Deck 14: Partial Derivatives

Question 1

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

A)
B)
C)
D)
E)

Answer 1

Question 2

At what point is the following function a local minimum?

A)
B)
C)
D)
E)

Answer 2

Question 3

Find the absolute minimum value of the function on the set D. D is the region bounded by the parabola and the line

A)0
B)
C)
D)30
E)

Answer 3

Question 4

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

Question 5

Find the points on the surface that are closest to the origin.

A)
B)
C)
D)
E)

Question 6

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

Question 7

Find and classify the relative extrema and saddle points of the function for and .

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

Question 8

Find all the saddle points of the function.

A)
B)
C)
D)
E)

Question 9

Find the direction in which the maximum rate of change of f at the given point occurs.

A)
B)
C)
D)
E)

Question 10

Find the shortest distance from the point to the plane .

A)
B)
C)
D)
E)

Question 11

Find the critical points of the function.

A)
B)
C)
D)
E)

Question 12

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 ?

A)f has a local maximum at (1,1)
B)f has a local minimum at (1,1)
C)f has a saddle point at (1,1)

Question 13

At what point is the following function a local maximum?

A)
B)
C)
D)
E)

Question 14

Find the absolute extrema of the function on the closed triangular region with vertices , and .

A)Absolute minimum 5, Absolute maximum 17
B)Absolute minimum -5, Absolute maximum 5
C)Absolute minimum 0, Absolute maximum 5
D)Absolute minimum -5, Absolute maximum 17

Question 15

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

Question 16

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is

A) , ,
B) , ,
C)32, , 16
D)32, 32, 32
E)4, 8, 16

Question 17

Find the dimensions of the rectangular box with largest volume if the total surface area is given as .

A) cm, cm, cm
B) cm, 1.75 cm, 1.75 cm
C) cm, cm, cm
D) cm, cm, 3.5 cm
E) cm, cm, 1.75 cm

Question 18

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

A)
B)
C)
D)
E)

Question 19

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

Question 20

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

A)
B)
C)
D)
E)

Question 21

Evaluate the gradient of f at the point P.

Question 22

Which of the given points are the points on the hyperboloid where the normal line is parallel to the line that joins the points and .
Select all that apply.

A)
B)
C)
D)
E)

Question 23

Find the gradient of the function .

A)
B)
C)
D)
E)

Question 24

Find the maximum rate of change of at the point (2,1). In what direction does it occur?

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

Question 25

Find equations for the tangent plane and the normal line to the surface with equation at the point

A) ,
B) ,
C) ,
D) ,

Question 26

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

Question 27

Find and classify the relative extrema and saddle points of the function .

Question 28

Find the maximum rate of change of f at the given point.

Question 29

Find equations for the tangent plane and the normal line to the surface with equation at the point

A) ,
B) ,
C) ,
D) ,

Question 30

If use the gradient vector to find the tangent line to the level curve at the point .

A)
B)
C)
D)
E)

Question 31

Find the absolute extrema of the function on the region bounded by the disk defined by .

Question 32

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

Question 33

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)28
D)
E)none of these

Question 34

A cardboard box without a lid is to have a volume of cm . Find the dimensions that minimize the amount of cardboard used.

Question 35

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)11
B)
C)1
D)

Question 36

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

A)
B)
C)
D)
E)

Question 37

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

Question 38

Find the direction in which the function decreases fastest at the point .

Question 39

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)44
B)
C)-2.91
D)20
E)

Question 40

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

Question 41

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.?

A)
B)
C)
D)
E)

Question 42

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)113.33
C)113.81
D)113.89
E)113.23

Question 43

Use the Chain Rule to find and if and

A)
B)
C)
D)

Question 44

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

Question 45

Use partial derivatives to find the implicit partial derivatives and

Question 46

Find the differential of the function

A)
B)
C)
D)

Question 47

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

A)
B)
C)
D)
E)

Question 48

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.

Question 49

Find the gradient of the function .

Question 50

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)6.7
B)6.91
C)6.99
D)8.34
E)

Question 51

Find the limit if .

Question 52

Find the gradient of at the point

Question 53

Use the Chain Rule to find .

A)
B)
C)
D)
E)

Question 54

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

Question 55

Use the Chain Rule to find

A)
B)
C)
D)

Question 56

Use the Chain Rule to find where .

Question 57

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

Question 58

Use the Chain Rule to find .

A)
B)
C)
D)
E)

Question 59

Use implicit differentiation to find .

Question 60

Use the equation to find .

A)
B)
C)
D)
E)

Question 61

Find for .

A)
B)
C)
D)
E)0

Question 62

Find for the function

A)
B)
C)
D)

Question 63

Use implicit differentiation to find

A)
B)
C)
D)

Question 64

Use the definition of partial derivatives as limits to find if .

A)
B)
C)
D)
E)

Question 65

Find for the function .

A)
B)
C)
D)
E)

Question 66

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 67

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

Question 68

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

Question 69

How many nth-order partial derivatives does a function of two variables have?

A)
B)
C)
D)
E)

Question 70

Let and suppose that changes from to (a) Compute (b) Compute

Question 71

Find the differential of the function.

Question 72

Find the differential of the function

Question 73

Use implicit differentiation to find

A)
B)
C)
D)

Question 74

Find for the function

A)
B)
C)
D)

Question 75

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

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 77

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 78

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height cm if the tin is 0.04 cm thick.

Question 79

Find the differential of the function

Question 80

If and changes from (2, 1) to find dz.