . .. Notice the pattern repeats every 4 terms. So f ′ (0) = 0, f ′ (0) = −1, f ′′ (0) = 0, f ′′′ (0) = 1, f (4) (0) = 0, x, f (5) (0) = −1 . . .. The Maclaurin series of f (x) = cos x is

M (x) = 1 −

∑ 1 1 1 2 1 4 x + x − x 6 + x8 − . . . = ∞ n=0 2! 4! 6! 8!

(−1)n 2n x . (2n)!

Exercise Find the Taylor series of following functions at the given x0 . 1 1. f (x) = 1−x at x = 0 x 2. f (x) = e at x = 1 3. f (x) = x1 at x = 2

Convergence of Taylor and Maclaurin Series Since f (n) (x) is deﬁned for all functions f in this text, the Taylor series T (x) of f is always deﬁned. As for power series in general, the ﬁrst question is: Is T (x) convergent at x = a ? There is no guarantee except at a = x0 . The second question is:

457

www.ck12.org

If T (x) converges at x = a, does it equal f (a)? The answer is negative as show by the function: 1 f1 (x) = e− x2 x ̸= 0 0x=0 Then ( ) 1 ′ 2 f1 (x) = · e− x2 x ̸= 0 3 x 0x=0 It can be veriﬁed that f1′ (0) = 0, f1′′ (0) = 0, f1′′′ (0) = 0, . . .. So the Maclaurin series is 0, clearly diﬁerent from f except at x = 0. Nevertheless, here is a positive result. Theorem If f has a power series representation at x = x0 , i.e. ∑ n f (x) = ∞ n=0 an (x − s0 ) for |x − x0 | < Rc , then the coeﬀicients are given by an =

f (n) (x0 ) . n!

So any power series representation at x = x0 has the form:

f (x) =

∞ ∑ f (n) (x0 ) n=0

n!

(x − x0 )n

Exercise 1. Find the higher order derivatives of the function f1 (x) above thus recursively showing (n) f1 (0) = 0 for n ≥ 0 2. Verify the Theorem using term-by-term diﬁerentiation.

Taylors Formula with Remainder, Remainder Estimation, Truncation Error Recall the remainder Rn (x) of the nth − degree Taylor polynomial at x = x0 is given by Rn (x) = f (x) − Tn (x). www.ck12.org

458

Theorem (Convergence of Taylor series) If limn→∞ Rn (x) = 0 for |x − x0 | < Rc , then f is equal to its Taylor Series on the interval |x − x0 | < Rc . The above condition limn→∞ Rn (x) = 0 could be achieved through the following bound. Theorem (Remainder Estimation) If |f (n+1) (x)| ≤ M for |x − x0 | ≤ r, then we have the following bound for Rn (x): |Rn (x)| ≤

M |x (n+1)!

− x0 |n+1 for |x − x0 | ≤ r.

Example 1 The function ex is equal to its Maclaurin series for all x. Proof. Let f (x) = ex . We need to ﬁnd the above bound on Rn (x). If |x| ≤ r, f (n) (x) = ex ≤ er for n ≥ 0 and the remainder estimation gives |Rn (x)| ≤ er |x|n+1 for |x| ≤ r. (n+1)! |x| e Since limn→∞ (n+1)! |x|n+1 = er limn→∞ (n+1)! = 0, limn→∞ |Rn (x)| = 0 by the squeeze Theo∑ xn x rem. So limn→∞ Rn (x) = 0. Hence e is equal to its Taylor series ∞ n=0 n! for all x. n+1

r

Example 2 (Truncation Error) What is the truncation error of approximating f (x) = √ 1 + x by its third-degree Maclaurin polynomial in for |x| ≤ 0.1. Solution. ( ) 3 3 1 1 1 1 − 12 ′′ f (x) = (1 + x) , f (x) = − (1 + x)− 2 = − (1 + x)− 2 , 2 2 2 4 ( )( ) ( )( ) 7 7 3 1 3 3 5 15 ′′′ − 52 − 25 (4) f (x) = − − (1 + x) = (1 + x) , f (x) = − (1 + x)− 2 = (1 + x)− 2 . 4 2 8 8 2 16 ′

For |x| ≤ 0.1, |f (4) (x)| =

15 1 16 (1+x) 72

≤

15 . 16

5 So |R4 (x)| ≤ 15 · 1 |x|4 ≤ 128 (0.1)4 ≈ 4 × 10−6 . This is the truncation error of approximating 16 4! by the third-degree Maclaurin polynomial.

Multimedia Link The following applet illustrates approximating functions with Taylor Series. You can change the center of the series and also observe how the error changes for the estimation at a particular value of x where the error is f (x) − Tn (x). Taylor Series and Polynomials Applet. Exercise 1. Find the power series representation of f (x) = sin x at x = 0 for all x. Why is it the Maclaurin series?

459

www.ck12.org

2. Find the power series representation of f (x) = cos x at x = π3 for all x. Why is it the Taylor series at x = π3 ? √ 3. What is the truncation error of approximating f (x) = 1 + x by its fourth-degree Maclaurin series in for |x| ≤ 0.1.

Combining Series, Eulers Formula In many cases, we could ﬁnd Taylor (Maclaurin) series of functions from existing series without going through the proof that limn→∞ Rn (x) = 0 Examples are products, quotients and some sine and cosine functions. Example 1 Find the Maclaurin series of f (x) = x sin x for all x. ∑ (−1)n x2n+1 Solution. The Maclaurin series of sin x is sin x = ∞ n=0 (2n+1)! . So the Maclaurin series ∑ ∑ (−1)n x2n+1 (−1)n x2n+2 of x sin x is x sin x = x ∞ = ∞ n=0 (2n+1)! n=0 (2n+1)! Example 2 Find the Maclaurin series of f (x) = cos2 x for all x. Solution. We could avoid multiplying the Maclaurin series of cos x with itself, by applying: cos2 x = 12 + 12 cos 2x ∑ (−1)n (2x)2n on the Maclaurin series of cos 2x = ∞ , giving n=0 (2n)!

cos2 x =

∞ ∞ ∑ 1 1 ∑ (−1)n (2x)2n (−1)n (2x)2n + =1+ 2 2 n=0 (2n)! 2 · (2n)! n=0

For any real number θ, ei θ = cos +i sin θ where i = Euler’s Formula.

√

−1 is the imaginary unit. This is the

Euler’s formula combines the complementary sine and cosine functions into the simpler exponential function and heavily applies the separation of real and imaginary parts of complex numbers. Example 3 Find the Maclaurin series of cos x and sin x for all x through eix . Solution. cos x + i sin x = eix which has a Maclaurin series

ix

e =

∞ ∑ (ix)n n=0

n!

∞ ∞ ∑ (ix)2m ∑ (ix)2m+1 = + (2m)! m=0 (2m + 1)! m=0

by dividing into sum of odd and even indices. So cos x + i sin x = ∑ 2m+1 m (x) i ∞ m=0 (−1) (2m+1)! www.ck12.org

460

∑∞

2m

m (x) m=0 (−1) (2m)!

+

The Maclaurin series of cos x and sin x follow by separately taking the real and imaginary parts. Exercise 1. Find and compare the Maclaurin series for sin x cos x and sin 2x. 2. Find the Maclaurin series of

x ex

for all x.

Hint: would you divide x by ex ? 3. Find the Maclaurin series of cos 3x and sin 3x for all x using Euler’s formula. ∑ ∑∞ n n 4. Find expressions for the series ∞ n=0 cos nθ x and n=0 sin nθ x for all θ and |x| < 1 using Euler’s formula.

Binomial Series We have learned the Binomial Theorem for positive integer exponents:

(a + b)n = an + nan−1 b + n ( ) ∑ n n−k k = a b k k=0

n(n − 1) n−2 2 a b + . . . + nabn−1 + bn (BE) 2!

where the Binomial coeﬀicients are denoted by (n) (n) n(n−1)...(n−k+1) = 1 and = for k ≥ 1 0 k k! AS a simple Binomial function, take a = 1 and b = x. Then (1 + x)n =

∑n k=0

(n) 0

xk

Let r be a real number and f (x) = (1 + x)r . Is f (x) equal to a series in the form of (BE) except that there may be an inﬁnite series? The answer is yes. Theorem (Binomial Series) Let r be a real number and |x| < 1. Then r(r − 1) 2 r(r − 1)(r − 2) 3 (1 + x)r = 1 + rex + x + x 2! 3! ( ) n ∑ r k = x k k=0 where the Binomial coeﬀicients are denoted by (n) ( ) = 1 and nk = n(n−1)...(n−k+1) for k ≥ 1 0 2!

461

www.ck12.org

As a simple Binomial function, take a = 1 and b = x. Then

n

(1 + x) =

n ( ) ∑ 1

n

k=0

(K)xk .

Let r be a real number and f (x) = (1 + x)r . Is f (x) equal to a series in the form of (BE) except that there may be an inﬁnite series? The answer is yes. Theorem (Binomial Series) Let r be a real number and |x| < 1. Then

r(r − 1) 2 r(r − 1)(r − 2) 3 (1 + x)r = 1 + rex + x + x ... 2! 3! n ( ) ∑ r k = x k k=0

where

(r ) 0

= 1 and

(r ) k

=

r(r−1)...(r−k+1) 2!

for k ≥ 1

Example 1 Find a power series representation of

√

1 + x.

Solution. We need to compute the Binomial coeﬀicients for r = − 12

(1) 2

k

(1) (

) ( 2k−3 ) 3 − ... − 2 (−1)k−1 1.3.5 . . . (2k − 3) 2 = = 2 k! 2k k! (−1)k−1 (2k − 2)! (−1)k−1 (2k − 2)! (−1)k−1 (2k − 2)! = = = 2.4.6 . . . (2k − 2))2k k! 2k−1 k!(k − 1)2k k! 22k−1 k!(k − 1)! − 12

)(

√ ∑ (1) k ∑∞ 2 x = 1 + So if |x| < 1, 1 + x = ∞ k=0 k k=0

(−1)k−1 (2k−2)! k x 22k−1 k!(k−1)!

Example 2 Find a power series representation of

1 (1−x)m

where m is a positive integer.

Solution. We need to compute the Binomial coeﬀicients for r = −m (and will replace x by −x) www.ck12.org

462

(

So

−m k

)

(−m)(−m − 1)(−m − 2) . . . (−m − k + 1) k! ) ( (−1)k m(m + 1)(m + 2) . . . (m + k − 1) k m+k−1 = = (−1) k! k ( ) ∞ ∑ m+k−1 = (−1)k (−x)k k k=0 ( ) ∞ ∑ k m+k−1 = (−1) (−1)k xk k k=0 ( ) ∞ ∑ m+k−1 k 2 = (−1) k x k k=0 ) ∞ ( ∑ m+k−1 k = x k k=0 =

1 (1 − x)m

Exercise 1 1. Find a power series representation of √1+x at x = 0. 1 2. Find a power series representation of (2−x)2 at ’x = 0. √ ( )2 3. Notice 1+x+x2 = x+ 34 + x + 12 . Find a power series representation of (1 + x + x2 ) at x = − 21 . In what interval is the equality true?

Choosing Centers Taylor Series (indeed Taylor polynomials of lower degrees) often provide good approximation of functions. However, the choice of center could determine 1. whether the intended value of x is inside the interval of convergence 2. rate of convergence, i.e. how many terms to take to achieve prescribed degree of accuracy For frequently used functions, the ﬁrst choice may be the standard center (see the list at the end of this section). Example 1 Approximate ln 0 : 99 Solution. Since .99 is close to the center x = 1, we use the standard Taylor series for ln(1−x).

ln 0.99 = ln(1 − 0.01) ≈ −0.01 −

463

(0.01)2 ≈ −0.01005 2 www.ck12.org

Then we may be able to deduce a useful Taylor Series centered close to the given x. Example 2 Approximate sin(1.1) to 4 decimal places. Since 1.1 is close to π3 , we would try to ﬁnd a Taylor Series of sin x at x0 = π3 . Let f (x) = sin x. Then √ ( ) ( ) f (x) = sin x, f√′ (x) = cos x, f ′′ (x) = − sin x, f ′′′ (x) = − cos x and f π3 = 23 , f ′ π3 = ( ) ( ) 3 1 ′′ π ′′′ π , f = − , f = − 12 . 2 3 2 3 This pattern repeats and limn→∞ Rn (x) = 0 can be checked as in the case x0 = 0. So the Taylor Series is √ ∞ ∞ ∑ (−1)n 3 ( π )2n ∑ (−1)n ( π )2n+1 sin x = x− + x− 2(2n)! 3 2(2n + 1)! 3 n=0 n=0 Hence √ ∞ ∞ ∑ ∑ (−1)n 3 (−1)n 2n sin(1.1) ≈ (0.0528) + (0.0528)2n+1 2(2n)! 2(2n + 1)! n=0 n=0 ≈ 0.86481823 + 0.02638773 taking 2 terms from each sum ≈ 0.8912 We may also apply algebraic manipulation to standard Taylor Series. Example 3 Approximate

1 1.92

to 4 decimal places.

Solution. There is standard Taylor Series: ∑∞ 1 n−1 = for |x| < 1 through term-by-term diﬀerentiation of the series for 2 k=1 nx (1−x) 1 ( (1−x)2 is inadequate at x = 0.9). Since 1.9 is close to 2, we consider ( x )n−1 ∑ 1 = 14 (1−1x )2 = 14 ∞ for |x| < 2. k=1 n 2 (2−x)2 2

So we take x = 0.1 and then 1 ∑ ( x )n−1 1 1 = ≈ . (1 + 2(0.05) + 3(0.05)2 + 4(0.05)3 ) ≈ 0.277. n 1.92 4 n=1 2 4 ∞

Exercise www.ck12.org

464

1 1−x

1. Approximate ln 0.9 to 4 decimal places. 2. Approximate sin(0.8) to 6 decimal places. Hint: consider center 3. Approximate

1 93

π 4

to 6 decimal places.

Evaluating Nonelementary Integrals There are many simple-looking functions that have no explicit formula for their integral in the form of elementary functions. We could write their in-tegrals as Taylor Series in their interval of convergence. ∫ x Example 1 Find a power series representation of ex dx x

Solution. Since ex is not deﬁned at x = 0, we apply the Taylor Series of ex at, say, x = 1 by writing ex = e.ex−1 with a change of variable u = x − 1. ex e.eu 1 ∑ (u)n = = e. x 1+u 1 + u n=0 n! ∞

(

)

(

u2 u3 u 4 = e 1 − u + u − u + ... 1 + u + + + + ... 2! 3! 2! ( ) 1 2 1 3 3 4 11 5 = e 1 + u − u + u − u + . . . for |u| < 1 2 3 8 30 3

)

) + . . . where u = x − 1 ∫ 2 Example 2 Find the power series representation of sinxx dx

so

∫

ex dx x

( = e u + 16 u3 −

2

1 4 u 12

+

3 5 u 40

−

11 6 u 180

Solution. Direct substitution of x2 in the Maclaurin Series of sin x gives sin x2 = ∑ 2 (−1)n 4n+1 and sin2x = ∞ n=0 (2n+1)! x

∑∞

(−1)n 2 2n+1 n=0 (2n+1)! (x )

So ∫

∑ (−1)n sin x2 dx = x (2n + 1)! n=0 ∞

=

∫ x4n+1 dx

∞ ∞ ∑ ∑ (−1)n x4n+2 (−1)n x4n+2 = (2n + 1)! 4n + 2 n=0 (4n + 2) (2n + 1)! n=0

Exercise

465

www.ck12.org

∫ 2 1. Find the power series representation (Maclaurin Series) of ex dx and approximate ∫1 −x2 e dx to 6 decimal places. 0 ∫ 2. Find the power series representation (Maclaurin Series) of sin x2 dx.

Frequently Used Maclaurin Series Some frequently used Maclaurin Series are listed below ∑

1 n = ∞ n=1 x 1−x ∑ ∞ 1 = n=1 nxn (1−x)2∑ xn ex = ∞ n=1∑ n! xn ln 1 − x = ∑∞ n=1 n , n+1 xn ln 1 + x = ∞ , n=1 (−1) ∑∞ (−1)n x2n+1 n sin x = n=1 (2n+1)! ∑ (−1)n x2n cos x = ∞ n=1 ∑ (2n)! (−1)n x2n+1 tan−1 x = ∞ n=1 (2n+1)

8.8

in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1)

Calculations with Series

Binomial Series We have learned the Binomial Theorem for positive integer exponents:

(a + b)n = an + nan−1 b + n ( ) ∑ n n−k k = a b k k=0

n(n − 1) n−2 2 a b + . . . + nabn−1 + bn (BE) 2!

where the Binomial coeﬀicients are denoted by (n) ( ) for k ≥ 1 = 1 and nk = n(n−1)...(n−k+1) 0 k! AS a simple Binomial function, take a = 1 and b = x. Then (1 + x)n =

∑n k=0

(n) 0

xk

Let r be a real number and f (x) = (1 + x)r . Is f (x) equal to a series in the form of (BE) except that there may be an inﬁnite series? The answer is yes. Theorem (Binomial Series) Let r be a real number and |x| < 1. Then www.ck12.org

466

r(r − 1) 2 r(r − 1)(r − 2) 3 (1 + x)r = 1 + rex + x + x 2! 3! n ( ) ∑ r k = x k k=0 where the Binomial coeﬀicients are denoted by (n) (n) n(n−1)...(n−k+1) = 1 and = for k ≥ 1 0 k 2! As a simple Binomial function, take a = 1 and b = x. Then

n

(1 + x) =

n ( ) ∑ 1

n

k=0

(K)xk .

Let r be a real number and f (x) = (1 + x)r . Is f (x) equal to a series in the form of (BE) except that there may be an inﬁnite series? The answer is yes. Theorem (Binomial Series) Let r be a real number and |x| < 1. Then r(r − 1) 2 r(r − 1)(r − 2) 3 x + x ... (1 + x)r = 1 + rex + 2! 3! n ( ) ∑ r k = x k k=0 where

(r ) 0

= 1 and

(r ) k

=

r(r−1)...(r−k+1) 2!

for k ≥ 1

Example 1 Find a power series representation of

√

1 + x.

Solution. We need to compute the Binomial coeﬀicients for r = − 12 (1) 2

k

(1) (

) 3 1 − (− 2 ) . . . (− 2k−3 ) (−1)k−1 1.3.5 . . . (2k − 3) 2 2 2 = = k! 2k k! (−1)k−1 (2k − 2)! (−1)k−1 (2k − 2)! (−1)k−1 (2k − 2)! = = = 2.4.6 . . . (2k − 2))2k k! 2k−1 k!(k − 1)2k k! 22k−1 k!(k − 1)!

√ ∑ (1) k ∑∞ 2 So if |x| < 1, 1 + x = ∞ k=0 k x = 1 + k=0

(−1)k−1 (2k−2)! k x 22k−1 k!(k−1)!

Example 2 Find a power series representation of

467

1 (1−x)m

where m is a positive integer. www.ck12.org

Solution. We need to compute the Binomial coeﬀicients for r = −m (and will replace x by −x) (

−m k

)

(−m)(−m − 1)(−m − 2) . . . (−m − k + 1) k! ( ) k (−1) m(m + 1)(m + 2) . . . (m + k − 1) k m+k−1 = = (−1) k k!

=

So ( ) ∞ ∑ 1 k m+k−1 = (−1) (−x)k m (1 − x) k k=0 ( ) ∞ ∑ k m+k−1 = (−1) (−1)k xk k k=0 ) ( ∞ ∑ m+k−1 k 2 x (−1) k = k k=0 ) ∞ ( ∑ m+k−1 k = x k k=0 Exercise 1 1. Find a power series representation of √1+x at x = 0. 1 2. Find a power series representation of (2−x)2 at x = 0. √ ( )2 3. Notice 1+x+x2 = x+ 34 + x + 12 . Find a power series representation of (1 + x + x2 ) at x = − 21 . In what interval is the equality true?

Choosing Centers Taylor Series (indeed Taylor polynomials of lower degrees) often provide good approximation of functions. However, the choice of center could determine 1. whether the intended value of x is inside the interval of convergence 2. rate of convergence, i.e. how many terms to take to achieve prescribed degree of accuracy www.ck12.org

468

For frequently used functions, the ﬁrst choice may be the standard center (see the list at the end of this section). Example 1 Approximate ln 0 : 99 Solution. Since .99 is close to the center x = 1, we use the standard Taylor series for ln(1−x).

(0.01)2 ln 0.99 = ln(1 − 0.01) ≈ −0.01 − ≈ −0.01005 2 Then we may be able to deduce a useful Taylor Series centered close to the given x. Example 2 Approximate sin(1.1) to 4 decimal places. Since 1.1 is close to π3 , we would try to ﬁnd a Taylor Series of sin x at x0 = π3 . Let f (x) = sin x. Then √ ( ) ( ) f (x) = sin x, f√′ (x) = cos x, f ′′ (x) = − sin x, f ′′′ (x) = − cos x and f π3 = 23 , f ′ π3 = ( ) ( ) 1 , f ′′ π3 = − 23 , f ′′′ π3 = − 12 . 2 This pattern repeats and limn→∞ Rn (x) = 0 can be checked as in the case x0 = 0. So the Taylor Series is √ ∞ ∞ ∑ (−1)n 3 ( π )2n ∑ (−1)n ( π )2n+1 sin x = x− x− + 2(2n)! 3 2(2n + 1)! 3 n=0 n=0 Hence √ ∞ ∞ ∑ ∑ (−1)n 3 (−1)n sin(1.1) ≈ (0.0528)2n + (0.0528)2n+1 2(2n)! 2(2n + 1)! n=0 n=0 ≈ 0.86481823 + 0.02638773 taking 2 terms from each sum ≈ 0.8912 We may also apply algebraic manipulation to standard Taylor Series. Example 3 Approximate

1 1.92

to 4 decimal places.

Solution. There is standard Taylor Series: ∑ 1 n−1 for |x| < 1 through term-by-term diﬀerentiation of the series for = ∞ k=1 nx (1−x)2 1 is inadequate at x = 0.9). (1−x)2

1 1−x

Since 1.9 is close to 2, we consider

469

www.ck12.org

1 (2−x)2

=

1 1 4 (1− x2 )2

=

1 4

∑∞ k=1

n

( x )n−1 2

for |x| < 2.

So we take x = 0.1 and then 1 1 ∑ ( x )n−1 1 n = ≈ . (1 + 2(0.05) + 3(0.05)2 + 4(0.05)3 ) ≈ 0.277. 2 1.9 4 n=1 2 4 ∞

Exercise 1. Approximate ln 0.9 to 4 decimal places. 2. Approximate sin(0.8) to 6 decimal places. Hint: consider center 3. Approximate

1 93

π 4

to 6 decimal places.

Evaluating Non-Elementary Integrals There are many simple-looking functions that have no explicit formula for their integral in the form of elementary functions. We could write their in- tegrals as Taylor Series in their interval of convergence. ∫ x Example 1 Find a power series representation of ex dx x

Solution. Since ex is not deﬁned at x = 0, we apply the Taylor Series of ex at, say, x = 1 by writing ex = e.ex−1 with a change of variable u = x − 1. ex e.eu 1 ∑ (u)n = = e. x 1+u 1 + u n=0 n! ( ) u 2 u3 u4 2 3 = e(1 − u + u − u + . . .) 1 + u + + + + ... 2! 3! 2! ) ( 1 2 1 3 3 4 11 5 = e 1 + u − u + u − u + . . . for |u| < 1 2 3 8 30 ∞

so ∫ ex

) + . . . where u = x − 1 ∫ 2 Example 2 Find the power series representation of sinxx dx x

( dx = e u + 16 u3 −

1 4 u 12

+

3 5 u 40

−

11 6 u 180

Solution. Direct substitution of x2 in the Maclaurin Series of sin x gives ∑ ∑ 2 (−1)n (−1)n 4n+1 2 2n+1 sin x2 = ∞ and sin2x = ∞ n=0 (2n+1)! (x ) n=0 (2n+1)! x www.ck12.org

470

So ∫

∑ (−1)n sin x2 dx = x (2n + 1)! n=0 ∞

∫ x4n+1 dx

∞ ∞ ∑ ∑ (−1)n x4n+2 (−1)n x4n+2 = = (2n + 1)! 4n + 2 n=0 (4n + 2) (2n + 1)! n=0

Exercise ∫ 2 1. Find the power series representation (Maclaurin Series) of ex dx and approximate ∫1 −x2 e dx to 6 decimal places. 0 ∫ 2. Find the power series representation (Maclaurin Series) of sin x2 dx.

Frequently Used Maclaurin Series Some frequently used Maclaurin Series are listed below ∑

1 n = ∞ n=1 x 1−x ∑ ∞ 1 = n=1 nxn (1−x)2∑ xn ex = ∞ n=1∑ n! xn ln 1 − x = ∑∞ n=1 n , n+1 xn ln 1 + x = ∞ , n=1 (−1) ∑∞ (−1)n x2n+1 n sin x = n=1 (2n+1)! ∑ (−1)n x2n cos x = ∞ n=1 ∑∞ (2n)! n x2n+1 −1 tan x = n=1 (−1) (2n+1)

471

in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1) in(−1, 1)

www.ck12.org

Copyright © 2019 PROPERTIBAZAR.COM. All rights reserved.