Stewart – Calculus ET 8e Chapter 14 Form A
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
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
Answer Key
Stewart – Calculus ET 8e Chapter 14 Form A
Stewart – Calculus ET 8e Chapter 14 Form B
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
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
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
Answer Key
Stewart – Calculus ET 8e Chapter 14 Form C
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.
6
Stewart – Calculus ET 8e Chapter 14 Form C
____ 3. Find the domain and range of the function .
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 14 Form C
____ 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.
1
8
c.
1
8
d.
1
2
Stewart – Calculus ET 8e Chapter 14 Form C
____ 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
____ 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
____ 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. –0.96
Stewart – Calculus ET 8e Chapter 14 Form C
____ 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 these
Stewart – Calculus ET 8e Chapter 14 Form C
____ 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
Answer Key
Stewart – Calculus ET 8e Chapter 14 Form D
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 determined
Stewart – Calculus ET 8e Chapter 14 Form D
____ 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
____ 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/mi