Deck 11: Partial Derivatives

Question 1

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

A)
B)
C)
D)
E)

Answer 1

Question 2

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

Answer 2

Relative minimum Saddle points and

Question 3

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

Answer 3

Question 4

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

Question 5

Find the shortest distance from the point to the plane .

A)
B)
C)
D)
E)

Question 6

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

A)
B)
C)
D)
E)

Question 7

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

A)
B)
C)
D)
E)

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 and whose product is a maximum.

A)
B)
C)
D)
E)

Question 10

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

A) , ,

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

E)32, 32, 32

Question 11

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

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

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

Question 13

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

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

Question 15

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

Question 16

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

Question 17

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

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

Question 18

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

Question 19

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

Question 20

Find the critical points of the function.

A)
B)
C)
D)
E)

Question 21

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 22

Find the differential of the function

A)
B)
C)
D)

Question 23

Use the Chain Rule to find

A)
B)
C)
D)

Question 24

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

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

Question 25

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 26

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)
D)11

Question 27

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

Question 28

Find for the function

A)
B)
C)
D)

Question 29

Use the Chain Rule to find .

A)
B)
C)
D)
E)

Question 30

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

Question 31

Find for the function .

A)
B)
C)
D)
E)

Question 32

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

Question 33

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 34

Use the Chain Rule to find where .

Question 35

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

A)
B)
C)
D)
E)

Question 36

Find the differential of the function

Question 37

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

A)
B)
C)
D)
E)

Question 38

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

Question 39

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

A)
B)
C)
D)
E)

Question 40

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

Question 41

Find the differential of the function.

Question 42

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 43

Find the indicated partial derivative.

A)
B)
C)
D)
E)

Question 44

Find all the second partial derivatives.

Question 45

Find for .

Question 46

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

A)
B)
C)
D)
E)

Question 47

Find the first partial derivatives of the function

Question 48

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

A)
B)
C)
D)
E)

Question 49

Use partial derivatives to find the implicit derivative

Question 50

Evaluate the limit.

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

Question 51

Use implicit differentiation to find

A)
B)
C)
D)

Question 52

Find the limit

A)
B)
C)
D)

Question 53

Find the limit

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

Question 54

Find .

Question 55

Use implicit differentiation to find

A)
B)
C)
D)

Question 56

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)0.291
B)0.476
C)0.347
D)0.957
E)0.098

Question 57

Find the first partial derivatives of the function.

Question 58

Find the limit

A)
B)
C)
D)

Question 59

Evaluate the limit.

A)
B)0
C)
D)
E)

Question 60

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 61

Find , if and .

Question 62

Describe the level surfaces of the function .

Question 63

Describe the level survaces of the function .

Question 64

Find and sketch the domain of the function

Question 65

Find the limit.

Question 66

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

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

Question 67

Determine where the function is continuous.

A)
B)
C)
D)

Question 68

Use spherical coordinates to find the limit.

Question 69

Find the limit

A)
B)
C)
D)

Question 70

Determine the largest set on which the function is continuous.

A)
B)
C)
D)
E)

Question 71

Determine where the function is continuous.

Question 72

Determine the largest set on which the function is continuous.

Question 73

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)impossible to determine
B)II
C)I