Deck 14: Functions of Several Variables

Question 1

Find the standard equation of the sphere whose center is $( - 6 , - 5,7 )$ and whose radius is 4.

A) $( x + 6 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 16$
B) $( x - 6 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 16$
C) $( x + 6 ) + ( y + 5 ) + ( z - 7 ) = 4$
D) $( x + 6 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 4$
E) $( x - 6 ) + ( y - 5 ) + ( z + 7 ) = 4$

Answer 1

$( x + 6 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 16$

Question 2

Find $( x , y , z )$ if the midpoint of the line segment joining the two points $( x , y , z )$ and $( 4 , - 2,4 )$ is $( - 1,3 , - 1 )$ .

A) $( 2,4,2 )$
B) $\left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$
C) $( 5 , - 5,5 )$
D) $\left( - \frac { 5 } { 2 } , \frac { 5 } { 2 } , - \frac { 5 } { 2 } \right)$
E) $( - 6,8 , - 6 )$

Answer 2

$( - 6,8 , - 6 )$

Question 3

Sketch the trace of the intersection of plane y = 4 with the sphere: $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ .

A)
B)
C)
D)
E)

Answer 3

Question 4

Describe the trace of the surface given by the function below $y ^ { 2 } + z ^ { 2 } - x ^ { 2 } = 1$ in the xz-plane.

A)circle
B)parabola
C)line
D)ellipse
E)hyperbola

Question 5

Find the center and radius of the sphere whose equation is $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 12 y - 12 z + 43 = 0$ . Round your answer to two decimal places, where applicable.

A)center: $( - 2,3 , - 3 )$ ; radius: 0.71
B)center: $( 2 , - 3,3 )$ ; radius: 0.71
C)center: $( 2 , - 3,3 )$ ; radius: 0.50
D)center: $( - 2 , - 3,3 )$ ; radius: 0.71
E)center: $( - 2,3 , - 3 )$ ; radius: 0.50

Question 6

Identify the quadric surface. $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } + z ^ { 2 } = 1$

A)The graph is hyperboloid of two sheets.
B)The graph is an ellipsoid.
C)The graph is an elliptic cone.
D)The graph is an elliptic paraboloid.
E)The graph is a hyperboloid of one sheet.

Question 7

Sketch the yz-trace of the equation: $$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 9$$

A)
B)
C)
D)
E)

Question 8

Evaluate the function at the given values of the independent variables. $z = x ^ { 3 } + 2 x y + 3 y ^ { 2 } , \quad x = - 9 , y = 7$

A) $z = - 708$
B) $z = 102$
C) $z = - 600$
D) $z = - 568$
E) $z = 174$

Question 9

Find the coordinates of the point that is located six units behind of the yz-plane, six units to the left of the xz-plane, and seven units below of the xy-plane.

A) $( - 6 , - 6 , - 7 )$
B) $( 6 , - 6 , - 7 )$
C) $( 6,6 , - 7 )$
D) $( - 6,6 , - 7 )$
E) $( - 6,6,7 )$

Question 10

Use the function $q \left( p _ { 1 } , p _ { 2 } \right) = \frac { 7 p _ { 1 } - 9 p _ { 2 } } { - 9 p _ { 1 } + 9 p _ { 2 } }$ to find $q ( - 5 , - 8 )$

A) $q ( - 5 , - 8 ) = - \frac { 27 } { 37 }$
B) $q ( - 5 , - 8 ) = \frac { 27 } { 37 }$
C) $q ( - 5 , - 8 ) = - \frac { 37 } { 27 }$
D) $q ( - 5 , - 8 ) = \frac { 37 } { 27 }$
E) $q ( - 5 , - 8 ) = - \frac { 54 } { 37 }$

Question 11

Describe the trace of the surface given by the function below in the xy-plane. $x ^ { 2 } - y - z ^ { 2 } = 0$

A)hyperbola
B)parabola
C)ellipse
D)line
E)circle

Question 12

Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. $( 0,0,0 ) , ( 2,2,1 ) , ( 2 , - 4,4 )$

A) $3 , \sqrt { 5 } , 5$ ; obtuse triangle
B) $3,3 \sqrt { 5 } , 6$ ; right triangle
C) $\sqrt { 6 } , 3,5$ ; right triangle
D) $2,4,3$ ; acute triangle
E) $2,4 , \sqrt { 3 }$ ; acute triangle

Question 13

Find the intercepts of the plane given by $3 x - 9 z = 18$ .

A)The $x$ -intercept is $( 0,0,6 )$ .The $z$ -intercept is $( - 2,0,0 )$ .
B)The $x$ -intercept is $( 0,6,0 )$ .The $z$ -intercept is $( - 2,0,0 )$ .
C)The $x$ -intercept is $( 0 , - 6,0 )$ .The $z$ -intercept is $( - 2,0,0 )$ .
D)The $x$ -intercept is $( - 6,0,0 )$ .The $z$ -intercept is $( 0,0 , - 2 )$ .
E)The $x$ -intercept is $( 6,0,0 )$ .The $z$ -intercept is $( 0,0 , - 2 )$ .

Question 14

Identify the quadric surface. $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1$

A)The graph is an elliptic paraboloid.
B)The graph is an elliptic cone.
C)The graph is a hyperboloid of two sheet.
D)The graph is hyperboloid of one sheet.
E)The graph is an ellipsoid.

Question 15

Find the the distance between the two points $( 0 , - 1,2 )$ and $( - 3 , - 2,5 )$ .

A)1 units
B)19 units
C) $\sqrt { 19 }$ units
D)3 units
E)7 units

Question 16

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0$

A)Center: $\left( \frac { 5 } { 2 } , 0,0 \right)$ Radius: $\frac { 5 } { 2 }$
B)Center: $\left( 0 , \frac { 3 } { 2 } , 0 \right)$ Radius: $\frac { 3 } { 2 }$
C)Center: $\left( 0 , \frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$ Radius: $\frac { \sqrt { 10 } } { 2 }$
D)Center: $\left( \frac { 1 } { 2 } , 0,0 \right)$ Radius: $\frac { 1 } { 2 }$
E)Center: $\left( \frac { 5 } { 2 } , 0,1 \right)$ Radius: $\frac { \sqrt { 29 } } { 2 }$

Question 17

The two planes $x - 8 y - 9 z = 4$ and $4 x - 21 y - 7 z = - 4$ are parallel.

A)true
B)false

Question 18

Find the equation of the sphere that has the points $( 8,2,2 )$ and $( 6,4,4 )$ as end points of a diameter.

A) $( x + 7 ) + ( y + 3 ) + ( z + 3 ) = 6$
B) $( x + 7 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z + 3 ) ^ { 2 } = 3$
C) $( x - 7 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 6$
D) $( x - 7 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 3$
E) $( x - 7 ) + ( y - 3 ) + ( z - 3 ) = 3$

Question 19

The two planes $4 x - 3 y + z = 6$ and $8 x + 7 y + 9 z = 1$ are perpendicular.

A)false
B)true

Question 20

Because of the forces caused by its rotation, a planet is actually an oblate ellipsoid rather than a sphere. The equatorial radius is 3961 miles and the polar radius is 3957 miles. Find an equation of the ellipsoid. Assume that the center of a planet is at the origin and the xy- trace $( z = 0 )$ corresponds to the equator.

A) $\frac { x ^ { 2 } } { 3961 ^ { 2 } } + \frac { y ^ { 2 } } { 3961 ^ { 2 } } - \frac { z ^ { 2 } } { 3957 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 3961 ^ { 2 } } + \frac { y ^ { 2 } } { 3957 ^ { 2 } } + \frac { z ^ { 2 } } { 3961 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 3961 ^ { 2 } } + \frac { y ^ { 2 } } { 3961 ^ { 2 } } + \frac { z ^ { 2 } } { 3957 ^ { 2 } } = 1$
D) $\frac { x ^ { 2 } } { 3961 ^ { 2 } } - \frac { y ^ { 2 } } { 3961 ^ { 2 } } - \frac { z ^ { 2 } } { 3957 ^ { 2 } } = 1$
E) $\frac { x ^ { 2 } } { 3961 ^ { 2 } } + \frac { y ^ { 2 } } { 3957 ^ { 2 } } + \frac { z ^ { 2 } } { 3957 ^ { 2 } } = 1$

Question 21

Evaluate the function $w = \frac { x ^ { 2 } - y z } { x y z }$ at $( - 8,6,3 )$

A) $w = \frac { 13 } { 72 }$
B) $w = - \frac { 29 } { 72 }$
C) $w = - \frac { 7 } { 9 }$
D) $w = - \frac { 41 } { 72 }$
E) $w = - \frac { 23 } { 72 }$

Question 22

Evaluate $f _ { x }$ and $f _ { y }$ for the function $f ( x , y ) = 7 x ^ { 3 } + x y - y ^ { 3 }$ at the point $( 4 , - 6 )$ .

A) $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -132
B) $f _ { x } ( 4 , - 6 ) =$ 360 and $f _ { y } ( 4 , - 6 ) =$ -104
C) $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -104
D) $f _ { x } ( 4 , - 6 ) =$ 546 and $f _ { y } ( 4 , - 6 ) =$ 344
E) $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ 112

Question 23

Use the function $f ( x , y ) = \frac { \ln ( 10 x y ) } { 8 x ^ { 2 } + 6 y ^ { 2 } }$ to find $f ( 2,7 )$

A) $f ( 2,7 ) = \frac { \ln 140 } { 58 }$
B) $f ( 2,7 ) = \frac { \ln 140 } { 416 }$
C) $f ( 2,7 ) = \frac { \ln 20 } { 326 }$
D) $f ( 2,7 ) = \frac { \ln 70 } { 424 }$
E) $f ( 2,7 ) = \frac { \ln 140 } { 326 }$

Question 24

Find the first partial derivatives with respect to x, y, and z. $w = \frac { 9 x z } { 8 x + 4 y }$

A) $\frac { \partial w } { \partial x } = \frac { 36 x z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial y } = - \frac { 36 y z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial z } = \frac { 9 x } { 8 x + 4 y }$
B) $\frac { \partial w } { \partial x } = \frac { 36 x z } { 8 x + 4 y } , \frac { \partial w } { \partial y } = - \frac { 36 y z } { 8 x + 4 y } , \frac { \partial w } { \partial z } = \frac { 9 x } { ( 8 x + 4 y ) ^ { 2 } }$
C) $\frac { \partial w } { \partial x } = \frac { 36 y z } { 8 x + 4 y } , \frac { \partial w } { \partial y } = - \frac { 36 x z } { 8 x + 4 y } , \frac { \partial w } { \partial z } = \frac { 9 x } { ( 8 x + 4 y ) ^ { 2 } }$
D) $\frac { \partial w } { \partial x } = \frac { 36 y z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial y } = - \frac { 36 x z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial z } = \frac { 9 x } { 8 x + 4 y }$
E) $\frac { \partial w } { \partial x } = \frac { 36 y z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial y } = - \frac { 36 x z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial z } = \frac { 9 x } { ( 8 x + 4 y ) ^ { 2 } }$

Question 25

Find the domain and range of the function. $f ( x , y ) = e ^ { \frac { x } { y } }$

A)Domain: all point $( x , y )$ such that $y \neq 0$ Range: $( - \infty , 0 )$
B)Domain: all point $( x , y )$ such that $y \neq 0$ Range: $( 0 , \infty )$
C)Domain: all point $( x , y )$ such that $y \neq 0,1$ Range: $( 0 , \infty )$
D)Domain: all point $( x , y )$ such that $y \neq 0,1$ Range: $( - \infty , \infty )$
E)Domain: all point $( x , y )$ such that $y \neq 0$ Range: $( - \infty , \infty )$

Question 26

Find the slopes of the surface $h ( x , y ) = 3 y ^ { 2 } - x ^ { 2 }$ in the x- and y- directions at the point $( - 1,3,26 )$ .

A)slope in x-direction: 29slope in y-direction: 17
B)slope in x-direction: 2slope in y-direction: 18
C)slope in x-direction: 17slope in y-direction: 29
D)slope in x-direction: 20slope in y-direction: 20
E)slope in x-direction: 18slope in y-direction: 2

Question 27

The utility function $U = f ( x , y )$ is a measure of utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is $U = - 7 x ^ { 2 } + 5 x y - 2 y ^ { 2 }$ . Determine the marginal utility of product x.

A) $5 x - 4 y$
B) $- 14 x + 5 y$
C) $- 14 x + 5 y - 2 y ^ { 2 }$
D) $- 7 x ^ { 2 } + 5 x - 4 y$
E) $- 14 x + 5 - 4 y$

Question 28

For $f ( x , y )$ , find all values of x and y such that $f _ { x } ( x , y ) = 0$ and $f _ { y } ( x , y ) = 0$ simultaneously. $f ( x , y ) = 16 x ^ { 3 } - 4 x y + 16 y ^ { 3 }$

A) $\left( - \frac { 1 } { 12 } , - \frac { 1 } { 12 } \right)$
B) $\left( \frac { 1 } { 12 } , \frac { 1 } { 12 } \right)$
C)(0,0)
D)Both B and C
E)Both A and B

Question 29

Describe the level curves for the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ for the c-values given by $c = 0,2,4,6,8$ .

A) $c = 0$ $8 = x ^ { 2 } + y ^ { 2 }$ $c = 2$ $6 = x ^ { 2 } + y ^ { 2 }$ $c = 4$ $4 = x ^ { 2 } + y ^ { 2 }$ $c = 6$ $2 = x ^ { 2 } + y ^ { 2 }$ $c = 8$ $0 = x ^ { 2 } + y ^ { 2 }$
B) $c = 0$ $0 = x ^ { 2 } + y ^ { 2 }$ $c = 2$ $2 = x ^ { 2 } + y ^ { 2 }$ $c = 4$ $4 = x ^ { 2 } + y ^ { 2 }$ $c = 6$ $6 = x ^ { 2 } + y ^ { 2 }$ $c = 8$ $8 = x ^ { 2 } + y ^ { 2 }$
C) $c = 0$ $2 = x ^ { 2 } + y ^ { 2 }$ $c = 2$ $4 = x ^ { 2 } + y ^ { 2 }$ $c = 4$ $6 = x ^ { 2 } + y ^ { 2 }$ $c = 6$ $8 = x ^ { 2 } + y ^ { 2 }$ $c = 8$ $0 = x ^ { 2 } + y ^ { 2 }$
D) $c = 0$ $4 = x ^ { 2 } + y ^ { 2 }$ $c = 2$ $6 = x ^ { 2 } + y ^ { 2 }$ $c = 4$ $8 = x ^ { 2 } + y ^ { 2 }$ $c = 6$ $0 = x ^ { 2 } + y ^ { 2 }$ $c = 8$ $2 = x ^ { 2 } + y ^ { 2 }$
E) $c = 0$ $0 = x ^ { 2 } + y ^ { 2 }$ $c = 2$ $4 = x ^ { 2 } + y ^ { 2 }$ $c = 4$ $16 = x ^ { 2 } + y ^ { 2 }$ $c = 6$ $36 = x ^ { 2 } + y ^ { 2 }$ $c = 8$ $64 = x ^ { 2 } + y ^ { 2 }$

Question 30

Test for relative extrema and saddle points. $z = x ^ { 2 } - 8 x y + y ^ { 2 } + 120 x$

A)saddle point at $( - 4 , - 16 , - 720 )$
B)saddle point at $( 4,16,240 )$
C)saddle point at $( 0,0,0 )$
D)relative minimum at $( 4 , - 16,1264 )$
E)relative minimum at $( 16,128,2176 )$

Question 31

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-insert stoves is $C = 30 \sqrt { x y } + 160 x + 195 y + 1050$ . Find the marginal costs ( $\partial C / \partial x$ and $\partial c / \partial y$ ) when $x = 60$ and $y = 10$ . Round your answers to two decimal places.

A) $\partial C / \partial x = 160.61 , \partial C / \partial y = 195.61$
B) $\partial C / \partial x = 172.25 , \partial C / \partial y = 268.48$
C) $\partial C / \partial x = 161.94 , \partial C / \partial y = 199.74$
D) $\partial C / \partial x = 166.12 , \partial C / \partial y = 231.74$
E) $\partial C / \partial x = 254.87 , \partial C / \partial y = 427.38$

Question 32

Describe the level curves of the function. Sketch the level curves for the given c-values. $z = 12 - 4 x - 5 y$ , c = 0, 2, 4, 6

A)
B)
C)
D)
E)

Question 33

The value $A ( t , r )$ of an investment of $4,000 after t years in an account for which the interest rate 100r% is compounded continuously is given by the function $A ( t , r ) = 4,000 e ^ { r t }$ dollars. Write the partial derivative $\frac { \partial A } { \partial t }$

A) $\frac { \partial A } { \partial t } = 4,000 r e ^ { n t }$
B) $\frac { \partial A } { \partial t } = 4,000 e ^ { rt }$
C) $\frac { \partial A } { \partial t } = 4,000 t e ^ {rt }$
D) $\frac { \partial A } { \partial t } = 4,000 r e ^ { r ( t - 1 ) }$
E) $\frac { \partial A } { \partial t } = 4,000 r e ^ { t }$

Question 34

If $f ( x , y ) = \sqrt { x ^ { 5 } - 5 y ^ { 10 } },$ find $f _ { x }$ and $f _ { y }.$

A) $f _ { x } = \sqrt { 5 x ^ { 4 } - 5 y ^ { 10 } } , \quad f _ { y } = \sqrt { x ^ { 5 } - 50 y ^ { 9 } }$
B) $f _ { x } = \frac { 1 } { 2 \sqrt { 5 x ^ { 4 } - 5 y ^ { 10 } } } , \quad f _ { y } = \frac { 1 } { 2 \sqrt { x ^ { 5 } - 50 y ^ { 9 } } }$
C) $f _ { x } = \frac { 5 x ^ { 4 } } { \sqrt { x ^ { 5 } - 5 y ^ { 10 } } } , \quad f _ { y } = - \frac { - 50 y ^ { 9 } } { \sqrt { x ^ { 5 } - 5 y ^ { 10 } } }$
D) $f _ { x } = \frac { 5 x ^ { 4 } } { 2 \sqrt { x ^ { 5 } - 5 y ^ { 10 } } } , \quad f _ { y } = - \frac { 25 y ^ { 9 } } { \sqrt { x ^ { 5 } - 5 y ^ { 10 } } }$
E) $f _ { x } = 5 x ^ { 4 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } } , \quad f _ { y } = - 50 y ^ { 9 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } }$

Question 35

If $f ( x , y ) = \ln \left( x y ^ { 4 } + 7 \right),$ find $\frac { \partial f } { \partial x } \text { and } \frac { \partial f } { \partial y }$

A) $\frac { \partial f } { \partial x } = \frac { x } { x y ^ { 4 } + 7 } , \frac { \partial f } { \partial y } = \frac { y ^ { 4 } } { x y ^ { 4 } + 7 }$
B) $\frac { \partial f } { \partial x } = \frac { y ^ { 4 } } { x y ^ { 4 } + 7 } , \frac { \partial f } { \partial y } = \frac { 4 x y ^ { 3 } } { x y ^ { 4 } + 7 }$
C) $\frac { \partial f } { \partial x } = \frac { 1 } { y ^ { 4 } } , \quad \frac { \partial f } { \partial y } = \frac { 1 } { 4 x y ^ { 3 } }$
D) $\frac { \partial f } { \partial x } = \frac { x y ^ { 4 } + 7 } { y ^ { 4 } } , \quad \frac { \partial f } { \partial y } = \frac { 4 x y ^ { 4 } + 7 } { x y ^ { 3 } }$
E) $\frac { \partial f } { \partial x } = \ln \left( \frac { 1 } { y ^ { 4 } } \right) , \quad \frac { \partial f } { \partial y } = \ln \left( \frac { 1 } { 4 x y ^ { 3 } } \right)$

Question 36

Sketch the level curves for the function below for the given $c -$ values $c = 0,1,2,3,4,5$ . $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$

A)
B)
C)
D)
E)

Question 37

Describe the domain and range of the function. $f ( x , y ) = \sqrt { 49 - x ^ { 2 } - y ^ { 2 } }$

A)domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval (0,7)
B)domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7]
C)domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7)
D)domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7]
E)domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7)

Question 38

Evaluate $f _ { x }$ and $f _ { y }$ for the function $f ( x , y ) = \frac { 7 x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ at the point $( 4,8 )$ . Round your answer to two decimal places.

A) $f _ { x } ( 4,8 ) =$ 5.01 and $f _ { y } ( 4,8 ) =$ 0.63
B) $f _ { x } ( 4,8 ) =$ 4.51 and $f _ { y } ( 4,8 ) =$ 0.63
C) $f _ { x } ( 4,8 ) =$ 3.13 and $f _ { y } ( 4,8 ) =$ 6.26
D) $f _ { x } ( 4,8 ) =$ 5.01 and $f _ { y } ( 4,8 ) =$ 0.13
E) $f _ { x } ( 4,8 ) =$ 3.13 and $f _ { y } ( 4,8 ) =$ 6.26.

Question 39

A manufacturer estimates the Cobb-Douglas production function to be given by $f ( x , y ) = 100 x ^ { 0.75 } y ^ { 0.25 }$ . Estimate the production levels when $x = 1500$ and $y = 1000$ .

A)135,540 units
B)122,560 units
C)131,601 units
D)145,330 units
E)112,745 units

Question 40

Find the four second partial derivatives. Observe that the second mixed partials are equal. $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$

A) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 2 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 6 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 0$
B) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 6 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
C) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 2 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 12 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
D) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
E) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 0$

Question 41

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from x1 units of running shoes and y1 units of basketball shoes is: $R = - 3 x _ { 1 } ^ { 2 } - 10 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 2 } + 50 x _ { 1 } + 96 x _ { 2 }$ , where x1 and x2 are in thousands of units. Find x1 and x2 so as to maximize the revenue.

A) $x _ { 1 } = \frac { 202 } { 29 } , x _ { 2 } = \frac { 119 } { 29 }$
B) $x _ { 1 } = \frac { 119 } { 29 } , x _ { 2 } = \frac { 202 } { 29 }$
C) $x _ { 1 } = \frac { 404 } { 59 } , x _ { 2 } = \frac { 238 } { 59 }$
D) $x _ { 1 } = \frac { 238 } { 59 } , x _ { 2 } = \frac { 404 } { 59 }$
E) $x _ { 1 } = \frac { 119 } { 29 } , x _ { 2 } = \frac { 101 } { 29 }$

Question 42

Examine the function $f ( x , y ) = x ^ { 3 } - 12 x y + y ^ { 3 } + 4$ for relative extrema and saddle points.

A)saddle point: $( 0,0,4 )$ ; relative minimum: $( 4,4 , - 60 )$
B)relative minimum: $( 0,0,4 )$ ; relative maximum: $( 4,4 , - 60 )$
C)saddle points: $( 0,0,4 )$ , $( 4,4 , - 60 )$
D)saddle point: $( 4,4 , - 60 )$ ; relative minimum: $( 0,0,4 )$
E)relative minimum: $( 4,4 , - 60 )$ ; relative maximum: $( 0,0,4 )$

Question 43

Use Lagrange multipliers to find the minimum distance from the circle $x ^ { 2 } + ( y - 4 ) ^ { 2 } = 49$ to the point $( - 10 , - 5 )$ Round your answer to the nearest tenth.

A)20.4
B)132.0
C)418.1
D)6.5
E)41.6

Question 44

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $\frac { x y } { 2 }$

A)The function has a saddle point at $( 0,0,0 )$ .
B)The function has a relative maximum at $( 0,0,0 )$ .
C)The function has a relative minimum at $( 0,0,0 )$ .
D)The function has a saddle point at $( 0,0,2 )$ .
E)The function has a relative maximum at $( 0,0,2 )$ .

Question 45

Use Lagrange multipliers to find the given extremum. In each case, assume that $x$ and $y$ are positive. Maximize $f ( x , y ) = x y$ Constraint $x + y = 10$

A) $f ( 7,3 ) = 21$
B) $f ( 5,5 ) = 25$
C) $f ( 6,4 ) = 24$
D) $f ( 2,8 ) = 16$
E) $f ( 1,9 ) = 9$

Question 46

Use Lagrange multipliers to find the given extremum. Assume that $x$ and $y$ are positive. Minimize $f ( x , y ) = e ^ { xy }$ Constraint $x ^ { 2 } + y ^ { 2 } - 8 = 0$

A) $f ( 2,2 ) = e ^ { 4 }$
B) $f ( 1,1 ) = e ^ { 1 }$
C) $f ( 3,1 ) = e ^ { 3 }$
D) $f ( 4,2 ) = e ^ { 8 }$
E) $f ( 2,1 ) = e ^ { 2 }$

Question 47

Find three positive numbers x, y, and z whose sum is 24 and the sum of the squares is a maximum.

A)x = y = z = 8
B)x = 4, y = 4, z = 16
C)x = 6, y = 6, z = 12
D)x = 3.2, y = 4.8, z = 8
E)x = 1.6, y = 9.6, z = 12.8

Question 48

A rectangular box is resting on the $x y$ -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize $V = x y z$ subject to the constraint $2 x + 3 y + 5 z - 90 = 0$ ).

A)15 units $\times$ 12 units $\times$ 5 units
B)11 units $\times$ 9 units $\times$ 5 units
C)10 units $\times$ 9 units $\times$ 6 units
D)15 units $\times$ 10 units $\times$ 6 units
E)12 units $\times$ 11 units $\times$ 7 units

Question 49

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$

A)The function has a relative minimum at $( 1,3,0 )$ .
B)The function has a relative maximum at $( 1,3,0 )$
C)The function has a saddle point at $( 1,3,0 )$ .
D)The function has a saddle point at $( 0,1,3 )$ .
E)The function has a saddle point at $( 0,1,3 )$

Question 50

Find three positive numbers x, y, and z whose sum is 33 and product is a maximum.

A)x = 5.5, y = 5.5, z = 22
B)x = y = z = 11
C)x = 8.25, y = 8.25, z = 16.5
D)x = 4.4, y = 6.6, z = 11
E)x = 2.2, y = 13.2, z = 17.6

Question 51

Examine the function $f ( x , y ) = 5 x ^ { 2 } - 8 y ^ { 2 } - 3 x + 8 y + 4$ for relative extrema.

A)saddle point at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$
B)relative minimum at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$
C)saddle point at $\left( - \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$
D)relative maximum at $\left( \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$
E)saddle point at $\left( - \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$

Question 52

The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 60 inches. Find the dimensions of the rectangular package of largest volume that may be sent.

A)x = 7.5, y = 7.5, z = 20
B)x = 5, y = 5, z = 40
C)x = 10 , y = 10 , z = 20
D)x = 4, y = 6, z = 40
E)x = 2, y = 12, z = 32

Question 53

Find the critical points of the function $f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

A)relative minimum at $( - 4 , - 7 , - 2 )$
B)relative maximum at $( - 4 , - 7 , - 2 )$
C)relative minimum at $( 4,7,2 )$
D)relative maximum at $( 4,7,2 )$
E)no relative extrema

Question 54

A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by $S = 12 x y z$ where $x , y,$ and $z$ are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions $x , y,$ and $z$ cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

A) $\begin{array} { l } x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \\y = \sqrt [ 3 ] { 0.065 } \approx 0.201 L \\z = \sqrt [ 3 ] { 0.065 } \approx 0.134 L\end{array}$
B) $\begin{array} { l } x = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.015 } \approx 0.2466 L\end{array}$
C) $\begin{array} { l } x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.055 } \approx 0.380 L\end{array}$
D) $\begin{array} { l } x = \sqrt [ 3 ] { 0.025 } \approx 0.292 L \\y = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \\z = \sqrt [ 3 ] { 0.115 } \approx 0.486 L\end{array}$
E) $\begin{array} { l } x = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.265 } \approx 0.642 L\end{array}$

Question 55

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function $f ( x , y )$ at the critical point $\left( x _ { 0 } , y _ { 0 } \right)$ . Given: $\begin{array} { l } f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = - 1 \\f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 8 \\f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = 5\end{array}$

A) $\text { relative minimum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
B) $\text { saddle point }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
C) $\text { relative maximum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
D) $\text { insufficient information to determine the nature of the function }$ at $\left( x _ { 0 } , y _ { 0 } \right)$

Question 56

Find the critical points of the function $f ( x , y , z ) = - 3 - ( ( x + 8 ) ( y - 7 ) ( z - 3 ) ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

A)relative minima at $( - 8 , a , b )$ , $( c , 7 , d )$ , $( m , n , 3 )$ , where $a , b , c , d , m,$ and $n$ are arbitrary real numbers
B)relative maxima at $( - 8 , a , b )$ , $( c , 7 , d )$ , $( m , n , 3 )$ , where $a , b , c , d , m,$ and $n$ are arbitrary real numbers
C)relative minimum at $( - 8,7,3 )$
D)relative maximum at $( - 8,7,3 )$
E)no relative extrema

Question 57

Examine the function $f ( x , y ) = 4 x ^ { 2 } - 2 x y + 4 y ^ { 2 } + 120 x + 60 y$ for relative extrema.

A)relative $\text { minimum }$ at $( - 18 , - 12 )$
B)relative $\text { maximum }$ at $( - 18 , - 12 )$
C)relative $\text { minimum }$ at $( - 18,12 )$
D)relative $\text { maximum }$ at $( - 18,12 )$
E)no relative extrema

Question 58

Use Lagrange multipliers to minimize the function $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the following constraint. $x + y + z - 21 = 0$ Assume that x, y, and z are positive.

A)49
B)147
C)98
D)294
E)441

Question 59

Use Lagrange multipliers to maximize the function $f ( x , y ) = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ subject to the following constraint: $x + y - 4 = 0$ Assume that x, y, and z are positive.

A) $24$
B) $\sqrt { 24 }$
C) $\sqrt { 8 }$
D) $8$
E)no absolute maximum

Question 60

Use Lagrange multipliers to find the given extremum. In each case, assume that $x , y,$ and $z$ are positive. Maximize $f ( x , y , z ) = x + y + z$ Constraints $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$

A) $f \left( \frac { \sqrt { 2 } } { 3 } , \frac { \sqrt { 2 } } { 3 } , \frac { \sqrt { 2 } } { 3 } \right) = \sqrt { 2 }$
B) $f \left( \frac { \sqrt { 5 } } { 3 } , \frac { \sqrt { 5 } } { 3 } , \frac { \sqrt { 5 } } { 3 } \right) = \sqrt { 5 }$
C) $f \left( \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) = \sqrt { 3 }$
D) $f ( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } ) = 3 \sqrt { 3 }$
E) $f \left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right) = \sqrt { 3 }$

Question 61

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .

A) $\frac { 1 } { 180 }$
B) $\frac { 1 } { 19 }$
C) $\frac { 1 } { 144 }$
D) $- \frac { 1 } { 72 }$
E) $- \frac { 1 } { 108 }$

Question 62

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by $y = 3.29 x - 4.17$ have a negative correlation.

A)True
B)False; The data modeled by $y = 3.29 x - 4.17$ have a positive correlation.

Question 63

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } ( 3 x + 2 y ) d x d y$ . Round your answer to two decimal places, where applicable.

A)-8.83
B)11.17
C)2.00
D)1.17
E)5.00

Question 64

A store manager wants to know the demand y for an energy bar as a function of price x. The daily sales for three different prices of the energy bar are shown in the table. Price, x
$ 1.02
$ 1.23
$ 1.54
Demand, y
410
365
280
(i) Use the regression capabilities of a graphing utility to find the least squares regression line for the data.
(ii) Use the model to estimate the demand when the price is $1.38.

A)(i) $y = - 24.314583 x + 382.38409$ ; (ii)348.708392
B)(i) $y = - 24.314583 x + 382.38409$ ; (ii)-416.059787
C)(i) $y = 382.38409 x - 24.314583$ ; (ii)505.287381
D)(i) $y = - 24.314583 x - 382.38409$ ; (ii)-416.059787
E)none of the above

Question 65

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .

A)134.00
B)144.00
C)154.00
D)42.00
E)24.00

Question 66

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points $( 10,2 ) , ( 1,8 ) , ( 6,9 ) , ( 4,10 ) , ( 16,4 ) , ( 17,4 )$ . Round your answer to three decimal places.

A) $y = - 0.383 x + 9.524$
B) $y = - 0.373 x + 9.514$
C) $y = - 0.373 x + 9.524$
D) $y = - 0.363 x + 9.524$
E) $y = - 0.363 x + 9.514$

Question 67

Sketch the region $R$ whose area is given by the following double integral. $$\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$$

A)
B)
C)
D)
E)

Question 68

A manufacturer has an order for 1100 units of fine paper that can be produced at two locations. Let $x _ { 1 }$ and $x _ { 2 }$ be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by $C = 0.2 x _ { 1 } ^ { 2 } + 25 x _ { 1 } + 0.05 x _ { 2 } ^ { 2 } + 12 x _ { 2 }$ .

A) $x _ { 1 } = 388$ units and $x _ { 2 } = 906$ units
B) $x _ { 1 } = 906$ units and $x _ { 2 } = 194$ units
C) $x _ { 1 } = 194$ units and $x _ { 2 } = 906$ units
D) $x _ { 1 } = 194$ units and $x _ { 2 } = 1812$ units
E) $x _ { 1 } = 1812$ units and $x _ { 2 } = 388$ units

Question 69

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $( - 4 , - 1 ) , ( - 2,0 ) , ( 2,4 ) , ( 4,5 )$

A) $y = 1.8 x + 5$
B) $y = 1.2 x + 2$
C) $y = 0.5 x + 1$
D) $y = 0.8 x + 2$
E) $y = 0.2 x + 1$

Question 70

Evaluate the double integral $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { x } \frac { 2 } { x ^ { 2 } + 8 } d y d x$ .

A) $\ln 43$
B) $\ln 43 - \ln 8$
C) $\ln 44$
D) $\ln 44 + \ln 8$
E) $\ln 44 - \ln 8$

Question 71

Find the least squares regression line for the points (1,0) , (6,6) , (11,12).

A) $y = \frac { 6 } { 5 } x - \frac { 6 } { 5 }$
B) $y = \frac { 6 } { 5 } x - \frac { 30 } { 133 }$
C) $y = \frac { 30 } { 133 } x - \frac { 6 } { 5 }$
D) $y = \frac { 30 } { 133 } x - \frac { 30 } { 133 }$
E)none of the above

Question 72

Evaluate the following integral. $\int _ { 3 x } ^ { x ^ { 3 } } \frac { 4 y } { x } d y$

A) $2 \left( x ^ { 5 } - 9 x \right)$
B) $4 \left( x ^ { 5 } - 9 x \right)$
C) $2 \left( x ^ { 5 } - 3 x \right)$
D) $2 \left( 9 x - x ^ { 5 } \right)$
E)none of the above

Question 73

An agronomist used four test plots to determine the relationship between the wheat yield $y$ (in bushels per acre) and the amount of fertilizer $x$ (in pounds per acre). The results are shown in the table. $\begin{array} { | c | c | c | c | c | } \hline \text { Fertilizer, } x & 100 & 150 & 200 & 250 \\\hline \text { Yield, } y & 35 & 44 & 50 & 56 \\\hline\end{array}$ (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the yield for a fertilizer application of 160 pounds per acre.

A) $y = 0.15 x + 15.1$
B) $y = 0.155 x + 21.1$
C) $y = 0.138 x + 22.1$
D) $y = 0.052 x + 34$
E) $y = 0.234 x + 17.5$

Question 74

Evaluate the double integral $\int _ { 0 } ^ { 2 } \int _ { y } ^ { 2 y } \left( 1 + x ^ { 2 } + y ^ { 2 } \right) d x d y$ . Round your answer to two decimal places, where applicable.

A)7.33
B)14.33
C)16.33
D)15.33
E)14.83

Question 75

Find the sum of the squared errors for the linear model $f ( x )$ and the quadratic model $g ( x )$ using the given points. $\begin{array} { l } f ( x ) = 1.6 x + 6 , g ( x ) = 0.29 x ^ { 2 } + 2.2 x + 6 \\( - 3,2 ) , ( - 2,2 ) , ( - 1,4 ) , ( 0,6 ) , ( 18 )\end{array}$

A) $S = 1.5$ ; $S = 0.7159$
B) $S = 1.6$ ; $S = 0.8259$
C) $S = 1.2$ ; $S = 0.8623$
D) $S = 1.3$ ; $S = 0.4160$
E) $S = 1.1$ ; $S = 0.7621$

Question 76

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0.

A)True
B)False

Question 77

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. When the correlation coefficient is $r \approx - 0.98781$ , the model is a good fit.

A)False
B)True

Question 78

A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs 15 per foot. To separate the corrals, a fence that costs 6 per foot will divide the region. The total area of the two corrals is to be $6000$ square feet. Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing.

A)dimensions: $50$ feet by $60$ feet
B)dimensions: $60$ feet by $50$ feet
C)dimensions: $25$ feet by $60$ feet
D)dimensions: $30$ feet by $50$ feet
E)dimensions: $25$ feet by $30$ feet

Question 79

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .

A) $2$
B) $11$
C) $\frac { 13 } { 2 }$
D) $\frac { 13 } { 15 }$
E) $\frac { 11 } { 30 }$

Question 80

Find the least squares regression line for the given points. Then plot the points and sketch the regression line. $( - 2 , - 1 ) , ( 0,0 ) , ( 2,3 )$

A) $y = x + \frac { 2 } { 3 }$
B) $y = x - \frac { 2 } { 3 }$
C) $y = x - \frac { 1 } { 3 }$
D) $y = x - \frac { 1 } { 2 }$
E) $y = x + \frac { 1 } { 3 }$