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 
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)  $x = 20 ; \theta = 60 ^ { \circ }$ 
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) 
  
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)  $p = \frac { 17 } { 1750 } \left( \frac { V } { T } \right)$ 
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. 
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)  $U _ { x } = - 11 x + y$ 
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 $\pm ( \mathbf { i } - 10 \mathbf { j } )$. 
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)  $13 \%$ 
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)  $\frac { d w } { d t } = - 12 \sin \left( 4 t ^ { 9 } - 12 \right) t ^ { 8 }$ 
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: $( - 6 , \sqrt { 13 } , 18 )$ 
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: $( 0,0,0 )$ 
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 
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 
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 
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 
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 
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 
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)  $f _ { y } ( \pi , 0 ) = 0$ 
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)  $y = \frac { 8 } { 67 } x + \frac { 6 } { 67 }$ 
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)  $y = \frac { 7 } { 10 } x + 2$ 
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)  $y = \frac { 7 } { 10 } x + 2$

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

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

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 $f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }$

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

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

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

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

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

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.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

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 $f ( x , y ) = \frac { 9 x y } { x ^ { 2 } + y ^ { 2 } }$

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

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

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