HOMEWORK 1 Find the first derivative: 1.) f (t) = (3t2 + 2)4 2.) y = T 5 − 4T 4 + 3T 2 − 2T − 1 √ 3.) h(x) = (2x8 + 5 x + 4.) f (x) =
x2 + 1
Find the second derivative: 5.) p(t) = 5t10 + 5t3 + 4 √ 3 6.) h(x) = 2x 2 + 5 x +
7.) g(x) = (2x + 1)5 1
8.) f (x) = (3x + 2) 2
9.) The number of fruit flies at time t (in seconds) is given by f (t) = t2 + 4t + 1. a.) How many fruit flies are there at time 1? At time 2? What is the average rate of change from 1 to 2? b.) Find the instantaneous rate of change for the function. c.) What is the instantaneous rate of change in the population of fruit flies at time 1? At time 2?
10.) Bob’s Bread company makes, fittingly enough, loaves of bread. The total cost of making x loaves of bread is given by C(x) = 1000 − 50x + x2 a.) What is the marginal cost of each loaf? b.) If Bob makes $30 off of each loaf of bread, how many loaves should he make to maximize his profit? c.) How much profit would he make if he followed your advice in part b? d.) If Bob makes $4 off of each loaf, how many loaves would he make to maximize his profit? What is his profit in this case? What would you recommend to Bob? 1
11.) A toy rocket fired straight up into the air has height s(t) = 160t − 16t2 feet after t seconds. a.) What is the rocket’s initial velocity (i.e. when t = 0)? b.) What is the velocity after 2 seconds? c.) What is the acceleration when t = 3? d.) At what time will the rocket hit the ground? e.) At what velocity will the rocket be traveling just as it smashes into the ground.