Question 1

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

Accepted Answer

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. 
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 2

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. $( 0,0,0 ) , ( 2,2,1 ) , ( 2 , - 4,4 )$

Accepted Answer

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 
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 3

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.

Accepted Answer

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 
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 4

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

Accepted Answer

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)  $( x - 7 ) + ( y - 3 ) + ( z - 3 ) = 3$ 
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 5

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

Accepted Answer

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 
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 6

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

Accepted Answer

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

Question 7

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.

Accepted Answer

A) True 
B) False 
A) True 
B) False

Question 8

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 $x + y = 10$

Accepted Answer

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

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

Accepted Answer

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 $( 0,0 , - 2 )$ . 
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 10

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

Accepted Answer

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)  $( - 6,8 , - 6 )$ 
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)  $( - 6,8 , - 6 )$

Question 11

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

Accepted Answer

A) false 
B) true 
A) false 
B) true

Question 12

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

Accepted Answer

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: $\frac { \sqrt { 29 } } { 2 }$ 
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 13

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

Accepted Answer

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)  $y = 0.2 x + 1$ 
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 14

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

Accepted Answer

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)  $\frac { \partial w } { \partial x } = \frac { 36 y z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial y } = - \frac { 36 x z } { ( 8 x + 4 y ) ^ { 2 } } , \frac { \partial w } { \partial z } = \frac { 9 x } { ( 8 x + 4 y ) ^ { 2 } }$ 
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 15

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.

Accepted Answer

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

Question 16

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

Accepted Answer

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 
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 17

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.

Accepted Answer

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)  $\frac { x ^ { 2 } } { 3961 ^ { 2 } } + \frac { y ^ { 2 } } { 3957 ^ { 2 } } + \frac { z ^ { 2 } } { 3957 ^ { 2 } } = 1$ 
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 18

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

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

Accepted Answer

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 
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 20

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

Accepted Answer

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)  $f _ { x } = 5 x ^ { 4 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } } , \quad f _ { y } = - 50 y ^ { 9 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } }$ 
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)  $f _ { x } = 5 x ^ { 4 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } } , \quad f _ { y } = - 50 y ^ { 9 } \sqrt { x ^ { 5 } - 5 y ^ { 10 } }$

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

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. $( 0,0,0 ) , ( 2,2,1 ) , ( 2 , - 4,4 )$

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.

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

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

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

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.

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 $x + y = 10$

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

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

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

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

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

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

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.

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

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

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

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

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Exam 14: Functions of Several Variables

Identify the quadric surface. x2+y24+z2=1x ^ { 2 } + \frac { y ^ { 2 } } { 4 } + z ^ { 2 } = 1x2+4y2​+z2=1

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)( 0,0,0 ) , ( 2,2,1 ) , ( 2 , - 4,4 )(0,0,0),(2,2,1),(2,−4,4)

Evaluate fxf _ { x }fx​ and fyf _ { y }fy​ for the function f(x,y)=7xyx2+y2f ( x , y ) = \frac { 7 x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }f(x,y)=x2+y2​7xy​ at the point (4,8)( 4,8 )(4,8) . Round your answer to two decimal places.

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

Find the slopes of the surface h(x,y)=3y2−x2h ( x , y ) = 3 y ^ { 2 } - x ^ { 2 }h(x,y)=3y2−x2 in the x- and y- directions at the point (−1,3,26)( - 1,3,26 )(−1,3,26) .

Evaluate the double integral ∫06∫0x2x2+8dydx\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { x } \frac { 2 } { x ^ { 2 } + 8 } d y d x∫06​∫0x​x2+82​dydx .

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.

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

Find the intercepts of the plane given by 3x−9z=183 x - 9 z = 183x−9z=18 .

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

The two planes 4x−3y+z=64 x - 3 y + z = 64x−3y+z=6 and 8x+7y+9z=18 x + 7 y + 9 z = 18x+7y+9z=1 are perpendicular.

Find the center and radius of the sphere. x2+y2+z2−5x=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0x2+y2+z2−5x=0

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)( - 4 , - 1 ) , ( - 2,0 ) , ( 2,4 ) , ( 4,5 )(−4,−1),(−2,0),(2,4),(4,5)

Find the first partial derivatives with respect to x, y, and z. w=9xz8x+4yw = \frac { 9 x z } { 8 x + 4 y }w=8x+4y9xz​

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.

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

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

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

If f(x,y)=x5−5y10,f ( x , y ) = \sqrt { x ^ { 5 } - 5 y ^ { 10 } },f(x,y)=x5−5y10​, find fxf _ { x }fx​ and fy.f _ { y }.fy​.

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

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. $( 0,0,0 ) , ( 2,2,1 ) , ( 2 , - 4,4 )$

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.

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

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

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

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 $x + y = 10$

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

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

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

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

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

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

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

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