Deck 13: Partial Derivatives

Question 1

Use the Lagrange multiplier method to find the volume of the largest rectangular box that can be inscribed in the ellipsoid 2x² + 3y² + 4z² = 12.

Answer 1

, , and z = 1, thus, the maximum volume is .

Question 2

Use the Lagrange multiplier method to find the point on the plane x + 2y + 8z = 1 that is closest to the point (1, 1, 0).

Answer 2

The closest point to (1, 1, 0) is .

Question 3

An open rectangular box is to contain 256 cubic inches. Use the Lagrange multiplier method to find the dimensions of the box which uses the least amount of material.

Answer 3

x = 8 in, y = 8 in, and z = 4 in.

Question 4

Use Lagrange multipliers to find the volume of the largest rectangular box that can be inscribed within the sphere .

A)
B)
C)
D)
E)

Question 5

Use the Lagrange multiplier method to find three positive numbers whose sum is 16 and whose product, x²yz is a maximum.

Question 6

Let . There is a critical point at

A) (10, 4)
B) (5, 2)
C) (-10, -4)
D) (0, 0)
E) (1, 1)

Question 7

Use the Lagrange multiplier method to find the maximum sum of 2(x² + y² + z²) if
2(x + 2y + 2z) = 24.

Question 8

An open rectangular box is to contain 864 cubic inches. Use the Lagrange multiplier method to find the dimensions of the box which uses the least amount of material.

Question 9

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

A) (0, 0)
B) ,

C) ,

,

D) ,

,

E) There are none

Question 10

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

A) (0, 0, 0)
B)
C)
D)
E) There are none

Question 11

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

A) No maximum or minimum values.
B) The maximum value is and the minimum value is

C) The maximum value is and the minimum value is

D) The maximum value is , and the minimum value is

E) The maximum value is , and the minimum value is

Question 12

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

A) The maximum value is 0, and there is no minimum value. 2
B) The maximum value is , and the minimum value is

C) The maximum value is , and the minimum value is

D) The maximum value is , and the minimum value is

E) There is no maximum value, and the minimum value is 0.

Question 13

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

A) (0, 0)
B)
C) ,

D) ,

,

E) There are none

Question 14

Use the Lagrange multiplier method to find the point on the surface z = xy + 10 that is closest to the origin.

Question 15

Use the Lagrange multiplier method to find three positive numbers whose sum is 12 and whose product, 2x²yz + 2 is a maximum.

Question 16

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

A) (0, 0, 0)
B)
C)
D)
E) There are none

Question 17

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

A) The maximum value is 0, and there is no minimum value.
B) The maximum value is and the minimum value is

C) The minimum value is 0, and there is no maximum value.
D) The minimum value is , and there is no maximum value.
E) The maximum value is and the minimum value is 0.

Question 18

Use Lagrange multipliers to find the volume of the largest rectangular box that can be inscribed within the ellipsoid .

A)
B)
C)
D)
E)

Question 19

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

A) The maximum value is 0, and there is no minimum value.
B) The maximum value is , and the minimum value is

C) The maximum value is , and the minimum value is

D) The maximum value is , and the minimum value is

E) There is no maximum value, and the minimum value is 0.

Question 20

Let . There is a critical point at

A) (10, 4)
B) (5, 2)
C) (-10, -4)
D) (0, 0)
E) (1, 1)

Question 21

A rectangular box is to contain cubic inches. Find the dimensions of the box for which the surface area is a minimum.

A)
B)
C)
D)
E)

Question 22

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 23

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 24

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 25

Let . is

A) A relative maximum
B) A relative minimum
C) A saddle point
D) Cannot be determined
E) Both a relative maximum and a saddle point

Question 26

Let . There is a critical point at

A) (0, 0)
B)
C)
D) none exist
E) (1, 1)

Question 27

A rectangular box, open at the top, is to contain cubic inches. Find the dimensions of the box for which the surface area is a minimum.

A) in
B)
C)

D)
E)

Question 28

Locate all relative maxima, relative minima, and saddle points for
f(x, y) = x² - xy + y ² + 2x + 2y - 3.

Question 29

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 30

Find the point on that is closest to the origin.

A)
B)
C)
D)
E)

Question 31

Let . is

A) A relative maximum
B) A relative minimum
C) A saddle point
D) Cannot be determined
E) Both a relative maximum and a saddle point

Question 32

Let . There is a critical point at

A) (0, 0)
B)
C)
D) none exist
E) (1, 1)

Question 33

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 34

Let . There is a critical point at

A) (10, 4)
B) (5, 2)
C)
D) (0, 0)
E) (1, 1)

Question 35

Locate all relative maxima, relative minima, and saddle points for
f(x, y) = x² - 2y² - 6x + 8y + 41.

Question 36

Let . There is a critical point at

A) (0, 0)
B)
C)
D) none exist
E) (1, 1)

Question 37

Find an equation for the tangent plane to at .

A)
B)
C)
D)
E)

Question 38

Find the minimum sum of 9x + 5y + 3z + 8 if x, y, and z are positive numbers such that xyz = 25.

Question 39

Find the point(s) on that are closest to .

A)
B)
C) , and

D) (0, 0, 0) and (1, 1, 1)
E)

Question 40

Let . There is a critical point at

A) (0, 0)
B)
C)
D) none exist
E) (1, 1)

Question 41

Find a point on the surface z = 16 - 12x² - y² at which the tangent plane is perpendicular to the line x = 3 + 12t, y = 2t, z = 2 - t.

Question 42

Find the equations of the tangent plane and normal line to z = xe^{sin y} at (2, $\pi$ , 5).Express the equation of the normal line parametrically.

Question 43

Let . Find if .

A)
B)
C)
D)
E)

Question 44

Let ; Find .

A)
B)
C)
D)
E)

Question 45

Let . Find at the point .

A)
B)
C)
D)
E)

Question 46

Find parametric equations for the normal line to at .

A)
B)
C)
D)
E)

Question 47

Find the equations of the tangent plane and normal line to at . Express the equation of the normal line parametrically.

Question 48

Let ; Find .

A)
B)
C)
D)
E)

Question 49

Let Find .

A)
B)
C)
D)
E)

Question 50

Let . Find if .

A)
B)
C)
D)
E)

Question 51

Find parametric equations for the normal line to at .

A)
B)
C)
D)
E)

Question 52

Find the equations of the tangent plane and normal line to at .Express the equation of the normal line parametrically.

Question 53

Let ; Find .

A)
B)
C)
D)
E)

Question 54

Find parametric equations for the normal line to at .

A)
B)
C)
D)
E)

Question 55

Find the equations of the tangent plane and normal line to x²z - xy² - yz² - 18 = 0 at (0, -2, 4).

Question 56

Let . Find if .

A)
B)
C)
D)
E)

Question 57

Let ; Find .

A)
B)
C)
D)
E)

Question 58

Find all points on the surface z = xe^-y + 8 at which the tangent plane is horizontal.

Question 59

Let ; Find .

A)
B)
C)
D)
E)

Question 60

Let ; Find .

A)
B)
C)
D)
E)

Question 61

Find the rate of change of at in the direction of .

Question 62

Find the rate of change of at (1, 6) in the direction of a vector making an angle of 120° with the positive x axis.

Question 63

A particle is located at the point (2, 7) on a metal surface whose temperature at a point (x, y) is T(x, y) = 16 - 2x² - 3y². Find the equation for the trajectory of a particle moving continuously in the direction of maximum temperature increase. y =

A)
B)
C)
D)
E) 1

Question 64

Let ; Find .

A)
B)
C)
D)
E)

Question 65

Find the gradient for f(x, y, z) = 7x⁴y³z.

Question 66

Let ; Find .

A)
B)
C)
D)
E)

Question 67

Let ; Find .

A)


B)


C)


D)


E)

Question 68

Let ; x = u + v , y = u - v. Find .

A)
B)
C)
D)
E)

Question 69

Let ; Find .

A)
B)
C)
D)
E)

Question 70

Let ; Find .

A)
B)
C)
D)
E)

Question 71

Let ; Find .

A)
B)
C)
D)
E)

Question 72

Let . Find if .

A)
B)
C)
D)
E)

Question 73

Let ; Find .

A)
B)
C)
D)
E)

Question 74

The sides of a rectangle are measured to be and cm with a maximum error of cm in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the area.

A) is the maximum error in the area.
B) is the maximum error in the area.
C) is the maximum error in the area.
D) is the maximum error in the area.
E) is the maximum error in the area.

Question 75

Let . Find if .

A)
B)
C)
D)
E)

Question 76

Let ; Find .

A)
B)
C)
D)
E)

Question 77

At t = 0, the position of a particle on a rectangular membrane is given by . Find the rate at which P changes if the particle moves from in a direction of a vector making an angle 30° with the positive x-axis.

Question 78

A particle is located at the point (5, 5) on a metal surface whose temperature at a point (x, y) is T(x, y) = 25 - 3x² - 2y². Find the equation for the trajectory of a particle moving continuously in the direction of maximum temperature increase. y =

A)
B)
C)
D)
E) 1

Question 79

Let . Find if .

A)
B)
C)
D)
E)

Question 80

Let ; . Using the chain rule, find .

A)
B)
C)
D)
E)