1. Find all the second partial derivatives.
2. Find the first partial derivatives of the function.
3. Find the differential of the function.
4. If and changes from (2, 1) to find dz.
5. The length l, width w and heig
...
1. Find all the second partial derivatives.
2. Find the first partial derivatives of the function.
3. Find the differential of the function.
4. If and changes from (2, 1) to find dz.
5. The length l, width w and height h of a box change with time. At a certain instant the dimensions are
and , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1
m/s. At that instant find the rates at which the surface area is changing.
6. Use implicit differentiation to find .
7. Find the equation of the normal line to the given surface at the specified point.
8. Find the local maximum, and minimum value and saddle points of the function.
9. Find three positive numbers whose sum is and whose product is a maximum.
10. Use Lagrange multipliers to find the maximum and minimum values of the function
subject to the constraints and .
11. Sketch the graph of the function
12. Describe the level survaces of the function .Stewart – Calculus ET 8e Chapter 14 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
13. Determine where the function is continuous.
14. Find the first partial derivatives of the function
15. Find the second partial derivatives of the function
16. Use partial derivatives to find the implicit derivative
17. Find the differential of the function
18. Find the differential of the function
19. Find the gradient of at the point
20. Find three positive real numbers whose sum is 388 and whose product is as large as possible.Stewart – Calculus ET 8e Chapter 14 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
Min at , , Max point - none, Saddle point - none
9.
10.
11.
12. A family of parallel planes with normal vector
13.
14.
15.Stewart – Calculus ET 8e Chapter 14 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
16.
17.
18.
19.
20. 388
3 ,
388
3 ,
388
3Stewart – Calculus ET 8e Chapter 14 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
1. The temperature-humidity index I (or humidex, for short) is the perceived air temperature when the
actual temperature is T and the relative humidity is h, so we can write I = f (T, h). The following
table of values of I is an excerpt from a table compiled by the National Oceanic and Atmospheric
Administration. For what value of T is ?
2. Find , if and .
3. Determine the largest set on which the function is continuous.
4. Find the limit.
5. Use polar coordinates to find the limit.Stewart – Calculus ET 8e Chapter 14 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
6. Find .
7. Find the first partial derivatives of the function.
8. Find the differential of the function.
9. Find for .
10. Find for the function .
11. Find an equation of the tangent plane to the given surface at the specified point.
12. Find the linearization L(x, y) of the function at the given point.
Round the answers to the nearest hundredth.
13. Use the linearization L(x, y) of the function.
at to approximate .Stewart – Calculus ET 8e Chapter 14 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
14. The wind-chill index I is the perceived temperature when the actual temperature is T and the wind
speed is v so we can write . The following table of values is an excerpt from a table
compiled by the National Atmospheric and Oceanic Administration. Use the table to find a linear
approximation to the wind chill index function when T is near and v is near 30 kmh.
10 20 30 40 50
20 18 17 14 13 12
16 15 12 10 7 8
12 8 6 2 2
8 5 1
15. Use the Chain Rule to find where .
16. Find the equation of the normal line to the given surface at the specified point.
17. Find three positive numbers whose sum is and whose product is a maximum.
18. Use Lagrange multipliers to find the maximum and the minimum of f subject to the given
constraint(s).
19. Use Lagrange multipliers to find the minimum value of the function subject to the given constraints.
20. Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.Stewart – Calculus ET 8e Chapter 14 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2.
3.
4.
5. 0
6.
7.
8.
9.
10.
11. z = - 1
2 x -
23
y +
17
6
12. L = 5x - 1
2 y +
25
2
13.
14.
15.
16.
17.
18. maximum = 1, minimum = 1
19. 4 14
20.Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Select the correct answer for each question.
____ 1. A contour map for a function f is shown. Use it to estimate the value of .
a. 78
b. 48
c.
d. 35
e. 28
____ 2. Let Find
a. 2
3
b.
c. 36
d. 6Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 3. Find the domain and range of the function .
a.
b.
c.
d.Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 4. Evaluate the limit.
a.
b. 0
c.
d.
e.
____ 5. Find the limit
a. 52
b. 16
c. 73
d. 19
____ 6. Find the limit
a. 1
2
b.
18
c. 1
8
d.
12Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 7. Find the limit Hint: If and then if
and only if
a. 9
2
b. 0
c. 2
9
d. 1
____ 8. Find for .
a.
b.
c.
d.
e. 0
____ 9. Find the indicated partial derivative.
a.
b.
c.
d.
e.
____ 10. The ellipsoid intersects the plane in an ellipse. Find parametric equations
for the tangent line to this ellipse at the point (1, 2, 2).
a.
b.
c.
d.
e.Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 11. Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8
cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01
cm thick. (rounded to the nearest hundredth.)
a. 8.34
b. 6.99
c. 6.91
d. 6.7
e.
____ 12. A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft.
Use differentials to approximate the number of square feet of paint in the stripe.
a. 113
b. 113.81
c. 113.22
d. 113.89
e. 113.33
____ 13. Use the Chain Rule to find .
a.
b.
c.
d.
e.Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. Find the directional derivative of the function at the point in the
direction of the unit vector that makes the angle with the positive x-axis.
a.
b. 13
c. 1
d.
____ 15. Find equations for the tangent plane and the normal line to the surface with equation
at the point
a.
,
b.
,
c.
,
d.
,
____ 16. Suppose that over a certain region of space the electrical potential V is given by
.
Find the rate of change of the potential at in the direction of the vector .
a.
b.
c. 44
d. 20
e. –0.96Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 17. Find the gradient of the function .
a.
b.
c.
d.
e.
____ 18. Find the maximum rate of change of at the point .
In what direction does it occur?
a.
b.
c.
d.
e. none of theseStewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 19. Find and classify the relative extrema and saddle points of the function
for and .
a. None
b.
Relative maximum
c. Saddle point
d.
Relative minimum
____ 20. Find three positive numbers whose sum is and whose product is a maximum.
a.
b.
c.
d.
e.Stewart – Calculus ET 8e Chapter 14 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. C
2. C
3. C
4. E
5. D
6. B
7. A
8. A
9. D
10. A
11. E
12. C
13. A
14. B
15. C
16. E
17. E
18. D
19. A
20. AStewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Select the correct answer for each question.
____ 1. Let Find
a.
b.
c.
d.
____ 2. Find the domain and range of the function .
a.
b.
c.
d.
____ 3. Use a table of numerical values of f (x, y) for (x, y) near the origin to make a conjecture about the
value of the limit of f (x, y) as .
a.
b.
c. 1
d.
e. the value cannot be determinedStewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 4. Evaluate the limit.
a. 2
b. the limit does not exist
c.
d. 1
e. 0
____ 5. Find the indicated partial derivative.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 6. Use implicit differentiation to find
a.
b.
c.
d.
____ 7. The height of a hill (in feet) is given by
where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are
at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the
height of the hill (a) in a northerly direction and (b) in an easterly direction?
a. (a) 570 ft/mi, (b) 690 ft/mi
b. (a) –570 ft/mi, (b) 690 ft/mi
c. (a) 690 ft/mi, (b) 570 ft/mi
d. (a) 690 ft/mi, (b) –570 ft/miStewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 8. If R is the total resistance of three resistors, connected in parallel, with resistances , then
.
If the resistances are measured in ohms as , with a possible error
of 0.8% in each case, estimate the maximum error in the calculated value of R.
a. 0.347
b. 0.476
c. 0.291
d. 0.098
e. 0.957
____ 9. Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8
cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01
cm thick. (rounded to the nearest hundredth.)
a. 8.34
b. 6.99
c. 6.91
d. 6.7
e.
____ 10. A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft.
Use differentials to approximate the number of square feet of paint in the stripe.
a. 113
b. 113.81
c. 113.22
d. 113.89
e. 113.33Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 11. Use the Chain Rule to find .
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 12. Use the Chain Rule to find .
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 13. Use the Chain Rule to find
a.
b.
c.
d.Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. Use the equation to find .
a.
b.
c.
d.
e.
____ 15. Find the directional derivative of at the point (1, 3) in the direction toward the
point (3, 1).
Select the correct answer.
a.
b.
c.
d. 28
e. none of theseStewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 16. Find equations for the tangent plane and the normal line to the surface with equation
at the point
a.
,
b.
,
c.
,
d.
,
____ 17. If use the gradient vector to find the tangent line to the level curve
at the point .
a.
b.
c.
d.
e.
____ 18. Find and classify the relative extrema and saddle points of the function
for and .
a. None
b.
Relative maximum
c. Saddle point
d.
Relative minimumStewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 19. At what point is the following function a local maximum?
a.
b.
c.
d.
e.
____ 20. Use Lagrange multipliers to find the maximum value of the function subject to the given constraint.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. D
2. B
3. A
4. E
5. C
6. B
7. B
8. D
9. E
10. C
11. A
12. B
13. A
14. E
15. B
16. C
17. C
18. A
19. B
20. CStewart - Calculus ET 8e Chapter 14 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Let Find
Select the correct answer.
a.
b.
c.
d.
____ 2. Let Find
Select the correct answer.
a. 9
10
b. 45
2
c. 225
d.
3. Evaluate the limit.
4. Evaluate the limit.
5. Find the limit Hint: If and then if
and only if
6. Determine where the function is continuous.Stewart - Calculus ET 8e Chapter 14 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 7. Find for .
Select the correct answer.
a. 0
b.
c.
d.
e.
8. How many nth-order partial derivatives does a function of two variables have?
9. Use the definition of partial derivatives as limits to find if .
____ 10. Find for the function
Select the correct answer.
a.
b.
c.
d.
11. A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft.
Use differentials to approximate the number of square feet of paint in the stripe.
12. Use the Chain Rule to find .Stewart - Calculus ET 8e Chapter 14 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
13. Use the equation to find .
14. Find equations for the tangent plane and the normal line to the surface with equation
at the point
15. Find the gradient of the function .
____ 16. If use the gradient vector to find the tangent line to the level curve
at the point . Select the correct answer.
a.
b.
c.
d.
e.
17. Find all the saddle points of the function.
18. Find the dimensions of the rectangular box with largest volume if the total surface area is given as
.
____ 19. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its
12 edges is . Select the correct answer.
a. 32, 32, 32
b. 32, , 16
c.
, ,
d.
, ,
e. 4, 8, 16Stewart - Calculus ET 8e Chapter 14 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
20. Use Lagrange multipliers to find the maximum value of the function subject to the given constraint.Stewart - Calculus ET 8e Chapter 14 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. C
2. C
3.
4. 0
5. 3
2
6.
7. B
8.
9.
10. B
11. 113.22
12.
13.
14. ,
15.
16. C
17.
18. cm, cm, cm
19. C
20.Stewart – Calculus ET 8e Chapter 14 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Let Find
Select the correct answer.
a. 2
3
b.
c. 36
d. 6
2. Find the domain and range of the function .
3. Evaluate the limit.
4. Evaluate the limit.
____ 5. Find for .
Select the correct answer.
a.
b.
c.
d.
e. 0
6. Use the definition of partial derivatives as limits to find if .Stewart – Calculus ET 8e Chapter 14 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
7. The height of a hill (in feet) is given by
where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are
at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the
height of the hill (a) in a northerly direction and (b) in an easterly direction?
8. Find the differential of the function
____ 9. Use the Chain Rule to find
Select the correct answer.
a.
b.
c.
d.
10. The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a
rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the
height is in.?
11. Use the equation to find .
12. Find equations for the tangent plane and the normal line to the surface with equation
at the pointStewart – Calculus ET 8e Chapter 14 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 13. Suppose that over a certain region of space the electrical potential V is given by
.
Find the rate of change of the potential at in the direction of the vector .
Select the correct answer.
a.
b.
c. 44
d. 20
e. –0.96
14. Find the gradient of the function .
____ 15. If use the gradient vector to find the tangent line to the level curve
at the point .
Select the correct answer.
a.
b.
c.
d.
e.
16. Find and classify the relative extrema and saddle points of the function
for and .
17. Find the absolute extrema of the function
on the closed triangular region with vertices , and .Stewart – Calculus ET 8e Chapter 14 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 18. Suppose (1, 1) is a critical point of a function f with continuous second derivatives.
In the case of , , what can you say about f ?
Select the correct answer.
a. f has a local maximum at (1,1)
b. f has a saddle point at (1,1)
c. f has a local minimum at (1,1)
19. Find the points on the surface that are closest to the origin.
20. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its
12 edges is .Stewart – Calculus ET 8e Chapter 14 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. C
2.
3.
4. 0
5. A
6.
7. (a) –570 ft/mi, (b) 690 ft/mi
8.
9. A
10.
11.
12. ,
13. E
14.
15. C
16. None
17. Absolute minimum –5, Absolute maximum 17
18. C
19.
20. , ,Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. A contour map for a function f is shown. Use it to estimate the value of .
Select the correct answer.
a.
b. 78
c. 35
d. 28
e. 48
2. Find the domain and range of the function .
3. Find the domain and range of the function .
____ 4. Let Find
Select the correct answer.
a. 24
b. 3
2
c.
d. 6Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
5. Determine the largest set on which the function is continuous.
6. Find the limit
____ 7. Find the limit Hint: If and then if
and only if
Select the correct answer.
a. 1
b. 0
c. 7
6
d. 6
7
8. How many nth-order partial derivatives does a function of two variables have?
____ 9. Use the definition of partial derivatives as limits to find if .
Select the correct answer.
a.
b.
c.
d.
e.
10. Find the indicated partial derivative.Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
11. Use implicit differentiation to find
____ 12. Find the equation of the tangent plane to the given surface at the specified point. \
Select the correct answer.
a.
b.
c.
d.
e.
____ 13. A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft.
Use differentials to approximate the number of square feet of paint in the stripe.
Select the correct answer.
a. 113
b. 113.81
c. 113.22
d. 113.89
e. 113.33
14. Find the equation of the normal line to the given surface at the specified point.Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 15. Suppose that over a certain region of space the electrical potential V is given by
.
Find the rate of change of the potential at in the direction of the vector .
Select the correct answer.
a. 44
b.
c. 20
d.
e. –0.96
16. Find and classify the relative extrema and saddle points of the function
for and .
17. At what point is the following function a local maximum?
____ 18. Find the absolute minimum value of the function
on the set D. D is the region bounded by the parabola and the line
Select the correct answer.
a.
b.
c. 30
d.
e. 0
19. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its
12 edges is .Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 20. Find the shortest distance from the point to the plane .
Select the correct answer.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Stewart - Calculus 8e Chapter 14 Form G
Answer Section
1. A
2.
3.
4. A
5.
6. 11
4
7. D
8.
9. C
10.
11.
12. E
13. C
14.
15. E
16. None
17.
18. B
19. , ,
20. CStewart - Calculus ET 8e Chapter 14 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
1. Find the domain and range of the function .
2. Find the domain and range of the function .
____ 3. Find for .
Select the correct answer.
a.
b.
c.
d.
e. 0
4. Find the indicated partial derivative.
5. Use implicit differentiation to find
6. Use implicit differentiation to findStewart - Calculus ET 8e Chapter 14 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 7. Find for the function
Select the correct answer.
a.
b.
c.
d.
____ 8. Find the differential of the function
Select the correct answer.
a.
b.
c.
d.
9. Use the Chain Rule to find
____ 10. Find the directional derivative of at the point (1, 3) in the direction toward the
point (3, 1).
Select the correct answer.
a.
b.
c.
d. 28
e. none of these
11. Find the equation of the normal line to the given surface at the specified point.Stewart - Calculus ET 8e Chapter 14 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 12. Find the directional derivative of the function at the point in the
direction of the unit vector that makes the angle with the positive x-axis.
Select the correct answer.
a.
b. 13
c. 1
d.
13. Find the gradient of the function .
14. Find the maximum rate of change of at the point .
In what direction does it occur?
15. If use the gradient vector to find the tangent line to the level curve
at the point .
____ 16. At what point is the following function a local minimum?
Select the correct answer.
a.
b.
c.
d.
e.
17. At what point is the following function a local maximum?Stewart - Calculus ET 8e Chapter 14 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 18. Find the points on the surface that are closest to the origin.
Select the correct answer.
a.
b.
c.
d.
e.
19. Find the dimensions of the rectangular box with largest volume if the total surface area is given as
.
____ 20. Find the shortest distance from the point to the plane .
Select the correct answer.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 14 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2.
3. A
4.
5.
6.
7. B
8. B
9.
10. B
11.
12. B
13.
14.
15.
16. E
17.
18. D
19. cm, cm, cm
20. C
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