Deck 14: Partial Derivatives

Question 1

If

Answer 1

Question 2

Find the second partial derivatives of the function

Answer 2

Question 3

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

Answer 3

Question 4

Find the differential of the function.

Question 5

Describe the level survaces of the function

Question 6

Find the local maximum, and minimum value and saddle points of the function.

Question 7

Use partial derivatives to find the implicit derivative

Question 8

Find the first partial derivatives of the function.

Question 9

Find the first partial derivatives of the function

Question 10

Find the differential of the function

Question 11

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the constraints

Question 12

Find the gradient of

Question 13

Find the equation of the normal line to the given surface at the specified point.

Question 14

Determine where the function is continuous.

Question 15

The length l, width w and height h of a box change with time.At a certain instant the dimensions are nd 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.

Question 16

Use implicit differentiation to find

Question 17

Sketch the graph of the function

Question 18

Find all the second partial derivatives.

Question 19

Find the differential of the function

Question 20

Find three positive numbers whose sum is and whose product is a maximum.

Question 21

Find an equation of the tangent plane to the given surface at the specified point.

Question 22

Find the limit.

Question 23

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints.

Question 24

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s).

Question 25

Find

Question 26

The (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 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

Question 27

Find the linearization L(x, y) of the function at the given point. Round theAnswers to the nearest hundredth.

Question 28

Find the differential of the function.

Question 29

Use the Chain Rule to find

Question 30

Find for the function

Question 31

Determine the largest set on which the function is continuous.

Question 32

Find the first partial derivatives of the function.

Question 33

Use polar coordinates to find the limit.

Question 34

Question 35

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.

Question 36

Use the linearization L(x, y) of the function.

Question 37

Find

Question 38

Find the equation of the normal line to the given surface at the specified point.

Question 39

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.

Question 40

Find three positive numbers whose sum is and whose product is a maximum.

Question 41

Select the correct Answer: for each question.
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

Question 42

Select the correct Answer: for each question.
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)

Question 43

Select the correct Answer: for each question.
Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 44

Select the correct Answer: for each question.
Find equations for the tangent plane and the normal line to the surface with equation at the point

A)
B)
C)
D)

Question 45

Select the correct Answer: for each question.
Find the limit

A)52
B)16
C)73
D)19

Question 46

Select the correct Answer: for each question.
Find the gradient of the function

A)
B)
C)
D)
E)

Question 47

Select the correct Answer: for each question.
Find three positive numbers whose sum is and whose product is a maximum.

A)
B)
C)
D)
E)

Question 48

Select the correct Answer: for each question.
Find and classify the relative extrema and saddle points of the function

A)None
B) Relative maximum
C)Saddle point Relative minimum
D)

Question 49

Select the correct Answer: for each question.
Use the Chain Rule to find

A)
B)
C)
D)
E)

Question 50

Select the correct Answer: for each question.
Evaluate the limit.

A)
B)0
C)
D)
E)

Question 51

Select the correct Answer: for each question.
Find

A)
B)
C)
D)
E)0

Question 52

Select the correct Answer: for each question.
Find the limit

A)
B)
C)
D)

Question 53

Select the correct Answer: for each question.
Find the maximum rate of change of at the point In what direction does it occur?

A)
B)
C)
D)
E)none of these

Question 54

Select the correct Answer: for each question.
Let Find

A)
B)
C)36
D)6

Question 55

Select the correct Answer: for each question.
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)

Question 56

Select the correct Answer: for each question.
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)

Question 57

Select the correct Answer: for each question.
Let

A)
B)
C)
D)

Question 58

Select the correct Answer: for each question.
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

Question 59

Select the correct Answer: for each question.
Find the limit Hint: If if and only if

A)
B)0
C)
D)1

Question 60

Select the correct Answer: for each question.
Find the domain and range of the function

A)
B)
C)
D)

Question 61

Select the correct Answer: for each question.
Find equations for the tangent plane and the normal line to the surface with equation at the point

A)
B)
C)
D)

Question 62

Select the correct Answer: for each question.
Use implicit differentiation to find

A)
B)
C)
D)

Question 63

Select the correct Answer: for each question.
Use the Chain Rule to find

A)
B)
C)
D)
E)

Question 64

Select the correct Answer: for each question.
At what point is the following function a local maximum?

A)
B)
C)
D)
E)

Question 65

Select the correct Answer: for each question.
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

Question 66

Select the correct Answer: for each question.
Find and classify the relative extrema and saddle points of the function

A)None
B) Relative maximum
C)Saddle point Relative minimum
D)

Question 67

Select the correct Answer: for each question.
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

Question 68

Select the correct Answer: for each question.
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)

Question 69

Select the correct Answer: for each question.
Use the equation

A)
B)
C)
D)
E)

Question 70

Select the correct Answer: for each question.
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

Question 71

Let Select the correct Answer

A)
B)
C)
D)

Question 72

Select the correct Answer: for each question.
Use the Chain Rule to find

A)
B)
C)
D)
E)

Question 73

Select the correct Answer: for each question.
If use the gradient vector to find the tangent line to the level curve at the point

A)
B)
C)
D)
E)

Question 74

Select the correct Answer: for each question.
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

Question 75

Select the correct Answer: for each question.
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

Question 76

Select the correct Answer: for each question.
Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 77

Select the correct Answer: for each question.
Use Lagrange multipliers to find the maximum value of the function subject to the given constraint.

A)
B)
C)
D)
E)

Question 78

Select the correct Answer: for each question.
Evaluate the limit.

A)2
B)the limit does not exist
C)
D)1
E)0

Question 79

Select the correct Answer: for each question.
Use the Chain Rule to find

A)
B)
C)
D)

Question 80

Select the correct Answer: for each question.
Find the domain and range of the function

A)
B)
C)
D)