Question 1

$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$

Accepted Answer

A)  $f _ { x } ( \pi , 0 ) = 3$ 
B)  $f _ { x } ( \pi , 0 ) = - 3$ 
C)  $f _ { x } ( \pi , 0 ) = - 11$ 
D)  $f _ { x } ( \pi , 0 ) = - 6$ 
E)  $f _ { x } ( \pi , 0 ) = 6$ 
A)  $f _ { x } ( \pi , 0 ) = 3$ 
B)  $f _ { x } ( \pi , 0 ) = - 3$ 
C)  $f _ { x } ( \pi , 0 ) = - 11$ 
D)  $f _ { x } ( \pi , 0 ) = - 6$ 
E)  $f _ { x } ( \pi , 0 ) = 6$

Question 2

$\text { For } f ( x , y ) = \sin ( 9 x y ) \text {, evaluate } f _ { y } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 9 \pi } { 4 }$ 
B)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 18$ 
C)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 9$ 
D)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 9 \pi } { 2 }$ 
E)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 0$ 
A)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 9 \pi } { 4 }$ 
B)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 18$ 
C)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 9$ 
D)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = \frac { 9 \pi } { 2 }$ 
E)  $f _ { y } \left( 2 , \frac { \pi } { 4 } \right) = 0$

Question 3

Differentiate implicitly to find $\frac { \partial w } { \partial x } \text {, given } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 7 y w + 2 w ^ { 2 } = 4$

Accepted Answer

A)  $\frac { x } { 4 w - 7 y }$ 
B)  $\frac { 2 x } { 7 w + 2 y }$ 
C)  $\frac { 2 x } { 7 y - 4 w }$ 
D)  $\frac { x } { 7 y - 2 w }$ 
E)  $\frac { 2 x } { 7 w + 4 y }$ 
A)  $\frac { x } { 4 w - 7 y }$ 
B)  $\frac { 2 x } { 7 w + 2 y }$ 
C)  $\frac { 2 x } { 7 y - 4 w }$ 
D)  $\frac { x } { 7 y - 2 w }$ 
E)  $\frac { 2 x } { 7 w + 4 y }$

Question 4

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $150.00, find theDimensions of the box of largest volume that can be made.

Accepted Answer

A)  base dimensions: $x = y = 5.77$; height $: z = 4.33$ 
B)  base dimensions: $x = y = 4.33$; height $: z = 5.77$ 
C)  base dimensions: $x = y = 7.75$; height $: z = 4.33$ 
D)  base dimensions: $x = y = 4.33$; height: $z = 7.75$ 
E)  base dimensions: $x = y = 7.75$; height: $z = 3.06$ 
A)  base dimensions: $x = y = 5.77$; height $: z = 4.33$ 
B)  base dimensions: $x = y = 4.33$; height $: z = 5.77$ 
C)  base dimensions: $x = y = 7.75$; height $: z = 4.33$ 
D)  base dimensions: $x = y = 4.33$; height: $z = 7.75$ 
E)  base dimensions: $x = y = 7.75$; height: $z = 3.06$

Question 5

Find the critical points of the function $f ( x , y , z ) = ( x + 7 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z + 6 ) ^ { 2 }$ and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Accepted Answer

A)  relative minimum at $( 7 , - 7,6 )$ 
B)  relative maximum at $( - 7,7 , - 6 )$ 
C)  relative maximum at $( 7 , - 7,6 )$ 
D)  relative minimum at $( - 7,7 , - 6 )$ 
E)  no relative extrema 
A)  relative minimum at $( 7 , - 7,6 )$ 
B)  relative maximum at $( - 7,7 , - 6 )$ 
C)  relative maximum at $( 7 , - 7,6 )$ 
D)  relative minimum at $( - 7,7 , - 6 )$ 
E)  no relative extrema

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 if the utility function is $U = - 3 x ^ { 2 } + x y - 8 y ^ { 2 }$

Accepted Answer

A)  $U _ { y } = x - 8 y$ 
B)  $U _ { y } = x - 32 y$ 
C)  $U _ { y } = x - 16 y$ 
D)  $U _ { y } = x + 8 y$ 
E)  $U _ { y } = x + 16 y$ 
A)  $U _ { y } = x - 8 y$ 
B)  $U _ { y } = x - 32 y$ 
C)  $U _ { y } = x - 16 y$ 
D)  $U _ { y } = x + 8 y$ 
E)  $U _ { y } = x + 16 y$

Question 7

Find the directional derivative of the function at P in the direction of $\overrightarrow { \mathbf { v } } \text {. }$ $f ( x , y ) = x ^ { 5 } - y ^ { 5 } , P ( 5,4 ) , \quad \overrightarrow { \mathbf { v } } = \frac { \sqrt { 2 } } { 2 } ( \hat { \mathbf { i } } + \hat { \mathbf { j } } )$

Accepted Answer

A)  $( 922.5 ) \sqrt { 2 }$ 
B)  $( - 5,252.5 ) \sqrt { 2 }$ 
C)  $( 1,845.0 ) \sqrt { 2 }$ 
D)  $( 5,252.5 ) \sqrt { 2 }$ 
E)  $( 1,677.5 ) \sqrt { 2 }$ 
A)  $( 922.5 ) \sqrt { 2 }$ 
B)  $( - 5,252.5 ) \sqrt { 2 }$ 
C)  $( 1,845.0 ) \sqrt { 2 }$ 
D)  $( 5,252.5 ) \sqrt { 2 }$ 
E)  $( 1,677.5 ) \sqrt { 2 }$

Question 8

Find and simplify the function $$f ( x , y ) = 7 - 2 x ^ { 2 } - 5 y ^ { 2 } \text { at the given value } ( 3,3 )$$

Accepted Answer

A) -56 
B) -26 
C) -14 
D) -44 
E) -27 
A) -56 
B) -26 
C) -14 
D) -44 
E) -27

Question 9

Find the total differential for the function $w = 4 z ^ { 3 } y \sin x$

Accepted Answer

A)  $d w = 4 z ^ { 3 } y \cos x d x - 4 z ^ { 3 } \cos x d y + 12 z ^ { 2 } y \sin x d z$ 
B)  $d w = 4 z ^ { 3 } y \sin x d x - 4 z ^ { 3 } \sin x d y - 8 z ^ { 2 } y \sin x d z$ 
C)  $d w = 4 z ^ { 3 } y \cos x d x + 8 z ^ { 3 } \sin x d y - 12 z ^ { 2 } y \sin x d z$ 
D)  $d w = 8 z ^ { 3 } y \sin x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$ 
E)  $d w = 4 z ^ { 3 } y \cos x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$ 
A)  $d w = 4 z ^ { 3 } y \cos x d x - 4 z ^ { 3 } \cos x d y + 12 z ^ { 2 } y \sin x d z$ 
B)  $d w = 4 z ^ { 3 } y \sin x d x - 4 z ^ { 3 } \sin x d y - 8 z ^ { 2 } y \sin x d z$ 
C)  $d w = 4 z ^ { 3 } y \cos x d x + 8 z ^ { 3 } \sin x d y - 12 z ^ { 2 } y \sin x d z$ 
D)  $d w = 8 z ^ { 3 } y \sin x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$ 
E)  $d w = 4 z ^ { 3 } y \cos x d x + 4 z ^ { 3 } \sin x d y + 12 z ^ { 2 } y \sin x d z$

Question 10

$\text { Let } w = x y \cos z \text {, where } x = t ^ { 3 } , y = t ^ { 5 } \text {, and } z = \arccos t \text {. Find } \frac { d w } { d t } \text {. }$

Accepted Answer

A)  $\frac { d w } { d t } = 8 t ^ { 8 } - \sqrt { 1 - t ^ { 2 } }$ 
B)  $\frac { d w } { d t } = 8 t ^ { 8 } + \sqrt { 1 - t ^ { 2 } }$ 
C)  $\frac { d w } { d t } = t ^ { 8 } \left( 8 + \sqrt { 1 - t ^ { 2 } } \right)$ 
D)  $\frac { d w } { d t } = 9 t ^ { 8 }$ 
E)  $\frac { d w } { d t } = t ^ { 8 } \left( 8 - \sqrt { 1 - t ^ { 2 } } \right)$ 
A)  $\frac { d w } { d t } = 8 t ^ { 8 } - \sqrt { 1 - t ^ { 2 } }$ 
B)  $\frac { d w } { d t } = 8 t ^ { 8 } + \sqrt { 1 - t ^ { 2 } }$ 
C)  $\frac { d w } { d t } = t ^ { 8 } \left( 8 + \sqrt { 1 - t ^ { 2 } } \right)$ 
D)  $\frac { d w } { d t } = 9 t ^ { 8 }$ 
E)  $\frac { d w } { d t } = t ^ { 8 } \left( 8 - \sqrt { 1 - t ^ { 2 } } \right)$

Question 11

Find the partial derivative $\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)$

Accepted Answer

A)  $\frac { \partial z } { \partial y } = - 5 y ^ { 4 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$ 
B)  $\frac { \partial z } { \partial y } = - 5 y ^ { 6 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$ 
C)  $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \sin \left( x ^ { 6 } + y ^ { 6 } \right)$ 
D)  $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \cos \left( x ^ { 6 } + y ^ { 6 } \right)$ 
E)  $\frac { \partial z } { \partial y } = 5 y ^ { 4 } \cos \left( x ^ { 5 } + y ^ { 5 } \right)$ 
A)  $\frac { \partial z } { \partial y } = - 5 y ^ { 4 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$ 
B)  $\frac { \partial z } { \partial y } = - 5 y ^ { 6 } \sin \left( x ^ { 5 } + y ^ { 5 } \right)$ 
C)  $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \sin \left( x ^ { 6 } + y ^ { 6 } \right)$ 
D)  $\frac { \partial z } { \partial y } = 5 y ^ { 6 } \cos \left( x ^ { 6 } + y ^ { 6 } \right)$ 
E)  $\frac { \partial z } { \partial y } = 5 y ^ { 4 } \cos \left( x ^ { 5 } + y ^ { 5 } \right)$

Question 12

Suppose the formula for wind chill C (in degrees Fahrenheit) is given by $C = 35.71 + 0.6217 T - 35.72 v ^ { 0.16 } + 0.4272 T v ^ { 0.16 }$ where $v$ is the wind speed in miles per hour and $T$ is the temperature in degrees Fahrenheit. The wind speed is $23 \pm 3$ miles per hour and the temperature is $8 ^ { \circ } \pm 1 ^ { \circ }$ Fahrenheit. Use $d C$ to estimate the maximum possible propagated error in calculating the wind chill. Round your answer to four decimal places.

Accepted Answer

A)  $d C = \pm 8.4405$ 
B)  $d C ^ { \prime } = \pm 9.8810$ 
C)  $d C = \pm 2.4405$ 
D)  $d C = \pm 8.8810$ 
E)  $d C = \pm 5.4405$ 
A)  $d C = \pm 8.4405$ 
B)  $d C ^ { \prime } = \pm 9.8810$ 
C)  $d C = \pm 2.4405$ 
D)  $d C = \pm 8.8810$ 
E)  $d C = \pm 5.4405$

Question 13

$\text { For } f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 5 x + y \text { find all values of } x \text { and } y \text { such that } f _ { x } ( x , y ) = 0$ and $f _ { y } ( x , y ) = 0$ simultaneously.

Accepted Answer

A)  $x = 4 ; y = 2$ 
B)  $x = 3 ; y = 1$ 
C)  $x = 5 ; y = 1$ 
D)  $x = 7 ; y = 3$ 
E)  $x = 2 , y = 0$ 
A)  $x = 4 ; y = 2$ 
B)  $x = 3 ; y = 1$ 
C)  $x = 5 ; y = 1$ 
D)  $x = 7 ; y = 3$ 
E)  $x = 2 , y = 0$

Question 14

Find three positive numbers x, y, and z whose sum is 30 and the sum of the squares is a minimum.

Accepted Answer

A)  $x = 10 , y = 10 , z = 10$ 
B)  $x = 2 , y = 12 , z = 16$ 
C)  $x = 7.5 , y = 7.5 , z = 15$ 
D)  $x = 4 , y = 6 , z = 10$ 
E)  $x = 5 , y = 5 , z = 20$ 
A)  $x = 10 , y = 10 , z = 10$ 
B)  $x = 2 , y = 12 , z = 16$ 
C)  $x = 7.5 , y = 7.5 , z = 15$ 
D)  $x = 4 , y = 6 , z = 10$ 
E)  $x = 5 , y = 5 , z = 20$

Question 15

$\text { Find } \frac { \partial w } { \partial s } \text { using the appropriate Chain Rule for } w = y ^ { 3 } - 8 x ^ { 2 } y \text { where } x = e ^ { s } \text { and }$ $y = e ^ { t }$, and evaluate the partial derivative at $s = - 2$ and $t = 3$. Round your answer to two decimal places.

Accepted Answer

A)  $- 8.83$ 
B)  $- 5.89$ 
C)  $- 6.62$ 
D)  $- 2.72$ 
E)  $- 8.09$ 
A)  $- 8.83$ 
B)  $- 5.89$ 
C)  $- 6.62$ 
D)  $- 2.72$ 
E)  $- 8.09$

Question 16

$\text { Given } f ( x , y ) = x \cos y \text {, use the total differential to approximate } \Delta z \text { at } ( 2,4 )$ towards $( 2.4,1.05 )$ . Round your answer to four decimal places.

Accepted Answer

A) -35.7266 
B) 30.5500 
C) 17.2734 
D) 15.5468 
E) -4.7266 
A) -35.7266 
B) 30.5500 
C) 17.2734 
D) 15.5468 
E) -4.7266

Question 17

The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What is the rate of change of the volumeWhen the radius is 14 inches and the height is 32 inches?

Accepted Answer

A)  $4480 \pi$ in $^ { 3 } / \mathrm { min }$ 
B)  $5488 \pi$ in $^ { 3 } / \mathrm { min }$ 
C)  $4920 \pi$ in $^ { 3 } / \mathrm { min }$ 
D)  $4592 \pi$ in $^ { 3 } / \mathrm { min }$ 
E)  $4928 \pi$ in $^ { 3 } / \mathrm { min }$ 
A)  $4480 \pi$ in $^ { 3 } / \mathrm { min }$ 
B)  $5488 \pi$ in $^ { 3 } / \mathrm { min }$ 
C)  $4920 \pi$ in $^ { 3 } / \mathrm { min }$ 
D)  $4592 \pi$ in $^ { 3 } / \mathrm { min }$ 
E)  $4928 \pi$ in $^ { 3 } / \mathrm { min }$

Question 18

Suppose the period T of a pendulum of length L is $T = 2 \pi \sqrt { \frac { L } { g } } \text { where } g \text { is the }$ acceleration due to gravity. A pendulum is moved from the Canal Zone, where $g = 32.01 \text { feet per }$ second per second, to Greenland, where $g = 32.83$ feet per second per second. Because of the change in temperature, the length of the pendulum changes from 2.6 feet to 2.45 feet. Approximate the Change in the period of the pendulum. Round your answer to four decimal places.

Accepted Answer

A) 5.8508 seconds 
B) -0.0746 second 
C) 0.9254 second 
D) 7.9254 seconds 
E) -7.1492 seconds 
A) 5.8508 seconds 
B) -0.0746 second 
C) 0.9254 second 
D) 7.9254 seconds 
E) -7.1492 seconds

Question 19

Find the limit  $\lim _ { \Delta y \rightarrow 0 } \frac { f ( x , y + \Delta y ) - f ( x , y ) } { \Delta y } \text { for the function } f ( x , y ) = 7 x + 5 x y - 2 y$

Accepted Answer

A)  $5 x - 2$ 
B)  $7 + 5 x$ 
C)  $7 + 5 y$ 
D)  0 
E)  $- 2 y$ 
A)  $5 x - 2$ 
B)  $7 + 5 x$ 
C)  $7 + 5 y$ 
D)  0 
E)  $- 2 y$

Question 20

$\text { Find } \lim _ { ( x , y ) \rightarrow ( a , b ) } [ f ( x , y ) - g ( x , y ) ) \text { by using the limits } \lim _ { ( x , y ) \rightarrow ( a , b ) } f ( x , y ) = 7 \text { and }$ $\lim _ { ( x , y ) \rightarrow ( a , b ) } g ( x , y ) = 4$

Accepted Answer

A)  7 
B)  2 
C)  3 
D)  $- 3$ 
E)  $- 4$ 
A)  7 
B)  2 
C)  3 
D)  $- 3$ 
E)  $- 4$

$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$

$\text { For } f ( x , y ) = \sin ( 9 x y ) \text {, evaluate } f _ { y } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Differentiate implicitly to find $\frac { \partial w } { \partial x } \text {, given } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 7 y w + 2 w ^ { 2 } = 4$

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $150.00, find theDimensions of the box of largest volume that can be made.

Find the critical points of the function $f ( x , y , z ) = ( x + 7 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z + 6 ) ^ { 2 }$ and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

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 if the utility function is $U = - 3 x ^ { 2 } + x y - 8 y ^ { 2 }$

Find the directional derivative of the function at P in the direction of $\overrightarrow { \mathbf { v } } \text {. }$ $f ( x , y ) = x ^ { 5 } - y ^ { 5 } , P ( 5,4 ) , \quad \overrightarrow { \mathbf { v } } = \frac { \sqrt { 2 } } { 2 } ( \hat { \mathbf { i } } + \hat { \mathbf { j } } )$

Find and simplify the function $f ( x , y ) = 7 - 2 x ^ { 2 } - 5 y ^ { 2 } \text { at the given value } ( 3,3 )$

Find the total differential for the function $w = 4 z ^ { 3 } y \sin x$

$\text { Let } w = x y \cos z \text {, where } x = t ^ { 3 } , y = t ^ { 5 } \text {, and } z = \arccos t \text {. Find } \frac { d w } { d t } \text {. }$

Find the partial derivative $\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)$

$\text { For } f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 5 x + y \text { find all values of } x \text { and } y \text { such that } f _ { x } ( x , y ) = 0$ and $f _ { y } ( x , y ) = 0$ simultaneously.

Find three positive numbers x, y, and z whose sum is 30 and the sum of the squares is a minimum.

$\text { Find } \frac { \partial w } { \partial s } \text { using the appropriate Chain Rule for } w = y ^ { 3 } - 8 x ^ { 2 } y \text { where } x = e ^ { s } \text { and }$ $y = e ^ { t }$ , and evaluate the partial derivative at $s = - 2$ and $t = 3$ . Round your answer to two decimal places.

$\text { Given } f ( x , y ) = x \cos y \text {, use the total differential to approximate } \Delta z \text { at } ( 2,4 )$ towards $( 2.4,1.05 )$ . Round your answer to four decimal places.

The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What is the rate of change of the volumeWhen the radius is 14 inches and the height is 32 inches?

Find the limit $\lim _ { \Delta y \rightarrow 0 } \frac { f ( x , y + \Delta y ) - f ( x , y ) } { \Delta y } \text { for the function } f ( x , y ) = 7 x + 5 x y - 2 y$

$\text { Find } \lim _ { ( x , y ) \rightarrow ( a , b ) } [ f ( x , y ) - g ( x , y ) ) \text { by using the limits } \lim _ { ( x , y ) \rightarrow ( a , b ) } f ( x , y ) = 7 \text { and }$ $\lim _ { ( x , y ) \rightarrow ( a , b ) } g ( x , y ) = 4$

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

For f(x,y)=eysin⁡3x, evaluate fx at the point (π,0). \text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. } For f(x,y)=eysin3x, evaluate fx​ at the point (π,0).

For f(x,y)=sin⁡(9xy), evaluate fy at the point (2,π4)\text { For } f ( x , y ) = \sin ( 9 x y ) \text {, evaluate } f _ { y } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right) For f(x,y)=sin(9xy), evaluate fy​ at the point (2,4π​)

Differentiate implicitly to find ∂w∂x, given x2+y2+z2−7yw+2w2=4\frac { \partial w } { \partial x } \text {, given } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 7 y w + 2 w ^ { 2 } = 4∂x∂w​, given x2+y2+z2−7yw+2w2=4

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $150.00, find theDimensions of the box of largest volume that can be made.

Find the critical points of the function f(x,y,z)=(x+7)2+(7−y)2+(z+6)2f ( x , y , z ) = ( x + 7 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z + 6 ) ^ { 2 }f(x,y,z)=(x+7)2+(7−y)2+(z+6)2 and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Find and simplify the function f(x,y)=7−2x2−5y2 at the given value (3,3)f ( x , y ) = 7 - 2 x ^ { 2 } - 5 y ^ { 2 } \text { at the given value } ( 3,3 )f(x,y)=7−2x2−5y2 at the given value (3,3)

Find the total differential for the function w=4z3ysin⁡xw = 4 z ^ { 3 } y \sin xw=4z3ysinx

Let w=xycos⁡z, where x=t3,y=t5, and z=arccos⁡t. Find dwdt. \text { Let } w = x y \cos z \text {, where } x = t ^ { 3 } , y = t ^ { 5 } \text {, and } z = \arccos t \text {. Find } \frac { d w } { d t } \text {. } Let w=xycosz, where x=t3,y=t5, and z=arccost. Find dtdw​.

Find the partial derivative ∂z∂y for the function z=cos⁡(x5+y5)\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)∂y∂z​ for the function z=cos(x5+y5)

Find three positive numbers x, y, and z whose sum is 30 and the sum of the squares is a minimum.

The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What is the rate of change of the volumeWhen the radius is 14 inches and the height is 32 inches?

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

$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$

$\text { For } f ( x , y ) = \sin ( 9 x y ) \text {, evaluate } f _ { y } \text { at the point } \left( 2 , \frac { \pi } { 4 } \right)$

Differentiate implicitly to find $\frac { \partial w } { \partial x } \text {, given } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 7 y w + 2 w ^ { 2 } = 4$

Find the critical points of the function $f ( x , y , z ) = ( x + 7 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z + 6 ) ^ { 2 }$ and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Find and simplify the function $f ( x , y ) = 7 - 2 x ^ { 2 } - 5 y ^ { 2 } \text { at the given value } ( 3,3 )$

Find the total differential for the function $w = 4 z ^ { 3 } y \sin x$

$\text { Let } w = x y \cos z \text {, where } x = t ^ { 3 } , y = t ^ { 5 } \text {, and } z = \arccos t \text {. Find } \frac { d w } { d t } \text {. }$

Find the partial derivative $\frac { \partial z } { \partial y } \text { for the function } z = \cos \left( x ^ { 5 } + y ^ { 5 } \right)$