Question 1

Sketch the level curves of the function $z = 8 - 2 x - 5 y \text { for the given } c \text {-values }$ $c = 0,2,4,6$

Accepted Answer

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

Question 2

$\text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 200 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$. Find the direction of greatest increase in heat from the point $( 4,15 )$.

Accepted Answer

A)  The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
B)  The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$. 
C)  The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
D)  The greatest increase is in the direction of the gradient $- 8 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
E)  The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$. 
A)  The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
B)  The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$. 
C)  The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
D)  The greatest increase is in the direction of the gradient $- 8 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$. 
E)  The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$.

Question 3

Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 360 cubic feet. The cost of wall paint is $0.054 per Square foot and the cost of ceiling paint is $0.18 per square foot. Let x, y, and z be the length, width, And height of a rectangular room respectively. Identify the room dimensions that result in a minimum
Cost for the paint and use these dimensions to find the minimum cost for the paint. Round your Answer to the nearest cent.

Accepted Answer

A) $54.00 
B) $8.64 
C) $7.00 
D) $19.44 
E) $21.60 
A) $54.00 
B) $8.64 
C) $7.00 
D) $19.44 
E) $21.60

Question 4

Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 640 cubic feet. The cost of wall paint is $0.076 per Square foot and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width, And height of a rectangular room respectively. Find the room dimensions that result in a minimum Cost for the paint. Round your answers to two decimal places.

Accepted Answer

A)  $x = y = 8.00$ and $z = 10.00$ 
B)  $x = y = 10.00$ and $z = 12.80$ 
C)  $y = z = 10.00$ and $x = 12.80$ 
D)  $x = z = 8.00$ and $y = 10.00$ 
E)  $y = z = 8.00$ and $x = 10.00$ 
A)  $x = y = 8.00$ and $z = 10.00$ 
B)  $x = y = 10.00$ and $z = 12.80$ 
C)  $y = z = 10.00$ and $x = 12.80$ 
D)  $x = z = 8.00$ and $y = 10.00$ 
E)  $y = z = 8.00$ and $x = 10.00$

Question 5

Find three positive numbers x, y, and z whose sum is 48 and $P = x y ^ { 2 } z \text { is a maximum. }$

Accepted Answer

A) x = 10, y = 7, z = 31 
B) x = 7, y = 31, z = 10 
C) x = 31, y = 10, z = 7 
D) x = 12, y = 24, z = 12 
E) x = 12, y = 12, z = 24 
A) x = 10, y = 7, z = 31 
B) x = 7, y = 31, z = 10 
C) x = 31, y = 10, z = 7 
D) x = 12, y = 24, z = 12 
E) x = 12, y = 12, z = 24

Question 6

The two radii of the frustum of a right circular cone are increasing at a rate of 7 centimeters per minute, and the height is increasing at a rate of 11 centimeters per minute (see figure).
Find the rate at which the surface area is changing when the two radii are 16 centimeters and 26 Centimeters, and the height is 13 centimeters. [Note: The surface area does not include the top andBottom circles.] Round your answer to two decimal places.

Accepted Answer

A)  $1871.79 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
B)  $2077.89 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
C)  $1511.11 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
D)  $2185.54 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
E)  $1445.22 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
A)  $1871.79 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
B)  $2077.89 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
C)  $1511.11 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
D)  $2185.54 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ 
E)  $1445.22 \mathrm {~cm} ^ { 2 } / \mathrm { min }$

Question 7

The temperature at the point $( x , y ) \text { on a metal plate is } T = \frac { 9 x } { x ^ { 2 } + y ^ { 2 } } \text {. Find the }$ direction of greatest increase in heat from the point $( 7,9 )$ . Round all numerical values in your answer to three decimal places.

Accepted Answer

A)  $0.101 \hat { \mathbf { i } } - 0.023 \hat { \mathbf { j } }$ 
B)  $0.034 \hat { \mathbf { i } } - 0.017 \hat { \mathbf { j } }$ 
C)  $0.023 \hat { \mathbf { i } } - 0.034 \hat { \mathbf { j } }$ 
D)  $0.038 \hat { \mathbf { i } } - 0.101 \hat { \mathbf { j } }$ 
E)  $0.017 \hat { \mathbf { i } } - 0.067 \hat { \mathbf { j } }$ 
A)  $0.101 \hat { \mathbf { i } } - 0.023 \hat { \mathbf { j } }$ 
B)  $0.034 \hat { \mathbf { i } } - 0.017 \hat { \mathbf { j } }$ 
C)  $0.023 \hat { \mathbf { i } } - 0.034 \hat { \mathbf { j } }$ 
D)  $0.038 \hat { \mathbf { i } } - 0.101 \hat { \mathbf { j } }$ 
E)  $0.017 \hat { \mathbf { i } } - 0.067 \hat { \mathbf { j } }$

Question 8

Find and simplify the function $$f ( x , y ) = \int _ { x } ^ { y } \frac { 9 } { t } d t \text { at the given value } ( 10,5 )$$

Accepted Answer

A)  $- 9 \ln ( 10 )$ 
B)  $- 18 \ln ( 2 )$ 
C)  $\quad 9 \ln ( 2 )$ 
D)  $- 9 \ln ( 2 )$ 
E)  $18 \ln ( 10 )$ 
A)  $- 9 \ln ( 10 )$ 
B)  $- 18 \ln ( 2 )$ 
C)  $\quad 9 \ln ( 2 )$ 
D)  $- 9 \ln ( 2 )$ 
E)  $18 \ln ( 10 )$

Question 9

$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 4 x ^ { 2 } - 5 y$

Accepted Answer

A)  $- 5 y$ 
B)  0 
C)  $4 x$ 
D)  $5 y$ 
E)  $8 x$ 
A)  $- 5 y$ 
B)  0 
C)  $4 x$ 
D)  $5 y$ 
E)  $8 x$

Question 10

$\text { Given } f ( x , y ) = 16 - x ^ { 2 } - y ^ { 2 } \text {, calculate } \Delta z \text { by evaluating } f ( 6,5 ) \text { and } f ( 6.4,5.29 ) \text {. }$ Round your answer to four decimal places.

Accepted Answer

A) -7.9441 
B) -45.9441 
C) -19.9441 
D) 0.1118 
E) -4.8882 
A) -7.9441 
B) -45.9441 
C) -19.9441 
D) 0.1118 
E) -4.8882

Question 11

$\text { Find } \nabla f ( x , y ) \text { for function } f ( x , y ) = 8 - \frac { x } { 8 } - \frac { y } { 2 } \text {. }$

Accepted Answer

A)  $\nabla f ( x , y ) = \frac { 1 } { 8 } \hat { \mathbf { i } } + \frac { 1 } { 2 } \hat { \mathbf { j } }$ 
B)  $\nabla f ( x , y ) = \frac { 1 } { 8 } \hat { \mathbf { i } } - \frac { 1 } { 2 } \hat { \mathbf { j } }$ 
C)  $\nabla f ( x , y ) = - 8 \hat { \mathbf { i } } - 2 \hat { \mathbf { j } }$ 
D)  $\nabla f ( x , y ) = 8 \hat { \mathbf { i } } - 2 \hat { \mathbf { j } }$ 
E)  $\nabla f ( x , y ) = - \frac { 1 } { 8 } \hat { \mathbf { i } } - \frac { 1 } { 2 } \hat { \mathbf { j } }$ 
A)  $\nabla f ( x , y ) = \frac { 1 } { 8 } \hat { \mathbf { i } } + \frac { 1 } { 2 } \hat { \mathbf { j } }$ 
B)  $\nabla f ( x , y ) = \frac { 1 } { 8 } \hat { \mathbf { i } } - \frac { 1 } { 2 } \hat { \mathbf { j } }$ 
C)  $\nabla f ( x , y ) = - 8 \hat { \mathbf { i } } - 2 \hat { \mathbf { j } }$ 
D)  $\nabla f ( x , y ) = 8 \hat { \mathbf { i } } - 2 \hat { \mathbf { j } }$ 
E)  $\nabla f ( x , y ) = - \frac { 1 } { 8 } \hat { \mathbf { i } } - \frac { 1 } { 2 } \hat { \mathbf { j } }$

Question 12

Use the gradient to find the directional derivative of the function at P in the direction of Q. $f ( x , y ) = \sin ( 7 x ) \cos y , P ( 0,0 ) , Q \left( \frac { \pi } { 7 } , \pi \right)$

Accepted Answer

A)  $\pi$ 
B)  $\pi ( \sin ( 7 x ) \cos y + \cos ( 7 x ) \sin y )$ 
C)  $\pi ( \cos ( 7 x ) \cos y - \sin ( 7 x ) \sin y )$ 
D)  $\frac { 1 } { \sqrt { 50 } }$ 
E)  $\frac { 7 } { \sqrt { 50 } }$ 
A)  $\pi$ 
B)  $\pi ( \sin ( 7 x ) \cos y + \cos ( 7 x ) \sin y )$ 
C)  $\pi ( \cos ( 7 x ) \cos y - \sin ( 7 x ) \sin y )$ 
D)  $\frac { 1 } { \sqrt { 50 } }$ 
E)  $\frac { 7 } { \sqrt { 50 } }$

Question 13

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: $R = - 6 x _ { 1 } ^ { 2 } - 9 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 2 } + 37 x _ { 1 } + 95 x _ { 2 }$
where $x _ { 1 }$ and $x _ { 2 }$ are in thousands of units. Find $x _ { 1 }$ and $x _ { 2 }$ so as to maximize the revenue.

Accepted Answer

A)  $x _ { 1 } = \frac { 119 } { 53 } , x _ { 2 } = \frac { 533 } { 106 }$ 
B)  $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 53 }$ 
C)  $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 106 }$ 
D)  $x _ { 1 } = \frac { 238 } { 107 } , x _ { 2 } = \frac { 533 } { 107 }$ 
E)  $x _ { 1 } = \frac { 533 } { 107 } , x _ { 2 } = \frac { 238 } { 107 }$ 
A)  $x _ { 1 } = \frac { 119 } { 53 } , x _ { 2 } = \frac { 533 } { 106 }$ 
B)  $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 53 }$ 
C)  $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 106 }$ 
D)  $x _ { 1 } = \frac { 238 } { 107 } , x _ { 2 } = \frac { 533 } { 107 }$ 
E)  $x _ { 1 } = \frac { 533 } { 107 } , x _ { 2 } = \frac { 238 } { 107 }$

Question 14

$\text { Find } \frac { \partial w } { \partial s } \text { using the appropriate Chain Rule for } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \text { where }$ $x = 8 t \sin s , y = 8 t \cos s$, and $z = 8 s t ^ { 2 }$

Accepted Answer

A)  $64 s t ^ { 4 }$ 
B)  $128 s ^ { 4 } t$ 
C)  $16 s ^ { 4 } t$ 
D)  $1285 t ^ { 4 }$ 
E)  $16 s t ^ { 4 }$ 
A)  $64 s t ^ { 4 }$ 
B)  $128 s ^ { 4 } t$ 
C)  $16 s ^ { 4 } t$ 
D)  $1285 t ^ { 4 }$ 
E)  $16 s t ^ { 4 }$

Question 15

$$\text { Find the partial derivative } \frac { \partial z } { \partial y } \text { for the function } z = 16 y ^ { 2 } \sqrt { x }$$

Accepted Answer

A)  $\frac { \partial z } { \partial y } = 16 y \sqrt { x }$ 
B)  $\frac { \partial z } { \partial y } = 129 y \sqrt { x }$ 
C)  $\frac { \partial z } { \partial y } = 32 y \sqrt { x }$ 
D)  $\frac { \partial z } { \partial y } = 193 y \sqrt { x }$ 
E)  $\frac { \partial z } { \partial y } = 192 y \sqrt { x }$ 
A)  $\frac { \partial z } { \partial y } = 16 y \sqrt { x }$ 
B)  $\frac { \partial z } { \partial y } = 129 y \sqrt { x }$ 
C)  $\frac { \partial z } { \partial y } = 32 y \sqrt { x }$ 
D)  $\frac { \partial z } { \partial y } = 193 y \sqrt { x }$ 
E)  $\frac { \partial z } { \partial y } = 192 y \sqrt { x }$

Question 16

$\text { For } f ( x , y ) = \sin ( 7 x y ) \text {, evaluate } f _ { x } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 7 \pi } { 4 }$ 
B)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 0$ 
C)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 7$ 
D)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 7 \pi } { 2 }$ 
E)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 14$ 
A)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 7 \pi } { 4 }$ 
B)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 0$ 
C)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 7$ 
D)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 7 \pi } { 2 }$ 
E)  $f _ { x } \left( 2 , \frac { \pi } { 4 } \right) = 14$

Question 17

Find the partial derivative $f _ { y } \text { for the function } f ( x , y ) = \sqrt { 8 x + y ^ { 9 } } \text {. }$

Accepted Answer

A) 
$f _ { y } ( x , y ) = \frac { 9 y ^ { 8 } } { 2 \sqrt { 8 x + y ^ { 9 } } }$ 
B) 
$f _ { y } ( x , y ) = \frac { 9 y ^ { 9 } } { \sqrt { 8 x + y ^ { 9 } } }$ 
C) 
$f _ { y } ( x , y ) = \frac { y ^ { 9 } } { \sqrt { 8 x + y ^ { 9 } } }$ 
D) 
$f _ { y } ( x , y ) = \frac { 1 } { 2 \sqrt { 8 x + y ^ { 9 } } }$ 
E) 
$f _ { y } ( x , y ) = \frac { y ^ { 8 } } { 2 \sqrt { 8 x + y ^ { 9 } } }$ 
A) 
$f _ { y } ( x , y ) = \frac { 9 y ^ { 8 } } { 2 \sqrt { 8 x + y ^ { 9 } } }$ 
B) 
$f _ { y } ( x , y ) = \frac { 9 y ^ { 9 } } { \sqrt { 8 x + y ^ { 9 } } }$ 
C) 
$f _ { y } ( x , y ) = \frac { y ^ { 9 } } { \sqrt { 8 x + y ^ { 9 } } }$ 
D) 
$f _ { y } ( x , y ) = \frac { 1 } { 2 \sqrt { 8 x + y ^ { 9 } } }$ 
E) 
$f _ { y } ( x , y ) = \frac { y ^ { 8 } } { 2 \sqrt { 8 x + y ^ { 9 } } }$

Question 18

Find the partial derivative $f _ { x } \text { for the function } f ( x , y ) = \sqrt { 28 x + y ^ { 3 } } \text {. }$

Accepted Answer

A) 
$f _ { x } ( x , y ) = \frac { 14 } { \sqrt { 28 x + y ^ { 3 } } }$ 
B) 
$f _ { x } ( x , y ) = \frac { 393 x } { \sqrt { 28 x + y ^ { 3 } } }$ 
C) 
$f _ { x } ( x , y ) = \frac { 197 } { \sqrt { 28 x + y ^ { 3 } } }$ 
D) 
$f _ { x } ( x , y ) = \frac { 1 } { 2 \sqrt { 28 x + y ^ { 3 } } }$ 
E) 
$f _ { x } ( x , y ) = \frac { 28 x } { \sqrt { 28 x + y ^ { 3 } } }$ 
A) 
$f _ { x } ( x , y ) = \frac { 14 } { \sqrt { 28 x + y ^ { 3 } } }$ 
B) 
$f _ { x } ( x , y ) = \frac { 393 x } { \sqrt { 28 x + y ^ { 3 } } }$ 
C) 
$f _ { x } ( x , y ) = \frac { 197 } { \sqrt { 28 x + y ^ { 3 } } }$ 
D) 
$f _ { x } ( x , y ) = \frac { 1 } { 2 \sqrt { 28 x + y ^ { 3 } } }$ 
E) 
$f _ { x } ( x , y ) = \frac { 28 x } { \sqrt { 28 x + y ^ { 3 } } }$

Question 19

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the surface Area when the radius is 12 inches and the height is 32 inches?

Accepted Answer

A)  $936 \pi \mathrm { in. } ^ { 2 } / \mathrm { min }$ 
B)  $984 \pi$ in. $^ { 2 } / \mathrm { min }$ 
C)  $876 \pi$ in. $^ { 2 } / \mathrm { min }$ 
D)  $840 \pi$ in. $^ { 2 } / \mathrm { min }$ 
E)  $968 \pi$ in. $^ { 2 } / \mathrm { min }$ 
A)  $936 \pi \mathrm { in. } ^ { 2 } / \mathrm { min }$ 
B)  $984 \pi$ in. $^ { 2 } / \mathrm { min }$ 
C)  $876 \pi$ in. $^ { 2 } / \mathrm { min }$ 
D)  $840 \pi$ in. $^ { 2 } / \mathrm { min }$ 
E)  $968 \pi$ in. $^ { 2 } / \mathrm { min }$

Question 20

$\text { Differentiate implicitly to find } \frac { d y } { d x } \text {. }$ $x ^ { 2 } - 4 x y + y ^ { 2 } - 10 x + y - 9 = 0$

Accepted Answer

A)  $\frac { d y } { d x } = - \frac { 2 x - 4 y - 10 } { 2 y - 4 x + 1 }$ 
B)  $\frac { d y } { d x } = \frac { 2 x + 4 y + 10 } { 2 y + 4 x + 1 }$ 
C)  $\frac { d y } { d x } = \frac { 2 x - 4 y - 10 } { 2 y - 4 x + 1 }$ 
D)  $\frac { d y } { d x } = \frac { 2 x + 4 y - 10 } { 2 y + 4 x - 1 }$ 
E)  $\frac { d y } { d x } = - \frac { 2 x + 4 y + 10 } { 2 y + 4 x + 1 }$ 
A)  $\frac { d y } { d x } = - \frac { 2 x - 4 y - 10 } { 2 y - 4 x + 1 }$ 
B)  $\frac { d y } { d x } = \frac { 2 x + 4 y + 10 } { 2 y + 4 x + 1 }$ 
C)  $\frac { d y } { d x } = \frac { 2 x - 4 y - 10 } { 2 y - 4 x + 1 }$ 
D)  $\frac { d y } { d x } = \frac { 2 x + 4 y - 10 } { 2 y + 4 x - 1 }$ 
E)  $\frac { d y } { d x } = - \frac { 2 x + 4 y + 10 } { 2 y + 4 x + 1 }$

Sketch the level curves of the function $z = 8 - 2 x - 5 y \text { for the given } c \text {-values }$ $c = 0,2,4,6$

$\text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 200 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$ . Find the direction of greatest increase in heat from the point $( 4,15 )$ .

Find three positive numbers x, y, and z whose sum is 48 and $P = x y ^ { 2 } z \text { is a maximum. }$

The temperature at the point $( x , y ) \text { on a metal plate is } T = \frac { 9 x } { x ^ { 2 } + y ^ { 2 } } \text {. Find the }$ direction of greatest increase in heat from the point $( 7,9 )$ . Round all numerical values in your answer to three decimal places.

Find and simplify the function $f ( x , y ) = \int _ { x } ^ { y } \frac { 9 } { t } d t \text { at the given value } ( 10,5 )$

$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 4 x ^ { 2 } - 5 y$

$\text { Given } f ( x , y ) = 16 - x ^ { 2 } - y ^ { 2 } \text {, calculate } \Delta z \text { by evaluating } f ( 6,5 ) \text { and } f ( 6.4,5.29 ) \text {. }$ Round your answer to four decimal places.

$\text { Find } \nabla f ( x , y ) \text { for function } f ( x , y ) = 8 - \frac { x } { 8 } - \frac { y } { 2 } \text {. }$

Use the gradient to find the directional derivative of the function at P in the direction of Q. $f ( x , y ) = \sin ( 7 x ) \cos y , P ( 0,0 ) , Q \left( \frac { \pi } { 7 } , \pi \right)$

$\text { Find } \frac { \partial w } { \partial s } \text { using the appropriate Chain Rule for } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \text { where }$ $x = 8 t \sin s , y = 8 t \cos s$ , and $z = 8 s t ^ { 2 }$

$\text { Find the partial derivative } \frac { \partial z } { \partial y } \text { for the function } z = 16 y ^ { 2 } \sqrt { x }$

$\text { For } f ( x , y ) = \sin ( 7 x y ) \text {, evaluate } f _ { x } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Find the partial derivative $f _ { y } \text { for the function } f ( x , y ) = \sqrt { 8 x + y ^ { 9 } } \text {. }$

Find the partial derivative $f _ { x } \text { for the function } f ( x , y ) = \sqrt { 28 x + y ^ { 3 } } \text {. }$

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the surface Area when the radius is 12 inches and the height is 32 inches?

$\text { Differentiate implicitly to find } \frac { d y } { d x } \text {. }$ $x ^ { 2 } - 4 x y + y ^ { 2 } - 10 x + y - 9 = 0$

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

Sketch the level curves of the function z=8−2x−5y for the given c-values z = 8 - 2 x - 5 y \text { for the given } c \text {-values }z=8−2x−5y for the given c-values c=0,2,4,6c = 0,2,4,6c=0,2,4,6

Find three positive numbers x, y, and z whose sum is 48 and P=xy2z is a maximum. P = x y ^ { 2 } z \text { is a maximum. }P=xy2z is a maximum.

Find and simplify the function f(x,y)=∫xy9tdt at the given value (10,5)f ( x , y ) = \int _ { x } ^ { y } \frac { 9 } { t } d t \text { at the given value } ( 10,5 )f(x,y)=∫xy​t9​dt at the given value (10,5)

Find ∇f(x,y) for function f(x,y)=8−x8−y2. \text { Find } \nabla f ( x , y ) \text { for function } f ( x , y ) = 8 - \frac { x } { 8 } - \frac { y } { 2 } \text {. } Find ∇f(x,y) for function f(x,y)=8−8x​−2y​.

Use the gradient to find the directional derivative of the function at P in the direction of Q. f(x,y)=sin⁡(7x)cos⁡y,P(0,0),Q(π7,π)f ( x , y ) = \sin ( 7 x ) \cos y , P ( 0,0 ) , Q \left( \frac { \pi } { 7 } , \pi \right)f(x,y)=sin(7x)cosy,P(0,0),Q(7π​,π)

Find the partial derivative ∂z∂y for the function z=16y2x\text { Find the partial derivative } \frac { \partial z } { \partial y } \text { for the function } z = 16 y ^ { 2 } \sqrt { x } Find the partial derivative ∂y∂z​ for the function z=16y2x​

For f(x,y)=sin⁡(7xy), evaluate fx at the point (2,π4)\text { For } f ( x , y ) = \sin ( 7 x y ) \text {, evaluate } f _ { x } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right) For f(x,y)=sin(7xy), evaluate fx​ at the point (2,4π​)

Find the partial derivative fy for the function f(x,y)=8x+y9. f _ { y } \text { for the function } f ( x , y ) = \sqrt { 8 x + y ^ { 9 } } \text {. }fy​ for the function f(x,y)=8x+y9​.

Find the partial derivative fx for the function f(x,y)=28x+y3. f _ { x } \text { for the function } f ( x , y ) = \sqrt { 28 x + y ^ { 3 } } \text {. }fx​ for the function f(x,y)=28x+y3​.

The radius of a right circular cylinder is increasing at a rate of 9 inches per minute, and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the surface Area when the radius is 12 inches and the height is 32 inches?

Differentiate implicitly to find dydx. \text { Differentiate implicitly to find } \frac { d y } { d x } \text {. } Differentiate implicitly to find dxdy​. x2−4xy+y2−10x+y−9=0x ^ { 2 } - 4 x y + y ^ { 2 } - 10 x + y - 9 = 0x2−4xy+y2−10x+y−9=0

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

Sketch the level curves of the function $z = 8 - 2 x - 5 y \text { for the given } c \text {-values }$ $c = 0,2,4,6$

Find three positive numbers x, y, and z whose sum is 48 and $P = x y ^ { 2 } z \text { is a maximum. }$

Find and simplify the function $f ( x , y ) = \int _ { x } ^ { y } \frac { 9 } { t } d t \text { at the given value } ( 10,5 )$

$\text { Find } \nabla f ( x , y ) \text { for function } f ( x , y ) = 8 - \frac { x } { 8 } - \frac { y } { 2 } \text {. }$

Use the gradient to find the directional derivative of the function at P in the direction of Q. $f ( x , y ) = \sin ( 7 x ) \cos y , P ( 0,0 ) , Q \left( \frac { \pi } { 7 } , \pi \right)$

$\text { Find the partial derivative } \frac { \partial z } { \partial y } \text { for the function } z = 16 y ^ { 2 } \sqrt { x }$

$\text { For } f ( x , y ) = \sin ( 7 x y ) \text {, evaluate } f _ { x } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Find the partial derivative $f _ { y } \text { for the function } f ( x , y ) = \sqrt { 8 x + y ^ { 9 } } \text {. }$

Find the partial derivative $f _ { x } \text { for the function } f ( x , y ) = \sqrt { 28 x + y ^ { 3 } } \text {. }$

$\text { Differentiate implicitly to find } \frac { d y } { d x } \text {. }$ $x ^ { 2 } - 4 x y + y ^ { 2 } - 10 x + y - 9 = 0$