13.5 Chain Rules for Functions of Several Variables
808
a.
b.
c.
d.
e.
809 Chapter 13: Functions of Several Variables
13.5 Chain Rules for Functions of Several
Variables Answer Section
13.6 Directional Derivatives and Gradients
810
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.
811 Chapter 13: Functions of Several Variables
____ 3. Find the directional derivative of the function at P in the direction of .
4/5 2
–8/5 2
12/5 2
24/5 2
–24/5 2
____ 4. Find the directional derivative of the function at in the direction
of . Round your answer to two decimal places.
–11.64
–1.87
–5.86
–0.70
–5.62
____ 5. Find the gradient of the function at the given point.
a.
b.
c.
d.
e.
13.6 Directional Derivatives and Gradients
812
____ 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
813
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.
899.28
814.59
922.05
933.23
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).
15 /8
113 /56
113 /224
113 /112
113 /168
13.6 Directional Derivatives and Gradients
814
____ 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.
815 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 .
There will be no change in directions perpendicular to the gradient .
There will be no change in directions parallel to the gradient .
There will be no change in directions parallel to the gradient .
There will be no change in directions perpendicular to the gradient .
There will be no change in directions perpendicular to the gradient .
____ 17. The temperature at the point on a metal plate is modeled by
, . Find the direction of greatest increase in heat from the point
.
13.6 Directional Derivatives and Gradients
816
The greatest increase is in the direction of the gradient .
The greatest increase is in the direction of the gradient .
The greatest increase is in the direction of the gradient .
The greatest increase is in the direction of the gradient .
The greatest increase is in the direction of the gradient .
817 Chapter 13: Functions of Several Variables
13.6 Directional Derivatives and Gradients
Answer Section
13.7 Tangent Planes and Normal Lines
818
13.7 Tangent Planes and Normal Lines
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. For the function given by describe the level surface
given by .
sphere of radius 4 centered at the origin
circle of radius 4 centered at the origin
elliptic cone centered at the origin
sphere of radius 16 centered at the origin
right circular cone centered at the origin
____
2.
Find the unit normal vector to the surface
at the point
.
a.
b.
c.
d.
e.
____
3.
Find a unit normal vector to the surface
at the point
.
a.
b.
c.
819
Chapter 13: Functions of Several Variables
d.
e.
____
4.
Find a unit normal vector to the surface
at the point
.
a.
b.
c.
d.
e.
____
5.
Find a unit normal vector to the surface
at the point
.
a.
b.
c.
d.
e.
13.7 Tangent Planes and Normal Lines
820
____
6.
Find an equation of the tangent plane to the surface
at the point
.
a.
b.
c.
d.
e.
____
7.
Find an equation of the tangent plane to the surface
at the point
.
a.
b.
c.
d.
e.
____ 8. Find an equation of the tangent plane to the surface at the point
.
a.
b.
c.
d.
e.
821 Chapter 13: Functions of Several Variables
____ 9. Find symmetric equations of the normal line to the surface at the point
.
a.
b.
c.
d.
e.
____ 10. Find symmetric equations of the normal line to the surface at the point
.
a.
b.
c.
d.
e.
13.7 Tangent Planes and Normal Lines
822
____
11.
Find symmetric equations of the normal line to the surface
at the
point
.
a.
b.
c.
d.
e.
____
12.
Find symmetric equations of the tangent line to the curve of intersection of the
surfaces
at the point
.
a.
b.
c.
d.
e.
823 Chapter 13: Functions of Several Variables
____ 13. Find symmetric equations of the tangent line to the curve of intersection of the
surfaces at the point .
a.
b.
c.
d.
e.
____ 14. Find the angle of inclination of the tangent plane to the surface at the
point . Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 15. Identify the point(s) on the surface where the tangent plane is horizontal.
a.
b.
c.
d.
e.
13.7 Tangent Planes and Normal Lines
824
____ 16.
Find the point(s) on the hyperboloid
where the tangent plane is
perpendicular to the line with parametric equations
and
.
a.
b.
c.
d.
e.
825 Chapter 13: Functions of Several Variables
13.7 Tangent Planes and Normal Lines
Answer Section
13.8 Extrema of Functions of Two Variables
826
13.8 Extrema of Functions of Two Variables
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Identify any extrema of the function by recognizing its
given form or its form after completing the square.
relative maximum:
relative maximum:
relative minimum:
relative minimum:
relative maximum:
____ 2. Identify any extrema of the function by recognizing
its given form or its form after completing the square.
relative maximum:
relative minimum:
relative minimum:
relative maximum:
relative maximum:
____ 3. Examine the function for relative extrema.
relative minimum:
relative maximum:
relative maximum:
827 Chapter 13: Functions of Several Variables
relative maximum:
relative minimum:
____ 4. Examine the function
relative minimum:
relative maximum:
relative minimum:
relative minimum:
relative maximum:
____ 5. Examine the function
relative maximum:
relative minimum:
relative maximum:
relative minimum:
no relative extrema
____ 6. Examine the function
saddle points.
saddle point:
relative minimum:
relative minimum:
for relative extrema.
for relative extrema.
for relative extrema and