Deck 11: Partial Derivatives

Question 1

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

Answer 1

Question 2

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is 84

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

Answer 2

, ,

Question 3

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

A)
B)
C)
D)
E)

Answer 3

Question 4

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

A)
B)
C)
D)
E)

Question 5

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

Question 6

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

Question 7

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

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

Question 8

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)
E) 30

Question 9

Find three positive numbers whose sum is 291 and whose product is a maximum.

A)
B)
C)
D)
E)

Question 10

Find three positive numbers whose sum is 400 and whose product is a maximum.

Question 11

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

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

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

Question 13

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 14

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

Question 15

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

Question 16

Find the critical points of the function.

A)
B)
C)
D)
E)

Question 17

Find the shortest distance from the point to the plane .

A)
B)
C)
D)
E)

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

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

Question 20

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

Question 21

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

A)
B)
C)
D)
E)

Question 22

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) 1
B)
C) 11
D)

Question 23

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

A)
B)
C)
D)
E)

Question 24

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 25

Use the Chain Rule to find where .

Question 26

Find for the function .

A)
B)
C)
D)
E)

Question 27

Find the differential of the function

Question 28

Use the Chain Rule to find .

A)
B)
C)
D)
E)

Question 29

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 10 cm if the tin is 0.04 cm thick.

Question 30

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

A)
B)
C)
D)
E)

Question 31

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

Question 32

Use the Chain Rule to find

A)
B)
C)
D)

Question 33

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

Question 34

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

Question 35

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

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

Question 36

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 108 in.and the height is 132 in.?

A)
B)
C)
D)
E)

Question 37

Find for the function

A)
B)
C)
D)

Question 38

Find the differential of the function

A)
B)
C)
D)

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)
D) -2.91
E) 20

Question 40

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

A)
B)
C)
D)
E)

Question 41

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 42

Find the limit

A)
B)
C)
D)

Question 43

Use implicit differentiation to find

A)
B)
C)
D)

Question 44

Find the limit

A) 34
B) 16
C) 80
D) 260

Question 45

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

A)
B)
C)
D)
E)

Question 46

Find the differential of the function.

Question 47

Find all the second partial derivatives.

Question 48

Use implicit differentiation to find

A)
B)
C)
D)

Question 49

Find for .

Question 50

Determine the largest set on which the function is continuous.

A)
B)
C)
D)
E)

Question 51

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 52

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 53

Find the first partial derivatives of the function.

Question 54

Use partial derivatives to find the implicit derivative

Question 55

Find .

Question 56

Find the limit

A)
B)
C)
D)

Question 57

Evaluate the limit.

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

Question 58

Evaluate the limit.

A)
B)
C) 0
D)
E)

Question 59

Find the first partial derivatives of the function

Question 60

Find the limit

A)
B)
C)
D)

Question 61

Find and sketch the domain of the function

Question 62

Describe the level surfaces of the function .

Question 63

Two contour maps are shown.One is for a function f whose graph is a cone.The other is for a function g whose graph is a paraboloid.Which is the contour map of a cone?

A) II
B) I
C) impossible to determine

Question 64

Determine where the function is continuous.

A)
B)
C)
D)

Question 65

Determine the largest set on which the function is continuous.

Question 66

Find the limit.

Question 67

Find ,if and .

Question 68

A contour map for a function f is shown.Use it to estimate the value of .

A) 78
B) 35
C) 48
D) 28
E) 71

Question 69

Determine where the function is continuous.

Question 70

Describe the level surfaces of the function .

Question 71

Use spherical coordinates to find the limit.