13.6 Directional Derivatives and Gradients 781
13.6 Directional Derivatives and Gradients
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the directional derivative of the function at P in the direction of .
a.
b.
c.
d.
e.
____ 2. Find the directional derivative of the function at P in the direction of .
a.
b.
c.
d.
e.
782 Chapter 13: Functions of Several Variables
____ 3. Find the directional derivative of the function at P in the direction of .
a. 4/
52
b. –8/
52
c. 12/
52
d. 24/
52
e. –24/
52
____ 4. Find the directional derivative of the function at in the
direction of . Round your answer to two decimal places.
a. –11.64
b. –1.87
c. –5.86
d. –0.70
e. –5.62
____ 5. Find the gradient of the function at the given point.
a.
b.
c.
d.
e.
13.6 Directional Derivatives and Gradients 783
____ 6. Use the gradient to find the directional derivative of the function at P in the direction
of Q.
a.
b.
c.
d.
e.
____ 7. Use the gradient to find the directional derivative of the function at P in the direction
of Q.
a.
b.
c.
d.
e.
____ 8. Find the maximum value of the directional derivative at the point of the
function . Round your answer to two decimal places.
a. 0.55
b. 0.14
c. 0.56
d. 0.10
e. 0.06
784 Chapter 13: Functions of Several Variables
____ 9. Find the maximum value of the directional derivative at the point of the
function . Round your answer to two decimal places.
a. 899.28
b. 814.59
c. 922.05
d. 933.23
e. 538.80
____ 10. Find for function .
a.
b.
c.
d.
e.
____ 11. For function , find the maximum value of the directional
derivative at (3,2).
a.
15
/8
b.
113
/56
c.
113
/224
d.
113
/112
e.
113
/168
13.6 Directional Derivatives and Gradients 785
____ 12. Use the gradient to find a normal vector to the graph of the equation at the given
point.
a.
b.
c.
d.
e.
____ 13. The temperature at the point on a metal plate is . Find the
direction of greatest increase in heat from the point . Round all numerical values in your answer
to three decimal places.
a.
b.
c.
d.
e.
____ 14. The surface of a mountain is modeled by the equation
. A mountain climber is at the point . In what
direction should the climber move in order to ascend at the greatest rate? Round all numerical values
in your answer to one decimal place.
a.
b.
c.
786 Chapter 13: Functions of Several Variables
d.
e.
____ 15. Find the path of a heat-seeking particle placed at point on a metal plate
with a temperature field .
a.
b.
c.
d.
e.
____ 16. The temperature at the point on a metal plate is modeled by
, . Find the directions of no change in heat on the plate from the
point .
a. There will be no change in directions perpendicular to the gradient .
b. There will be no change in directions parallel to the gradient .
c. There will be no change in directions parallel to the gradient .
d. There will be no change in directions perpendicular to the gradient .
e. There will be no change in directions perpendicular to the gradient .
13.6 Directional Derivatives and Gradients 787
____ 17. The temperature at the point on a metal plate is modeled by
, . Find the direction of greatest increase in heat from the point
.
a. The greatest increase is in the direction of the gradient .
b. The greatest increase is in the direction of the gradient .
c. The greatest increase is in the direction of the gradient .
d. The greatest increase is in the direction of the gradient .
e. The greatest increase is in the direction of the gradient .
788 Chapter 13: Functions of Several Variables
13.6 Directional Derivatives and Gradients – Answer Section