Question 1

Determine the continuity of the composite function $f \circ g \text { where } f ( t ) = \frac { 1 } { t } \text { and }$ $g ( x , y ) = 5 x - 8 y$

Accepted Answer

A) continuous except at $\left( 0 , \frac { 5 } { 8 } \right)$ 
B)  continuous for all $y \neq \frac { 5 } { 8 } x$ 
C)  continuous for all $y = \frac { 5 } { 8 } x$ 
D)  continuous except at $\left( \frac { 8 } { 5 } , \frac { 5 } { 8 } \right)$ 
E)  continuous everywhere

Question 2

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure). Find the cross section of maximum area.

Accepted Answer

A)  $x = 12 ; \theta = 45 ^ { \circ }$ 
B)  $x = 20 ; \theta = 55 ^ { \circ }$ 
C)  $x = 10 ; \theta = 45 ^ { \circ }$ 
D)  $x = 10 ; \theta = 60 ^ { \circ }$ 
E)  $x = 20 ; \theta = 60 ^ { \circ }$

Question 3

Sketch the level curves for the function $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } \text { for the given } c \text {-values }$ $c = 0,1,2,3,4,5$

Accepted Answer

A) 
 
B) 
 
C) 
 
D) 
 
E)

Question 4

According to the Ideal Gas Law, $P V = k T \text {. where } P \text { is pressure. } V \text { is yolume. } T \text { is }$ temperature (in Kelvins), and k is a constant of proportionality. A tank contains 3500 cubic inches of nitrogen at a pressure of 34 pounds per square inch and a temperature of 300 K. Write P as a function Of V and T after evaluating k.

Accepted Answer

A)  $P = \frac { 1190 } { 3 } \left( \frac { T } { V } \right)$ 
B)  $P = \frac { 3 } { 35 } \left( \frac { V } { T } \right)$ 
C)  $P = \frac { 35 } { 102 } \left( \frac { T } { V } \right)$ 
D)  $P = \frac { 595 } { 2 } \left( \frac { V } { T } \right)$ 
E)  $p = \frac { 17 } { 1750 } \left( \frac { V } { T } \right)$

Question 5

Discuss the continuity of the function. $$f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 36 }$$

Accepted Answer

A)  $f$ is continuous everywhere except at those points $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } = 36$. 
B)  $f$ is continuous everywhere except at those points $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \leq 36$. 
C)  $f$ is continuous at those points $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \leq 36$. 
D)  $f$ is continuous only when $x ^ { 2 } + y ^ { 2 } = 36$. 
E)  $f$ is continuous everywhere in the $x y$-plane.

Question 6

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

Accepted Answer

A)  $U _ { x } = - 7 x + y$ 
B)  $U _ { x } = - 9 x - y$ 
C)  $U _ { x } = - 10 x + y$ 
D)  $U _ { x } = - 14 x - y$ 
E)  $U _ { x } = - 11 x + y$

Question 7

The temperature at the point $( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 300 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$. Find the directions of no change in heat on the plate from the point $( 5,7 )$.

Accepted Answer

A)  There will be no change in directions perpendicular to the gradient $\pm ( 15 \mathbf { i } - \mathbf { j } )$. 
B)  There will be no change in directions parallel to the gradient $\pm ( 10 \mathbf { i } - \mathbf { j } )$. 
C)  There will be no change in directions parallel to the gradient $\pm ( \mathbf { i } + 10 \mathbf { j } )$. 
D)  There will be no change in directions perpendicular to the gradient $\pm ( \mathbf { i } + 15 \mathbf { j } )$. 
E)  There will be no change in directions perpendicular to the gradient $\pm ( \mathbf { i } - 10 \mathbf { j } )$.

Question 8

Suppose the centripetal acceleration of a particle moving in a circle is $a = \frac { v ^ { 2 } } { r } , \text { where }$ $v$ is the velocity and $r$ is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of $4 \%$ in $v$ and $2 \%$ in $r$.

Accepted Answer

A)  $11 \%$ 
B)  $16 \%$ 
C)  $10 \%$ 
D)  $21 \%$ 
E)  $13 \%$

Question 9

$\text { Let } w = \cos ( 4 x - 4 y ) \text {, where } x = t ^ { 9 } \text { and } y = 3 \text {. Find } \frac { d w } { d t } \text {. }$

Accepted Answer

A)  $\frac { d w } { d t } = 12 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$ 
B)  $\frac { d w } { d t } = - 36 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$ 
C)  $\frac { d w } { d t } = 9 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$ 
D)  $\frac { d w } { d t } = 36 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$ 
E)  $\frac { d w } { d t } = - 12 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$

Question 10

Find the absolute extrema of $$f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }$$ $$R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 49 \right\} .$$

Accepted Answer

A)  absolute minimum: $( - 3,0 , - 44 )$
 absolute maximum: $( - 6 , - \sqrt { 13 } , 306 )$ 
B)  absolute minimum: $( - 3,0 , - 44 )$ 
absolute maximum: $( 0,0 , - 8 )$ 
C)  absolute minimum: $( 0,0 , - 8 )$ 
absolute maximum: $( 3,0,100 )$ 
D)  absolute minimum: $( - 3,0 , - 44 )$
 absolute maximum: $( 7,0,356 )$ 
E)  absolute minimum: $( - 3,0 , - 44 )$
 absolute maximum: $( - 6 , \sqrt { 13 } , 18 )$

Question 11

Find the absolute extrema of $f ( x , y ) = 4 x ^ { 2 } + 8 x y - 48 x - 8 y \text { on the region bounded }$ by the square with vertices $( 0,0 ) , ( 7,0 ) , ( 0,7 )$, and $( 7,7 )$.

Accepted Answer

A)  absolute minimum: $( 0,7 , - 56 )$
absolute maximum: $( 0,0,0 )$ 
B)  absolute minimum: $( 0,7 , - 56 )$
absolute maximum: $( 7,7,196 )$ 
C)  absolute minimum: $( 1,5 , - 44 )$
absolute maximum: $( 0,0,0 )$ 
D)  absolute minimum: $( 6,0 , - 144 )$
absolute maximum: $( 7,7,196 )$ 
E)  absolute minimum: $( 6,0 , - 144 )$
absolute maximum: $( 0,0,0 )$

Question 12

$\text { Find } \frac { d w } { d t } \text { using the appropriate Chain Rule for } w = x ^ { 2 } + y ^ { 2 } \text { where } x = 7 t \text { and } y = 3 t \text {. }$

Accepted Answer

A) 116t 
B) 58t 
C) 104t 
D) 32t 
E) 174t

Question 13

Sketch the surface given by the function. $$z = 9 - x ^ { 2 } - y ^ { 2 }$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D) 
 
E)  none of the above

Question 14

Use polar coordinates and L'Hopital's Rule to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  10 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  0

Question 15

Find the limit. $$\lim _ { ( x , y ) \rightarrow ( 1 , - 4 ) } \frac { 3 x ^ { 2 } y } { 1 + x y ^ { 2 } }$$

Accepted Answer

A)  1 
B)  $\frac { 12 } { 19 }$ 
C)  $- \frac { 12 } { 17 }$ 
D)  $- \frac { 3 } { 4 }$ 
E)  0

Question 16

Determine the continuity of the function $$f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

A)  continuous except at $( 4,0 )$ 
B)  continuous except at $( 4,2 )$ 
C)  continuous except at $( 0,0 )$ 
D)  continuous except at $( 0,2 )$ 
E)  continuous everywhere

Question 17

Examine the function $z = e ^ { - x } \sin 2 y \text { for relative extrema and saddle points. }$

Accepted Answer

A)  saddle point: $( 0,0,0 )$ 
B)  relative minimum: $( 0,0,1 )$ 
C)  relative maximum: $( 0,0,0 )$ 
D)  relative minimum: $( 0,0,0 )$ 
E)  no critical points

Question 18

$\text { For } f ( x , y ) = e ^ { y } \sin 11 x \text {, evaluate } f _ { y } \text { at the point } ( \pi , 0 ) \text {. }$

Accepted Answer

A)  $f _ { y } ( \pi , 0 ) = 244$ 
B)  $f _ { y } ( \pi , 0 ) = 23$ 
C)  $f _ { y } ( \pi , 0 ) = 11$ 
D)  $f _ { y } ( \pi , 0 ) = 22$ 
E)  $f _ { y } ( \pi , 0 ) = 0$

Question 19

Find the least squares regression line for the points $$( 1,0 ) , ( 2,2 ) , ( 9,4 )$$

Accepted Answer

A)  $y = \frac { 8 } { 19 } x + \frac { 6 } { 19 }$ 
B)  $y = \frac { 8 } { 19 } x + \frac { 3 } { 19 }$ 
C)  $y = \frac { 8 } { 19 } x + \frac { 6 } { 67 }$ 
D)  $y = \frac { 8 } { 67 } x + \frac { 6 } { 19 }$ 
E)  $y = \frac { 8 } { 67 } x + \frac { 6 } { 67 }$

Question 20

Find the least squares regression line for the points shown in the graph.

Accepted Answer

A)  $y = \frac { 2 } { 5 } x + 1$ 
B)  $y = \frac { 2 } { 5 } x + 2$ 
C)  $y = \frac { 7 } { 10 } x + 1$ 
D)  $y = \frac { 3 } { 10 } x + 2$ 
E)  $y = \frac { 7 } { 10 } x + 2$

Exam 13: Introduction to Functions of Several Variables

Determine the continuity of the composite function $f \circ g \text { where } f ( t ) = \frac { 1 } { t } \text { and }$ $g ( x , y ) = 5 x - 8 y$

Sketch the level curves for the function $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } \text { for the given } c \text {-values }$ $c = 0,1,2,3,4,5$

Discuss the continuity of the function. $f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 36 }$

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

The temperature at the point $( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 300 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$ . Find the directions of no change in heat on the plate from the point $( 5,7 )$ .

Suppose the centripetal acceleration of a particle moving in a circle is $a = \frac { v ^ { 2 } } { r } , \text { where }$ $v$ is the velocity and $r$ is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of $4 \%$ in $v$ and $2 \%$ in $r$ .

$\text { Let } w = \cos ( 4 x - 4 y ) \text {, where } x = t ^ { 9 } \text { and } y = 3 \text {. Find } \frac { d w } { d t } \text {. }$

Find the absolute extrema of $f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }$ $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 49 \right\} .$

Find the absolute extrema of $f ( x , y ) = 4 x ^ { 2 } + 8 x y - 48 x - 8 y \text { on the region bounded }$ by the square with vertices $( 0,0 ) , ( 7,0 ) , ( 0,7 )$ , and $( 7,7 )$ .

$\text { Find } \frac { d w } { d t } \text { using the appropriate Chain Rule for } w = x ^ { 2 } + y ^ { 2 } \text { where } x = 7 t \text { and } y = 3 t \text {. }$

Sketch the surface given by the function. $z = 9 - x ^ { 2 } - y ^ { 2 }$

Use polar coordinates and L'Hopital's Rule to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 1 , - 4 ) } \frac { 3 x ^ { 2 } y } { 1 + x y ^ { 2 } }$

Determine the continuity of the function $f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }$

$\text { For } f ( x , y ) = e ^ { y } \sin 11 x \text {, evaluate } f _ { y } \text { at the point } ( \pi , 0 ) \text {. }$

Find the least squares regression line for the points $( 1,0 ) , ( 2,2 ) , ( 9,4 )$

Exam 13: Introduction to Functions of Several Variables

Determine the continuity of the composite function f∘g where f(t) =1t and f \circ g \text { where } f ( t ) = \frac { 1 } { t } \text { and }f∘g where f(t) =t1​ and g(x,y) =5x−8yg ( x , y ) = 5 x - 8 yg(x,y) =5x−8y

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure) . Find the cross section of maximum area.

Sketch the level curves for the function z=16−x2−y2 for the given c-values z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } \text { for the given } c \text {-values }z=16−x2−y2​ for the given c-values c=0,1,2,3,4,5c = 0,1,2,3,4,5c=0,1,2,3,4,5

Discuss the continuity of the function. f(x,y,z) =zx2+y2−36f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 36 }f(x,y,z) =x2+y2−36z​

Let w=cos⁡(4x−4y) , where x=t9 and y=3. Find dwdt. \text { Let } w = \cos ( 4 x - 4 y ) \text {, where } x = t ^ { 9 } \text { and } y = 3 \text {. Find } \frac { d w } { d t } \text {. } Let w=cos(4x−4y) , where x=t9 and y=3. Find dtdw​.

Find the absolute extrema of f(x,y) =4x2+2y2+24x−8 on the region f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }f(x,y) =4x2+2y2+24x−8 on the region R={(x,y) :x2+y2≤49}.R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 49 \right\} .R={(x,y) :x2+y2≤49}.

Sketch the surface given by the function. z=9−x2−y2z = 9 - x ^ { 2 } - y ^ { 2 }z=9−x2−y2

Use polar coordinates and L'Hopital's Rule to find the limit. lim⁡(x,y) →(0,0) 10sin⁡(x2+y2) x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }lim(x,y) →(0,0) ​x2+y210sin(x2+y2) ​

Find the limit. lim⁡(x,y) →(1,−4) 3x2y1+xy2\lim _ { ( x , y ) \rightarrow ( 1 , - 4 ) } \frac { 3 x ^ { 2 } y } { 1 + x y ^ { 2 } }lim(x,y) →(1,−4) ​1+xy23x2y​

Determine the continuity of the function f(x,y) =9xyx2+y2f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }f(x,y) =x2+y29xy​

Examine the function z=e−xsin⁡2y for relative extrema and saddle points. z = e ^ { - x } \sin 2 y \text { for relative extrema and saddle points. }z=e−xsin2y for relative extrema and saddle points.

For f(x,y) =eysin⁡11x, evaluate fy at the point (π,0) . \text { For } f ( x , y ) = e ^ { y } \sin 11 x \text {, evaluate } f _ { y } \text { at the point } ( \pi , 0 ) \text {. } For f(x,y) =eysin11x, evaluate fy​ at the point (π,0) .

Find the least squares regression line for the points (1,0) ,(2,2) ,(9,4) ( 1,0 ) , ( 2,2 ) , ( 9,4 ) (1,0) ,(2,2) ,(9,4)

Find the least squares regression line for the points shown in the graph.

Determine the continuity of the composite function $f \circ g \text { where } f ( t ) = \frac { 1 } { t } \text { and }$ $g ( x , y ) = 5 x - 8 y$

Sketch the level curves for the function $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } \text { for the given } c \text {-values }$ $c = 0,1,2,3,4,5$

Discuss the continuity of the function. $f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 36 }$

$\text { Let } w = \cos ( 4 x - 4 y ) \text {, where } x = t ^ { 9 } \text { and } y = 3 \text {. Find } \frac { d w } { d t } \text {. }$

Find the absolute extrema of $f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }$ $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 49 \right\} .$

Sketch the surface given by the function. $z = 9 - x ^ { 2 } - y ^ { 2 }$

Use polar coordinates and L'Hopital's Rule to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 1 , - 4 ) } \frac { 3 x ^ { 2 } y } { 1 + x y ^ { 2 } }$

Determine the continuity of the function $f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }$

$\text { For } f ( x , y ) = e ^ { y } \sin 11 x \text {, evaluate } f _ { y } \text { at the point } ( \pi , 0 ) \text {. }$

Find the least squares regression line for the points $( 1,0 ) , ( 2,2 ) , ( 9,4 )$