Math 140 Written Homework 6 3 3 3 4 3 5 3 6 3 7 Page 1 of 2 From the questions below please solve 5 problems only The homework is worth 10 points Each question is worth 1 points Show all of your work and put a box around your nal answer Number each attempted question clearly Write legibly that is suitably large and suitably dark if the grader can t read your answer it s consider uncompleted Question 1 Calculate the derivative indicated using the given function a f cid 48 cid 48 1 f t t 1 t b d3x dt3 x t t 3 4 Question 2 Find the derivative of each function a f sec 3 b f z zez 1 4 1 16 ez c g y cos y y2 sin y 3ey d h x tan x x3 ex Question 3 Suppose that the function f x and f cid 48 x have the following values at x 0 and x 1 Find the rst derivatives of the following combination at the given value of x a f x x cos x x 0 Question 4 Use the quotient rule to derive the formula for csc x and for cot x Question 5 Some values of a di erentiable invertible function f and its derivative are given below x 0 1 f x 9 3 f cid 48 x 2 1 5 b ex f x 2 x 1 d dx d dx 0 15 8 x f x 2 f cid 48 x 1 2 2 5 3 5 4 10 2 13 6 6 Calculate the following a g cid 48 2 where g x f x e3x b h cid 48 6 where h x xef x Question 6 Use the table below to evaluate the derivatives below x 1 f x 2 f cid 48 x 3 g x 4 g cid 48 x 5 h x 2 h cid 48 x 1 a Let F x f x g x and nd F cid 48 1 b Let G x and nd G cid 48 1 c Let H x f x g x h x and nd H cid 48 1 d Let J x and nd J cid 48 1 g x f x f x g x h x Math 140 Written Homework 6 3 3 3 4 3 5 3 6 3 7 Page 2 of 2 Question 7 A mass is hanging from the ceiling on a spring and oscillating forever and without friction At its highest point the mass is 12 cm from the oor and at its lowest point the mass is 6 cm from the oor At t 0 the mass is at its equilibrium and moving downwards and a single period takes 0 5 seconds HINT Graph your y t to check your answers a Find a formula for y t the position of the mass from the oor as a function of time b What is the position velocity and acceleration of the object at t 2 c In its rst period during what time interval s is the speed v of the mass less than 1 cm s Question 8 The fuel e ciency of a car E s in miles per gallon is a function of the speed of the car s in miles per hour Suppose that E 60 40 and E cid 48 60 0 7 a What are the units of E cid 48 s b Let C s be the fuel consumption in gallons per mile of the car when it is driving s miles per hour Find a formula for C s in terms of E s Calculate C cid 48 60 c Let R s be the rate the car burns fuel in gallons per hour when the car is driving s miles per hour Find a formula for R s in terms of E s Calculate R cid 48 60 Question 9 In 1999 the population of Richmond Petersburg Virginia metropolitan area was 961 400 and was increasing at a at roughly 9200 people per year The average annual income in the area was 30 593 per capita and this average was increasing at about 1400 per year Use the product rule to estimate the rate at which total personal income was rising in the area at this time Explain the meaning of each term in the product rule Question 10 The revenue per month earned by the Couture clothing chain at time t is R t N t S t where N t is the number of stores and S t is the average revenue per store per month Couture releases a statement to their investors saying that currently they have 50 stores with a total revenue of 7 500 000 and expect to increase the number of stores by about 2 stores per month and increase total revenue by about 250 000 per month What does this statement imply about the current rate of change of the average revenue per store per month