Question 1

$\text { Find three positive numbers whose sum is } 400 \text { and whose product is a maximum. }$

Accepted Answer

$$\frac { 400 } { 3 } , \frac { 400 } { 3 } , \frac { 400 } { 3 }$$

Question 2

Find the gradient of the function $f ( x , y , z ) = z ^ { \gamma } e ^ { 2 x \sqrt { y } }$.

Accepted Answer

A)  
$e ^ { x ^ { 2 } \sqrt { y } } \left( 7 x z \sqrt { y } , \frac { x ^ { 2 } z } { 2 \sqrt { y } } , 2 \right)$ 
B) 
$z e ^ { x \sqrt { y } } \left( z \sqrt { y } , \frac { x z } { 2 \sqrt { y } } , 7 \right)$ 
C) 
$e ^ { x ^ { 2 } } \sqrt { y } \left( 2 x z \sqrt { y } , \frac { x ^ { 2 } z } { 7 \sqrt { y } } , 1 \right)$ 
D) 
$z e ^ { 2 x \sqrt { y } } \left( 2 z \sqrt { y } , \frac { x z } { \sqrt { y } } , 2 \right)$ 
E) 
$z e ^ { 2 x \sqrt { y } } \left( 2 z ^ { 6 } \sqrt { y } , \frac { x z ^ { 6 } } { \sqrt { y } } , 7 z ^ { 5 } \right)$ 
A)  
$e ^ { x ^ { 2 } \sqrt { y } } \left( 7 x z \sqrt { y } , \frac { x ^ { 2 } z } { 2 \sqrt { y } } , 2 \right)$ 
B) 
$z e ^ { x \sqrt { y } } \left( z \sqrt { y } , \frac { x z } { 2 \sqrt { y } } , 7 \right)$ 
C) 
$e ^ { x ^ { 2 } } \sqrt { y } \left( 2 x z \sqrt { y } , \frac { x ^ { 2 } z } { 7 \sqrt { y } } , 1 \right)$ 
D) 
$z e ^ { 2 x \sqrt { y } } \left( 2 z \sqrt { y } , \frac { x z } { \sqrt { y } } , 2 \right)$ 
E) 
$z e ^ { 2 x \sqrt { y } } \left( 2 z ^ { 6 } \sqrt { y } , \frac { x z ^ { 6 } } { \sqrt { y } } , 7 z ^ { 5 } \right)$ E

Question 3

The ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point $( 1,2,2 )$.

Accepted Answer

A)  $x = 1 + t , y = 2 , z = - 4 t + 2$ 
B)  $x = 1 + t , y = 2 , z = - 4 t - 2$ 
C)  $x = 1 + t , y = 1 , z = 4 t + 2$ 
D)  $x = t , y = 2 , z = - 4 t + 2$ 
E)  $x = 1 + t , y = 2 , z = 4 t - 2$ 
A)  $x = 1 + t , y = 2 , z = - 4 t + 2$ 
B)  $x = 1 + t , y = 2 , z = - 4 t - 2$ 
C)  $x = 1 + t , y = 1 , z = 4 t + 2$ 
D)  $x = t , y = 2 , z = - 4 t + 2$ 
E)  $x = 1 + t , y = 2 , z = 4 t - 2$ A

Question 4

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$

Accepted Answer

A)  12 
B)  - 4 
C)  - 12 
D)  4 
E)  0 
A)  12 
B)  - 4 
C)  - 12 
D)  4 
E)  0

Question 5

$\text { Find the second partial derivatives of the function } g ( x , y ) = 2 x ^ { 3 } y ^ { 2 } + 9 x y ^ { 3 } - 6 x + 8 y + 5$

Accepted Answer

The answer of \(\text { Find the second partial derivatives...

Question 6

$\text { Describe the level survaces of the function } f ( x , y , z ) = 3 x + 8 y - 4 z + 2 \text {. }$

Accepted Answer

A family o

Question 7

Evaluate the limit.
$$\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 5 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } } }$$

Accepted Answer

A)  $\sqrt { 4 }$ 
B)  0 
C)  $\frac { 5 } { \sqrt { 4 } }$ 
D)  $\frac { \sqrt { 5 } } { 5 }$ 
E)  $\sqrt { 5 }$ 
A)  $\sqrt { 4 }$ 
B)  0 
C)  $\frac { 5 } { \sqrt { 4 } }$ 
D)  $\frac { \sqrt { 5 } } { 5 }$ 
E)  $\sqrt { 5 }$

Question 8

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

Accepted Answer

The answer of Find three positive real numbers whose sum...

Question 9

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 9 x ^ { 2 } + 9 y ^ { 2 } \right) } { 2 x ^ { 2 } + 2 y ^ { 2 } }$. Hint: If $x = r \cos \theta$ and $y = r \sin \theta$, then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$.

Accepted Answer

A)  $\frac { 9 } { 2 }$ 
B)  0 
C)  $\frac { 2 } { 9 }$ 
D)  1 
A)  $\frac { 9 } { 2 }$ 
B)  0 
C)  $\frac { 2 } { 9 }$ 
D)  1

Question 10

A boundary stripe 2 in.wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Accepted Answer

A)  $\quad 113 \mathrm { ft } ^ { 2 }$ 
B)  $\quad 113.81 \mathrm { ft } ^ { 2 }$ 
C)  $\quad 113.22 \mathrm { ft } ^ { 2 }$ 
D)  $\quad 113.89 \mathrm { ft } ^ { 2 }$ 
E)  $\quad 113.33 \mathrm { ft } ^ { 2 }$ 
A)  $\quad 113 \mathrm { ft } ^ { 2 }$ 
B)  $\quad 113.81 \mathrm { ft } ^ { 2 }$ 
C)  $\quad 113.22 \mathrm { ft } ^ { 2 }$ 
D)  $\quad 113.89 \mathrm { ft } ^ { 2 }$ 
E)  $\quad 113.33 \mathrm { ft } ^ { 2 }$

Question 11

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$.
$8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Accepted Answer

The answer of Use partial derivatives to find the implicit...

Question 12

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$

Accepted Answer

A) 
$x + 9 y + 9 z = 22 , \quad \frac { x - 2 } { 2 } = y - 9 = z - 9$ 
B) 
$x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$ 
C) 
$2 x + 9 y + 9 z = 22 , \quad \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$ 
D) 
$2 x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = y - 9 = z - 9$ 
A) 
$x + 9 y + 9 z = 22 , \quad \frac { x - 2 } { 2 } = y - 9 = z - 9$ 
B) 
$x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$ 
C) 
$2 x + 9 y + 9 z = 22 , \quad \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$ 
D) 
$2 x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = y - 9 = z - 9$

Question 13

$\text { Find the first partial derivatives of the function } f ( x , y , z ) = 2 x ^ { 3 } + 9 x y + 3 y z - z ^ { 5 } \text {. }$

Accepted Answer

The answer of \(\text { Find the first partial derivatives...

Question 14

Find the limit $\lim _ { ( x , y ) \rightarrow \left( 0 ^ { + } , 0 ^ { + } \right) } \frac { e ^ { \sqrt { x + y } } } { 7 x + 3 y - 8 }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $- \frac { 1 } { 2 }$

Question 15

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$

Accepted Answer

The answer of Use Lagrange multipliers to find the maximum...

Question 16

Suppose that over a certain region of space the electrical potential $V$ is given by $V ( x , y , z ) = 8 x ^ { 2 } - 7 x y + 7 x y z$.
Find the rate of change of the potential at $( - 1,1 , - 1 )$ in the direction of the vector $\mathbf { v } = 7 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$.

Accepted Answer

A)  -15.099 
B)  14.569856 
C)  44 
D)  20 
E)  - 0.96 
A)  -15.099 
B)  14.569856 
C)  44 
D)  20 
E)  - 0.96

Question 17

$\text { Find the differential of the function } z = \frac { x + 5 y } { x - 6 y } \text {. }$

Accepted Answer

The answer of \(\text { Find the differential of the...

Question 18

Find the directional derivative of the function $f ( x , y ) = ( x + 6 ) e ^ { y }$ at the point $P ( 7,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive $x$-axis.

Accepted Answer

A)  $6 e ^ { 7 }$ 
B)  13 
C)  1 
D)  $e ^ { \gamma }$ 
A)  $6 e ^ { 7 }$ 
B)  13 
C)  1 
D)  $e ^ { \gamma }$

Question 19

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01
Cm thick.(rounded to the nearest hundredth.)

Accepted Answer

A)  $8.34 \mathrm {~cm} ^ { 3 }$ 
B)  $6.99 \mathrm {~cm} ^ { 3 }$ 
C)  $6.91 \mathrm {~cm} ^ { 3 }$ 
D)  $6.7 \mathrm {~cm} ^ { 3 }$ 
E)  $12.06 \mathrm {~cm} ^ { 3 }$ 
A)  $8.34 \mathrm {~cm} ^ { 3 }$ 
B)  $6.99 \mathrm {~cm} ^ { 3 }$ 
C)  $6.91 \mathrm {~cm} ^ { 3 }$ 
D)  $6.7 \mathrm {~cm} ^ { 3 }$ 
E)  $12.06 \mathrm {~cm} ^ { 3 }$

Question 20

The length $l$, width $w$ and height $h$ of a box change with time. At a certain instant the dimensions are $l = 3$ and $w = h = 4$, and $l$ and $w$ are increasing at a rate of $10 \mathrm {~m} / \mathrm { s }$ while $h$ is decreasing at a rate of 1 $\mathrm { m } / \mathrm { s }$. At that instant find the rates at which the surface area is changing.

Accepted Answer

The answer of The length $l$, width $w$ and height...

$\text { Find three positive numbers whose sum is } 400 \text { and whose product is a maximum. }$

Find the gradient of the function $f ( x , y , z ) = z ^ { \gamma } e ^ { 2 x \sqrt { y } }$ .

The ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point $( 1,2,2 )$ .

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$

$\text { Find the second partial derivatives of the function } g ( x , y ) = 2 x ^ { 3 } y ^ { 2 } + 9 x y ^ { 3 } - 6 x + 8 y + 5$

$\text { Describe the level survaces of the function } f ( x , y , z ) = 3 x + 8 y - 4 z + 2 \text {. }$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 5 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } } }$

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 9 x ^ { 2 } + 9 y ^ { 2 } \right) } { 2 x ^ { 2 } + 2 y ^ { 2 } }$ . Hint: If $x = r \cos \theta$ and $y = r \sin \theta$ , then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$ .

A boundary stripe 2 in.wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ . $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$

$\text { Find the first partial derivatives of the function } f ( x , y , z ) = 2 x ^ { 3 } + 9 x y + 3 y z - z ^ { 5 } \text {. }$

Find the limit $\lim _ { ( x , y ) \rightarrow \left( 0 ^ { + } , 0 ^ { + } \right) } \frac { e ^ { \sqrt { x + y } } } { 7 x + 3 y - 8 }$

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$

$\text { Find the differential of the function } z = \frac { x + 5 y } { x - 6 y } \text {. }$

Find the directional derivative of the function $f ( x , y ) = ( x + 6 ) e ^ { y }$ at the point $P ( 7,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive $x$ -axis.

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 Cm thick.(rounded to the nearest hundredth.)

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 14: Partial Derivatives

Find three positive numbers whose sum is 400 and whose product is a maximum. \text { Find three positive numbers whose sum is } 400 \text { and whose product is a maximum. } Find three positive numbers whose sum is 400 and whose product is a maximum.

Find the gradient of the function f(x,y,z)=zγe2xyf ( x , y , z ) = z ^ { \gamma } e ^ { 2 x \sqrt { y } }f(x,y,z)=zγe2xy​ .

The ellipsoid 8x2+3y2+z2=148 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 148x2+3y2+z2=14 intersects the plane y=2y = 2y=2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2)( 1,2,2 )(1,2,2) .

Find fy(−24,8)f _ { y } ( - 24,8 )fy​(−24,8) for f(x,y)=sin⁡(4x+12y)f ( x , y ) = \sin ( 4 x + 12 y )f(x,y)=sin(4x+12y)

Find the second partial derivatives of the function g(x,y)=2x3y2+9xy3−6x+8y+5\text { Find the second partial derivatives of the function } g ( x , y ) = 2 x ^ { 3 } y ^ { 2 } + 9 x y ^ { 3 } - 6 x + 8 y + 5 Find the second partial derivatives of the function g(x,y)=2x3y2+9xy3−6x+8y+5

Describe the level survaces of the function f(x,y,z)=3x+8y−4z+2. \text { Describe the level survaces of the function } f ( x , y , z ) = 3 x + 8 y - 4 z + 2 \text {. } Describe the level survaces of the function f(x,y,z)=3x+8y−4z+2.

Evaluate the limit. lim⁡(x,y)→(1,1)5x2yx2+4y2\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 5 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } } }lim(x,y)→(1,1)​x2+4y2​5x2y​

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

A boundary stripe 2 in.wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Use partial derivatives to find the implicit derivative dydx\frac { d y } { d x }dxdy​ . 8x2+9xy−7y=58 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 58x2+9xy​−7y=5

Find equations for the tangent plane and the normal line to the surface with equation x2+9y2+9z2=22x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22x2+9y2+9z2=22 at the point P(2,1,1)P ( 2,1,1 )P(2,1,1)

Find the first partial derivatives of the function f(x,y,z)=2x3+9xy+3yz−z5. \text { Find the first partial derivatives of the function } f ( x , y , z ) = 2 x ^ { 3 } + 9 x y + 3 y z - z ^ { 5 } \text {. } Find the first partial derivatives of the function f(x,y,z)=2x3+9xy+3yz−z5.

Find the limit lim⁡(x,y)→(0+,0+)ex+y7x+3y−8\lim _ { ( x , y ) \rightarrow \left( 0 ^ { + } , 0 ^ { + } \right) } \frac { e ^ { \sqrt { x + y } } } { 7 x + 3 y - 8 }lim(x,y)→(0+,0+)​7x+3y−8ex+y​​

Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,z)=5x−y−5zf ( x , y , z ) = 5 x - y - 5 zf(x,y,z)=5x−y−5z subject to the constraints x+2y−z=0x + 2 y - z = 0x+2y−z=0 and x2+4y2=1x ^ { 2 } + 4 y ^ { 2 } = 1x2+4y2=1

Find the differential of the function z=x+5yx−6y. \text { Find the differential of the function } z = \frac { x + 5 y } { x - 6 y } \text {. } Find the differential of the function z=x−6yx+5y​.

Find the directional derivative of the function f(x,y)=(x+6)eyf ( x , y ) = ( x + 6 ) e ^ { y }f(x,y)=(x+6)ey at the point P(7,0)P ( 7,0 )P(7,0) in the direction of the unit vector that makes the angle θ=π2\theta = \frac { \pi } { 2 }θ=2π​ with the positive xxx -axis.

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 Cm thick.(rounded to the nearest hundredth.)

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

$\text { Find three positive numbers whose sum is } 400 \text { and whose product is a maximum. }$

Find the gradient of the function $f ( x , y , z ) = z ^ { \gamma } e ^ { 2 x \sqrt { y } }$ .

The ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point $( 1,2,2 )$ .

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$

$\text { Find the second partial derivatives of the function } g ( x , y ) = 2 x ^ { 3 } y ^ { 2 } + 9 x y ^ { 3 } - 6 x + 8 y + 5$

$\text { Describe the level survaces of the function } f ( x , y , z ) = 3 x + 8 y - 4 z + 2 \text {. }$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 5 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } } }$

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ . $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$

$\text { Find the first partial derivatives of the function } f ( x , y , z ) = 2 x ^ { 3 } + 9 x y + 3 y z - z ^ { 5 } \text {. }$

Find the limit $\lim _ { ( x , y ) \rightarrow \left( 0 ^ { + } , 0 ^ { + } \right) } \frac { e ^ { \sqrt { x + y } } } { 7 x + 3 y - 8 }$

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$

$\text { Find the differential of the function } z = \frac { x + 5 y } { x - 6 y } \text {. }$

Find the directional derivative of the function $f ( x , y ) = ( x + 6 ) e ^ { y }$ at the point $P ( 7,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive $x$ -axis.