758 Chapter 13: Functions of Several Variables
13.3 Partial Derivatives
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the partial derivative for the function .
a.
b.
c.
d.
e.
____ 2. Find the partial derivative for the function .
a.
b.
c.
d.
e.
____ 3. Find the partial derivative for the function .
a.
b.
c.
d.
e.
13.3 Partial Derivatives 759
____ 4. Find the partial derivative for the function .
a.
b.
c.
d.
e.
____ 5.
Find the partial derivative for the function .
a.
b.
c.
d.
e.
760 Chapter 13: Functions of Several Variables
____ 6. Find the partial derivative for the function .
a.
b.
c.
d.
e.
____ 7. Find the partial derivative for the function .
a.
b.
c.
d.
e.
13.3 Partial Derivatives 761
____ 8. Find the partial derivative for the function .
a.
b.
c.
d.
e.
____ 9. For , evaluate at the point .
a.
b.
c.
d.
e.
____ 10. For , evaluate at the point .
a.
b.
c.
d.
e.
762 Chapter 13: Functions of Several Variables
____ 11. For , evaluate at the point .
a.
b.
c.
d.
e.
____ 12. For , evaluate at the point .
a.
b.
c.
d.
e.
13.3 Partial Derivatives 763
____ 13. Find the first partial derivative for the function with
respect to z.
a.
b.
c.
d.
e.
____ 14. For , evaluate at the point .
a.
b.
c.
d.
e.
____ 15. Find the second partial derivative for the function with respect to
x.
a.
b.
764 Chapter 13: Functions of Several Variables
c.
d.
e.
____ 16. For find all values of x and y such that
and simultaneously.
a.
b.
c.
d.
e.
____ 17. For find all values of x and y such that and
simultaneously.
a.
b.
c.
d.
e.
13.3 Partial Derivatives 765
____ 18. For , find all values of x and y such that and
simultaneously.
a.
b.
c.
d.
e.
____ 19. Suppose a pharmaceutical corporation has two plants that produce the same over-the-
counter medicine. If and are the numbers of units produced at plant 1 and plant 2, respectively,
then the total revenue for the product is given by . When
and find the marginal revenue for plant 1.
a. 174
b. 172
c. 80
d. 124
e. 93
____ 20. Suppose a company manufactures two types of wood-burning stoves: a freestanding
model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-
insert stoves is . Find the marginal costs when and
. Round your answer to the nearest integer.
a. 192
b. 234
c. 210
d. 402
e. 403
766 Chapter 13: Functions of Several Variables
____ 21. Suppose the temperature at any point in a steel plate is
where x and y are measured in meters. At the point find the rate of change of the temperature
with respect to the distance moved along the plate in the direction of the x-axis. Round your answer to
one decimal place.
a.
b.
c.
d.
e.
____ 22. Suppose the utility function is a measure of the utility (or satisfaction)
derived by a person from the consumption of two products x and y. Determine the marginal utility of
product x if the utility function is .
a.
b.
c.
d.
e.
____ 23. Suppose the utility function is a measure of the utility (or satisfaction)
derived by a person from the consumption of two products x and y. Determine the marginal utility of
product if the utility function is .
a.
b.
c.
d.
e.
13.3 Partial Derivatives 767
13.3 Partial Derivatives – Answer Section