Deck 11: Partial Derivatives

Question 1

Find the maximum value of the function $f ( x , y ) = 6 - 4 x ^ { 2 } - y ^ { 2 }$ subject to the constraint that $4 x + y = 5$ .

A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { \sqrt { 2 } }$
C)1
D) $\sqrt { 2 }$
E)2
F) $2 \sqrt { 2 }$
G)4
H) $4 \sqrt { 2 }$

Answer 1

1

Question 2

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the maximum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

A) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$

B)
$\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$
C) $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$
D) $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$
E) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$
F) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

G) $\left( - \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

H)None of these

Answer 2

$\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

Question 3

Find the maximum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

A) $\frac { 1 } { 2 }$
B)1
C) $\frac { 3 } { 2 }$
D)2
E) $- \frac { 1 } { 2 }$
F) $- 1$
G) $- \frac { 3 } { 2 }$
H) $- 2$

Answer 3

1

Question 4

Optimize subject to .

Question 5

Use the method of Lagrange multipliers to find points on the surface of where the function has
(a) a minimum
(b) a maximum.

Question 6

Find two positive numbers who sum is eighteen and whose product is a maximum, using the method of Lagrange multipliers

Question 7

Find the greatest product three numbers can have if the sum of their squares must be 48.

Question 8

Compute the minimum value of subject to the condition that .

Question 9

What is the shortest distance from the origin to the surface ?

Question 10

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the minimum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

A) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$

B) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$
C) $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$
D) $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$
E) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$
F) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

G) $\left( - \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

H)None of these

Question 11

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x + y = 3$ , what is the value of the multiplier $\lambda$ ?

A) $\frac { 1 } { 2 }$

B)1
C) $\frac { 3 } { 2 }$
D)2
E) $\frac { 5 } { 2 }$
F)3
G) $\frac { 7 } { 2 }$
H)4

Question 12

Solve completely, using Lagrange multipliers: Find the dimension of a box with volume 1000 which minimizes the total length of the 12 edges.

Question 13

Find the extreme values of on the circle .

Question 14

Find the maximum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

A) $- 4$
B) $- 5$
C) $- 6$
D) $- 7$
E)4
F)5
G)6
H)7

Question 15

Find the point on the plane where is minimum.

Question 16

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x y = 2$ , what is the value of the multiplier $\lambda$ ?

A) $\frac { 1 } { 2 }$
B)1
C) $\frac { 3 } { 2 }$
D)2
E)
$\frac { 5 } { 2 }$
F)3
G) $\frac { 7 } { 2 }$
H)4

Question 17

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x y = 2$ .

A) $\frac { 1 } { 2 }$
B)1

C) $\frac { 3 } { 2 }$
D)2
E) $\frac { 5 } { 2 }$

F)3
G) $\frac { 7 } { 2 }$
H)4

Question 18

Find the maximum and minimum values of the function on the ellipse given by the equation .

Question 19

Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

A) $- 4$
B) $- 5$
C) $- 6$
D) $- 7$
E)4
F)5
G)6
H)7

Question 20

Find the minimum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

A) $\frac { 1 } { 2 }$
B)1
C) $\frac { 3 } { 2 }$
D).2
E) $- \frac { 1 } { 2 }$
F) $- 1$
G) $- \frac { 3 } { 2 }$
H) $- 2$

Question 21

The quality of a good produced by a company is given by , where is the quantity of capital and is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value?
(b) Draw the level curves of and the graph of the budget constraint on the same set of axes.(c) Complete the value of . What does represent?

Question 22

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2 x ^ { 2 } y + 3$ has.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 23

The function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x y$ has one critical point. Determine its location and type.

A) $( 2,1 )$ , saddle point
B) $( 2,1 )$ , minimum point
C) $( 2,1 )$ , maximum point
D) $( \sqrt { 2 } , 1 )$ , saddle point
E) $( \sqrt { 2 } , 1 )$ , minimum point
F) $( \sqrt { 2 } , 1 )$ , maximum point
G) $( 0,0 )$ , saddle point
H) $( 0,0 )$ , minimum point

Question 24

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of . What does represent?

Question 25

Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$

A)90
B)160
C)50
D)15
E)80
F)135
G)16
H)5

Question 26

The function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$ has one critical point. Determine its location and type.

A) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , saddle point
B) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , minimum point
C) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , maximum point
D) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , saddle point
E) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , minimum point
F) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , maximum point
G) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , saddle point
H) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , minimum point

Question 27

The function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y$ has one critical point. Determine its location and type.

A) $( 2,1 )$ , saddle point
B) $( 2,1 )$ , minimum point
C) $( 2,1 )$ , maximum point
D) $( \sqrt { 2 } , 1 )$ , saddle point
E) $( \sqrt { 2 } , 1 )$ , minimum point
F) $( \sqrt { 2 } , 1 )$ , maximum point
G) $( 0,0 )$ , saddle point
H) $( 0,0 )$ , minimum point

Question 28

The level curves of a function and a curve with equation ( constant) are given below. Estimate the point where has a maximum value and the point where has a minimum value, subject to the constraint that . Indicate your answer in the figure.

Question 29

Suppose that the quantity produced of a certain good depends on the number of units of labor and the quantity of capital according to the function . Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?
(b) Draw the level curves of , the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of . What does represent?

Question 30

What point on the surface , , , is closest to the origin?

Question 31

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x ^ { 2 } y + 10$ has.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 32

A package in the shape of a cylindrical box can be mailed by the U.S. Postal Service if the sum of its height and girth (the circumference of the circular base) is at most 108 in. Find the dimension of the package with largest volume that can be mailed.

Question 33

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 } - 9 y - 4 x y + 50$

A)8.5
B)18
C)19
D)10
E)10.5
F)9
G)9.5
H)50

Question 34

A rectangular area of 3200 ft is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of . What does represent?

Question 35

The function $f ( x , y ) = x ^ { 2 } - 2 y ^ { 2 }$ has one critical point. Determine its location and type.

A) $( 2,1 )$ , saddle point
B) $( 2,1 )$ , minimum point
C) $( 2,1 )$ , maximum point
D) $( \sqrt { 2 } , 1 )$ , saddle point
E) $( \sqrt { 2 } , 1 )$ , minimum point
F) $( \sqrt { 2 } , 1 )$ , maximum point
G) $( 0,0 )$ , saddle point
H) $( 0,0 )$ , minimum point

Question 36

A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of . What does represent?

Question 37

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } - 2 x + y ^ { 3 } + y$ has.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 38

Determine how many critical points the function $f ( x , y ) = x y - x ^ { 2 } y + x y ^ { 2 }$ has.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 39

Find the maximum and minimum values of on the circle .

Question 40

Determine how many critical points the function $f ( x , y ) = x ^ { 4 } - 2 x ^ { 2 } + 3 y - y ^ { 3 }$ has.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 41

Find the local maximum and minimum values and saddle points of the function .

Question 42

Find the local maximum and minimum values and saddle points of the function .

Question 43

Find the point at which the function $f ( x , y ) = x y - x ^ { 2 } y - x y ^ { 2 }$ has a local maximum.

A) $( 0,0 )$
B) $( 1,1 )$
C) $( 0,2 )$
D) $( 2,0 )$
E) $( - 1,0 )$
F) $( 0 , - 1 )$

G) $\left( \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$

H) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Question 44

For each of the following functions, find the critical point, if there is one, and determine if it is a local maximum, local minimum, saddle point, or otherwise.(a) (b) (c) (d)

Question 45

Use the level curves of shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points.

Question 46

A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.

A)2

B) $5 / 2$
C)4
D)5
E) $25 ^ { 1 / 3 }$
F) $25 ^ { 2 / 3 }$
G) $200 ^ { 1 / 3 }$
H) $200 ^ { 2 / 3 }$

Question 47

Find the local maximum and minimum values and saddle points of the function .

Question 48

Find the local maximum and minimum values and saddle points of the function .

Question 49

The function has a maximum. Find the values of and at which it occurs.

Question 50

Use the level curves of shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points.

Question 51

Find the local maximum and minimum values and saddle points of the function .

Question 52

Compare the minimum value of and sketch a portion of the graph of near its lowest point.

Question 53

Find the absolute maximum and minimum values of over the region , where is a closed triangular region with vertices , , and .

Question 54

Find three positive numbers , , and whose sum is 48 and product is maximum.

Question 55

Find the shortest distance from the origin to the surface $z ^ { 2 } = 2 x y + 2$ .

A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { \sqrt { 2 } }$
C)1
D) $\sqrt { 2 }$
E)2
F) $2 \sqrt { 2 }$
G)4
H) $4 \sqrt { 2 }$

Question 56

Find the critical points (if any) for and determine if each is a local extreme value or a saddle point.

Question 57

Find the local maximum and minimum values and saddle points of the function .

Question 58

Find the absolute maximum and minimum value of on the square region with vertices , , , and .

Question 59

Find the absolute maximum and minimum value of on the square region with vertices , , , and .

Question 60

A company estimates that its annual profits for next year will be , where represents investment in research and represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit?

Question 61

Find an equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at the point $( 4,1,1 )$ .

A) $2 x + y - z = 1$

B) $x + 2 y + 3 z = 9$
C) $x - 2 y + 4 z = 0$
D) $4 x - y + 6 z = 1$
E) $x + 2 y + 2 z = 8$
F) $3 x + 2 y + z = 1$

G) $x + y + z = 6$

H) $2 x + y + z = 10$

Question 62

Let $f ( x , y , z ) = x e ^ { y \sin z }$ . Find the gradient vector $\nabla f ( 1,1,0 )$ at the point $( x , y , z ) = ( 1,1,0 )$ .

A)i
B)j
C)k
D) $\mathbf { j } + \mathbf { k }$
E) $\mathbf { i } + \mathbf { k }$
F) $\mathbf { i } + \mathbf { j }$
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$
H)0

Question 63

Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..

A) $\frac { 17 } { 3 }$
B) $6$
C) $\frac { 19 } { 3 }$
D) $\frac { 20 } { 3 }$
E)7
F) $\frac { 22 } { 3 }$
G) $\frac { 23 } { 3 }$
H)8

Question 64

Find the largest value of the directional derivative of the function $f ( x , y ) = \frac { y } { x + y }$ at the point $( x , y ) = ( 1,2 )$ .

A) $\sqrt { 5 }$

B) $\frac { 5 } { 9 }$
C) $- \sqrt { 5 }$
D) $- \frac { 5 } { 9 }$
E) $\frac { \sqrt { 5 } } { 3 }$
F) $\frac { \sqrt { 5 } } { 9 }$

G) $- \frac { \sqrt { 5 } } { 3 }$

H) $- \frac { \sqrt { 5 } } { 9 }$

Question 65

Let $f ( x , y ) = 3 x + 2 y$ . Find the gradient vector $\nabla f$ .

A) $3 x \mathbf { i } + 2 y \mathbf { j }$

B) $3 x ^ { 2 } \mathbf { i } + 2 y ^ { 2 } \mathbf { j }$
C)
$2 x \mathbf { i } + 3 y \mathbf { j }$
D) $2 x ^ { 2 } \mathbf { i } + 3 y ^ { 2 } \mathbf { j }$
E) $3 y \mathbf { i } + 2 x \mathbf { j }$
F) $2 y \mathbf { i } + 3 x \mathbf { j }$

G) $2 \mathbf { i } + 3 \mathbf { j }$

H) $3 \mathbf { i } + 2 \mathbf { j }$

Question 66

Find a normal vector to the surface $x y z = 8$ at the point $( 1,2,4 )$ .

A) $2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$

B) $\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$
C) $4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$
D) $\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$
E) $4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$
F) $2 \mathbf { k } + \mathbf { j } + 4 \mathbf { i }$

G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$

H) $\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

Question 67

Let $f ( x , y ) = \frac { x } { y } + \frac { y } { x }$ . Find the gradient vector $\nabla f$ .

A) $2 x \mathbf { i } + 2 y \mathbf { j }$

B) $2 y \mathbf { i } + 2 x \mathbf { j }$
C) $- \frac { y } { x ^ { 2 } } \mathbf { i } - \frac { x } { y ^ { 2 } } \mathbf { j }$
D) $\frac { 1 } { y } \mathbf { i } + \frac { 1 } { x } \mathbf { j }$
E) $x \mathbf { i } + y \mathbf { j }$
F) $y \mathbf { i } + x \mathbf { j }$

G) $\frac { y } { x ^ { 2 } } \mathbf { i } + \frac { x } { y ^ { 2 } } \mathbf { j }$

H) $\left( \frac { 1 } { y } - \frac { y } { x ^ { 2 } } \right) \mathbf { i } + \left( \frac { 1 } { x } - \frac { x } { y ^ { 2 } } \right) \mathbf { j }$

Question 68

Find the directional derivative of the function $f ( x , y ) = y ^ { 2 } \ln x$ at the point $( 1,2 )$ in the direction $( 3,4 )$ .

A) $\frac { 5 } { 16 }$

B)0
C) $\frac { 12 } { 5 }$
D)12
E)1
F) $\frac { 16 } { 5 }$

G) $\frac { 5 } { 12 }$
H) $\frac { 1 } { 12 }$

Question 69

Find the direction of maximum increase of the function $f ( x , y , z ) = x ^ { 2 } - 2 x y + z ^ { 2 }$ at the point $( 1,1,2 )$ .

A) $\{ 1,0 , - 2 \}$

B) $\{ 4,1,2 \}$
C) $\{ 2,1 , - 4 \}$
D) $\{ 0 , - 2,4 \}$
E) $\{ 2,0,3 \}$
F) $\{ 3,1,2 \}$

G) $\{ 0,2,1 \}$

H) $( 5 , - 1,6 )$

Question 70

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } y$ at the point $( - 1,1 )$ in the direction $( 3,4 )$ .

A) $\frac { 2 } { 5 }$

B) $\frac { 6 } { 5 }$
C)2
D)6
E) $- \frac { 2 } { 5 }$
F) $- \frac { 6 } { 5 }$
G) $- 2$
H) $- 6$

Question 71

Find three positive numbers , , and whose product is 343 and sum is minimum.

Question 72

Let $f ( x , y ) = \frac { 1 } { x + y ^ { 2 } }$ . Find the gradient vector $\nabla f ( 1,1 )$ at the point $( x , y ) = ( 1,1 )$ .

A) $- i - j$

B) $- \frac { 1 } { 4 } \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$
C) $- \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$
D) $- \frac { 1 } { 2 } \mathbf { i } - \mathbf { j }$
E) $\mathbf { i } + \mathbf { j }$
F) $\frac { 1 } { 4 } \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

G) $\mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

H) $\frac { 1 } { 2 } \mathbf { i } + \mathbf { j }$

Question 73

Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .

A) $3 x - y + z = 4$

B) $2 x - 4 y + z = 3$
C) $x + 2 y + 3 z = 4$
D) $2 x + y - z = 1$
E) $3 x + 2 y - z = 2$
F) $4 x - y + 2 z = 6$

G) $x + y - z = 0$

H) $5 x + 3 y - 2 z = 3$

Question 74

Find the directional derivative of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ at the point $( 1,1 )$ in the direction $\theta = \frac { \pi } { 4 }$ .

A) $\sqrt { 2 }$
B) $2 \sqrt { 2 }$
C) $4 \sqrt { 2 }$
D) $\frac { 1 } { \sqrt { 2 } }$
E)1
F)2
G)4
H)
$\frac { 1 } { 2 }$

Question 75

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } e ^ { y }$ at the point $( 2,0 )$ in the direction $(3 , - 4 )$ .

A) $\frac { 14 } { 5 }$

B) $\frac { 4 } { 5 }$
C) $\frac { 14 } { 25 }$
D)14
E) $- \frac { 14 } { 5 }$
F) $- \frac { 4 } { 5 }$

G) $- \frac { 14 } { 25 }$
H) $- 14$

Question 76

Find the direction $\theta$ in which the directional derivative of the function $f ( x , y ) = x y + y ^ { 2 }$ at the point $( 1,1 )$ is maximum.

A) $\cot ^ { - 1 } 2$

B)
$\cot ^ { - 1 } 3$
C) $\cos ^ { - 1 } \frac { 1 } { 2 }$
D) $\cos ^ { - 1 } \frac { 1 } { 3 }$
E) $\sin ^ { - 1 } \frac { 1 } { 2 }$
F)
$\sin ^ { - 1 } \frac { 1 } { 3 }$

G)
$\tan ^ { - 1 } 2$

H) $\tan ^ { - 1 } 3$

Question 77

Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .

A) $\{ 1,1,3 \}$

B) $\{ - 1 , - 1,3 \}$
C) $\{ - 1,1,3 \}$
D) $(1 , - 1,4 )$
E) $\{ 1,3,3 \}$
F) $\{ 1 , - 3,3 \}$

G) $\{ 1 , - 1,3 \}$

H) $\{ - 1,1,4 \}$

Question 78

Find an equation of the tangent plane to the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ at the point $( 1,2,2 )$ .

A) $2 x + y + z = 4$

B) $x + y + z = 5$
C) $x + 2 y + 2 z = 9$
D) $x + 4 y + 4 z = 17$
E) $2 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 10$
F) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$

G) $x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } = 17$

H) $x ^ { 2 } + 4 y ^ { 2 } + 4 z ^ { 2 } = 33$

Question 79

Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .

A) $- \frac { 1 } { 6 }$


B) $- \frac { 1 } { 4 }$
C) $- \frac { 1 } { 2 }$
D)0
E) $\frac { 1 } { 2 }$
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 6 }$
H) $\frac { 1 } { 8 }$

Question 80

Find the directional derivative of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ at the point $( 1,2 )$ in the direction $\theta = \frac { \pi } { 2 }$ .

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7