Question 1

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

Accepted Answer

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 
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 B

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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)  $f ( 2,7 ) = \frac { \ln 140 } { 326 }$ 
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 5

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $z = x y , z = 0 , y = 2 x , y = 0 , x = 0 , x = 3$

Accepted Answer

A)  $\frac { 71 } { 3 }$ 
B)  $\frac { 83 } { 3 }$ 
C)  $\frac { 79 } { 2 }$ 
D)  $\frac { 81 } { 2 }$ 
E)  $\frac { 62 } { 5 }$ 
A)  $\frac { 71 } { 3 }$ 
B)  $\frac { 83 } { 3 }$ 
C)  $\frac { 79 } { 2 }$ 
D)  $\frac { 81 } { 2 }$ 
E)  $\frac { 62 } { 5 }$

Question 6

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.

Accepted Answer

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)  $x _ { 1 } = \frac { 119 } { 29 } , x _ { 2 } = \frac { 101 } { 29 }$ 
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 7

Use a double integral to find the volume of the indicated solid. $z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2$

Accepted Answer

A)  $\frac { 3 } { 4 }$ 
B)  $\frac { 14 } { 11 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 11 } { 14 }$ 
E) none of the above 
A)  $\frac { 3 } { 4 }$ 
B)  $\frac { 14 } { 11 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 11 } { 14 }$ 
E) none of the above

Question 8

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.

Accepted Answer

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 
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 9

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 $y = 1000$ .

Accepted Answer

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

Question 10

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

Accepted Answer

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)  $y = x + \frac { 1 } { 3 }$ 
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)  $y = x + \frac { 1 } { 3 }$

Question 11

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

Accepted Answer

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

Question 12

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: $\begin{array} { l } 
f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = - 1 \
f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 8 \
f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = 5
\end{array}$

Accepted Answer

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 $\left( x _ { 0 } , y _ { 0 } \right)$ 
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 13

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

Accepted Answer

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. 
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 14

Sketch the region of integration $\int _ { 0 } ^ 2 \int _ { 0 } ^ { 4 - x ^ { 2 } } x y ^ { 2 } d y d x$ .

Accepted Answer

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

Question 15

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.

Accepted Answer

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

Question 16

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.

Accepted Answer

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 
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 17

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

Accepted Answer

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 $( 0,0,2 )$ . 
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 18

Find the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over the region R: square with vertices $( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )$ .

Accepted Answer

A)  $\frac { 8 } { 3 }$ 
B)  $\frac { 32 } { 3 }$ 
C)  $\frac { 16 } { 5 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 8 } { 5 }$ 
A)  $\frac { 8 } { 3 }$ 
B)  $\frac { 32 } { 3 }$ 
C)  $\frac { 16 } { 5 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 8 } { 5 }$

Question 19

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

Accepted Answer

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 
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 20

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

Accepted Answer

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

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

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

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

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

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $z = x y , z = 0 , y = 2 x , y = 0 , x = 0 , x = 3$

Use a double integral to find the volume of the indicated solid. $Use a double integral to find the volume of the indicated solid. z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2$ $z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2$

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 $y = 1000$ .

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

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

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: , =-1 , =-8 , =5

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

Sketch the region of integration $\int _ { 0 } ^ 2 \int _ { 0 } ^ { 4 - x ^ { 2 } } x y ^ { 2 } d y d x$ .

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

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

Find the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over the region R: square with vertices $( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )$ .

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

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

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Exam 14: Functions of Several Variables

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

Sketch the level curves for the function below for the given c−c -c− values c=0,1,2,3,4,5c = 0,1,2,3,4,5c=0,1,2,3,4,5 . z=25−x2−y2z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }z=25−x2−y2​

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

Use the function f(x,y)=ln⁡(10xy)8x2+6y2f ( x , y ) = \frac { \ln ( 10 x y ) } { 8 x ^ { 2 } + 6 y ^ { 2 } }f(x,y)=8x2+6y2ln(10xy)​ to find f(2,7)f ( 2,7 )f(2,7)

Use a double integral to find the volume of the solid bounded by the graphs of the equations. z=xy,z=0,y=2x,y=0,x=0,x=3z = x y , z = 0 , y = 2 x , y = 0 , x = 0 , x = 3z=xy,z=0,y=2x,y=0,x=0,x=3

Use a double integral to find the volume of the indicated solid. z=4−y2,z>0,x>0,3x<y<2z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2z=4−y2,z>0,x>0,3x<y<2

A manufacturer estimates the Cobb-Douglas production function to be given by f(x,y)=100x0.75y0.25f ( x , y ) = 100 x ^ { 0.75 } y ^ { 0.25 }f(x,y)=100x0.75y0.25 . Estimate the production levels when x=1500x = 1500x=1500 and y=1000y = 1000y=1000 .

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)( - 2 , - 1 ) , ( 0,0 ) , ( 2,3 )(−2,−1),(0,0),(2,3)

Sketch the trace of the intersection of plane y = 4 with the sphere: x2+y2+z2=25x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25x2+y2+z2=25 .

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)f ( x , y )f(x,y) at the critical point (x0,y0)\left( x _ { 0 } , y _ { 0 } \right)(x0​,y0​) . Given: , =-1 , =-8 , =5

Identify the quadric surface. x24+y2−z2=1\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 14x2​+y2−z2=1

Sketch the region of integration ∫02∫04−x2xy2dydx\int _ { 0 } ^ 2 \int _ { 0 } ^ { 4 - x ^ { 2 } } x y ^ { 2 } d y d x∫02​∫04−x2​xy2dydx .

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≈−0.98781r \approx - 0.98781r≈−0.98781 , the model is a good fit.

Examine the function given below for relative extrema and saddle points. f(x,y)=f ( x , y ) =f(x,y)= xy2\frac { x y } { 2 }2xy​

Find the average value of f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 over the region R: square with vertices (0,0),(2,0),(2,2),(0,2)( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )(0,0),(2,0),(2,2),(0,2) .

Examine the function f(x,y)=4x2−2xy+4y2+120x+60yf ( x , y ) = 4 x ^ { 2 } - 2 x y + 4 y ^ { 2 } + 120 x + 60 yf(x,y)=4x2−2xy+4y2+120x+60y for relative extrema.

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

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

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

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

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $z = x y , z = 0 , y = 2 x , y = 0 , x = 0 , x = 3$

Use a double integral to find the volume of the indicated solid. $Use a double integral to find the volume of the indicated solid. z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2$ $z = 4 - y ^ { 2 } , \quad z > 0 , \quad x > 0,3 x < y < 2$

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 $y = 1000$ .

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

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

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: , =-1 , =-8 , =5

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

Sketch the region of integration $\int _ { 0 } ^ 2 \int _ { 0 } ^ { 4 - x ^ { 2 } } x y ^ { 2 } d y d x$ .

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

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

Find the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over the region R: square with vertices $( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )$ .

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

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