Question 1 Some values of a differentiable, invertible function fand its derivative are given below.
x0 2 4 6 8
f(x) 15 13 10 6 2
f′(x) −1 −2 −2.5 −3 −5

Calculate the following.
(a) g′(2), where g(x) = f(x)e3x.
(b) h′(6), where h(x) = xef(x).

Question 2 Use the table below to evaluate the derivatives below.
xf(x) f′(x) g(x) g′(x) h(x) h′(x)
1 2 3 4 5 −2 −1

(a) Let F(x) = f(x)g(x) and find F′(1).
(b) Let G(x) = g(x)
f(x) and find G′(1).
(c) Let H(x) = f(x)g(x)h(x) and find H′(1).
(d) Let J(x) = f(x)g(x)
h(x) and find J′(1).

Question 3 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 floor, and at its lowest point, the mass is 6 cm from the
floor. 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 floor as a function of time.
(b) What is the position, velocity, and acceleration of the object at t= 2?
(c) In its first period, during what time interval(s) is the speed, |v|, of the mass less than 1 cm/s?

Question 4 The fuel eﬀiciency 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′(60) = −0.7.

(a) What are the units of E′(s)?
(b) Let C(s) be the fuel consumption, in gallons per mile, of the car when it is driving smiles per
hour. Find a formula for C(s) in terms of E(s). Calculate C′(60).
(c) Let R(s), be the rate the car burns fuel, in gallons per hour, when the car is driving smiles per
hour. Find a formula for R(s) in terms of E(s). Calculate R′(60).

Question 5 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 6 The revenue per month earned by the Couture clothing chain at time tis
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?

Question

Question 1 Some values of a differentiable, invertible function fand its derivative are given below.
x0 2 4 6 8
f(x) 15 13 10 6 2
f′(x) −1 −2 −2.5 −3 −5

Calculate the following.
(a) g′(2), where g(x) = f(x)e3x.
(b) h′(6), where h(x) = xef(x).

Question 2 Use the table below to evaluate the derivatives below.
xf(x) f′(x) g(x) g′(x) h(x) h′(x)
1 2 3 4 5 −2 −1

(a) Let F(x) = f(x)g(x) and find F′(1).
(b) Let G(x) = g(x)
f(x) and find G′(1).
(c) Let H(x) = f(x)g(x)h(x) and find H′(1).
(d) Let J(x) = f(x)g(x)
h(x) and find J′(1).

Question 3 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 floor, and at its lowest point, the mass is 6 cm from the
floor. 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 floor as a function of time.
(b) What is the position, velocity, and acceleration of the object at t= 2?
(c) In its first period, during what time interval(s) is the speed, |v|, of the mass less than 1 cm/s?

Question 4 The fuel eﬀiciency 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′(60) = −0.7.

(a) What are the units of E′(s)?
(b) Let C(s) be the fuel consumption, in gallons per mile, of the car when it is driving smiles per
hour. Find a formula for C(s) in terms of E(s). Calculate C′(60).
(c) Let R(s), be the rate the car burns fuel, in gallons per hour, when the car is driving smiles per
hour. Find a formula for R(s) in terms of E(s). Calculate R′(60).

Question 5 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 6 The revenue per month earned by the Couture clothing chain at time tis
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?

Accepted Answer

Answered: Image /qna-images/answer/b3cc566d-280c-4475-ae1c-0ed38dfa826b.jpg

Question 1 Some values of a differentiable, invertible function fand its derivative are given below. x0 2 4 6 8 f(x) 15 13 10 6 2 f′(x) −1 −2 −2.5 −3 −5 Calculate the following. (a) g′(2), where g(x) = f(x)e3x. (b) h′(6), where h(x) = xef(x).