Question 1

Find all the saddle points of the function.
$$f ( x , y ) = x \sin \frac { y } { 5 }$$

Accepted Answer

The answer of Find all the saddle points of the...

Question 2

If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R _ { 1 } , R _ { 2 } , R _ { 3 }$, then
$$\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } } + \frac { 1 } { R _ { 3 } }$$
If the resistances are measured in ohms as $R _ { 1 } = 30 \Omega , R _ { 2 } = 35 \Omega$ and $R _ { 3 } = 50 \Omega$, with a possible error of $0.8 \%$ in each case, estimate the maximum error in the calculated value of $R$.

Accepted Answer

A)  $0.347 \Omega$ 
B)  $0.476 \Omega$ 
C)  $0.291 \Omega$ 
D)  $0.098 \Omega$ 
E)  $0.957 \Omega$ 
A)  $0.347 \Omega$ 
B)  $0.476 \Omega$ 
C)  $0.291 \Omega$ 
D)  $0.098 \Omega$ 
E)  $0.957 \Omega$

Question 3

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

Accepted Answer

The answer of Evaluate the limit.
\[\lim _ { ( x...

Question 4

Use the equation $\frac { d y } { d x } = - \frac { \frac { \partial F } { \partial x } } { \frac { \partial F } { \partial y } } = - \frac { F _ { x } } { F _ { y } }$ to find $\frac { d y } { d x }$.
$$\cos ( x - 8 y ) = x e ^ { 4 y }$$

Accepted Answer

The answer of Use the equation \(\frac { d y...

Question 5

Determine the largest set on which the function is continuous.
$$f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$$

Accepted Answer

The answer of Determine the largest set on which the...

Question 6

$$\text { Find the domain and range of the function } h ( x , y ) = \sqrt { 2 x - 9 y } \text {. }$$

Accepted Answer

\[\begin{array} { l } 
D = \le

Question 7

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 8

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 9

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z \quad$ 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 10

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 11

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 12

Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point $( 1,3 )$ in the direction toward the point $( 3,1 )$.
Select the correct answer.

Accepted Answer

A)  $7 \sqrt { 2 }$ 
B)  $14 \sqrt { 2 }$ 
C)  $\sqrt { 2 }$ 
D)  28 
E)  none of these 
A)  $7 \sqrt { 2 }$ 
B)  $14 \sqrt { 2 }$ 
C)  $\sqrt { 2 }$ 
D)  28 
E)  none of these

Question 13

Find $h _ { z y y } ( x , y , z )$ for the function $h ( x , y , z ) = e ^ { 6 x } \cos ( y + 3 z )$
Select the correct answer.

Accepted Answer

A)  $18 e ^ { 6 x } \cos ( y + 3 z )$ 
B)  $9 e ^ { 6 x } \sin ( y + 3 z )$ 
C)  $- 9 e ^ { 6 x } \cos ( y + 3 z )$ 
D)  $- 18 e ^ { 6 x } \sin ( y + 3 z )$ 
A)  $18 e ^ { 6 x } \cos ( y + 3 z )$ 
B)  $9 e ^ { 6 x } \sin ( y + 3 z )$ 
C)  $- 9 e ^ { 6 x } \cos ( y + 3 z )$ 
D)  $- 18 e ^ { 6 x } \sin ( y + 3 z )$

Question 14

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

Accepted Answer

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

Question 15

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$. Find $f ( 3 h , 4 k )$.

Accepted Answer

A)  $12 h ^ { 2 } + 36 h k - 60 h - 84$ 
B)  $9 h ^ { 2 } + 36 h k - 20 h - 7$ 
C)  $16 h ^ { 2 } + 36 h k - 20 h - 7$ 
D)  $9 h ^ { 2 } + 36 h k - 15 h - 7$ 
A)  $12 h ^ { 2 } + 36 h k - 60 h - 84$ 
B)  $9 h ^ { 2 } + 36 h k - 20 h - 7$ 
C)  $16 h ^ { 2 } + 36 h k - 20 h - 7$ 
D)  $9 h ^ { 2 } + 36 h k - 15 h - 7$

Question 16

At what point is the following function a local maximum?
$$f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$$

Accepted Answer

A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 3,1 )$ 
D)  $( 0,1 )$ 
E)  $( 1 , - 1 )$ 
A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 3,1 )$ 
D)  $( 0,1 )$ 
E)  $( 1 , - 1 )$

Question 17

A boundary stripe $2 \mathrm { in }$. wide is painted around a rectangle whose dimensions are $100 \mathrm { ft }$ by $240 \mathrm { ft }$. Use differentials to approximate the number of square feet of paint in the stripe.
Select the correct answer.

Accepted Answer

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

Question 18

Use implicit differentiation to find $\frac { \partial z } { \partial x }$. $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

Accepted Answer

A) 
$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y } { 1 + 2 y }$ 
B) 
$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y + z } { x + 2 y z }$ 
C) 
$$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } } { 1 + 2 y }$$ 
D)  $\frac { \partial z } { \partial x } = \frac { 1 } { 1 + 2 y }$ 
A) 
$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y } { 1 + 2 y }$ 
B) 
$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y + z } { x + 2 y z }$ 
C) 
$$\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } } { 1 + 2 y }$$ 
D)  $\frac { \partial z } { \partial x } = \frac { 1 } { 1 + 2 y }$

Question 19

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 2,1 )$. In what direction does it occur?

Accepted Answer

The answer of Find the maximum rate of change of...

Question 20

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

Accepted Answer

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

Find all the saddle points of the function. $f ( x , y ) = x \sin \frac { y } { 5 }$

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

Use the equation $\frac { d y } { d x } = - \frac { \frac { \partial F } { \partial x } } { \frac { \partial F } { \partial y } } = - \frac { F _ { x } } { F _ { y } }$ to find $\frac { d y } { d x }$ . $\cos ( x - 8 y ) = x e ^ { 4 y }$

Determine the largest set on which the function is continuous. $f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$

$\text { Find the domain and range of the function } h ( x , y ) = \sqrt { 2 x - 9 y } \text {. }$

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

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 $f ( x , y , z ) = 5 x - y - 5 z \quad$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

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

Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point $( 1,3 )$ in the direction toward the point $( 3,1 )$ . Select the correct answer.

Find $h _ { z y y } ( x , y , z )$ for the function $h ( x , y , z ) = e ^ { 6 x } \cos ( y + 3 z )$ Select the correct answer.

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

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$ . Find $f ( 3 h , 4 k )$ .

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

A boundary stripe $2 \mathrm { in }$ . wide is painted around a rectangle whose dimensions are $100 \mathrm { ft }$ by $240 \mathrm { ft }$ . Use differentials to approximate the number of square feet of paint in the stripe. Select the correct answer.

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ . $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 2,1 )$ . In what direction does it occur?

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

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

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Vectors and the Geometry of Space

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Exam 14: Partial Derivatives

Find all the saddle points of the function. f(x,y)=xsin⁡y5f ( x , y ) = x \sin \frac { y } { 5 }f(x,y)=xsin5y​

Evaluate the limit. lim⁡(x,y)→(1,1)3x2yx2+2y2\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 3 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 2 y ^ { 2 } } }lim(x,y)→(1,1)​x2+2y2​3x2y​

Determine the largest set on which the function is continuous. f(x,y,z)=xyz6x2+2y2−zf ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }f(x,y,z)=6x2+2y2−zxyz​

Find the domain and range of the function h(x,y)=2x−9y. \text { Find the domain and range of the function } h ( x , y ) = \sqrt { 2 x - 9 y } \text {. } Find the domain and range of the function h(x,y)=2x−9y​.

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 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 z \quadf(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 .

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 the directional derivative of f(x,y)=2x−y3f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }f(x,y)=2x​−y3 at the point (1,3)( 1,3 )(1,3) in the direction toward the point (3,1)( 3,1 )(3,1) . Select the correct answer.

Find hzyy(x,y,z)h _ { z y y } ( x , y , z )hzyy​(x,y,z) for the function h(x,y,z)=e6xcos⁡(y+3z)h ( x , y , z ) = e ^ { 6 x } \cos ( y + 3 z )h(x,y,z)=e6xcos(y+3z) Select the correct answer.

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) . Select the correct answer.

Let f(x,y)=x2+3xy−5x−7f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7f(x,y)=x2+3xy−5x−7 . Find f(3h,4k)f ( 3 h , 4 k )f(3h,4k) .

At what point is the following function a local maximum? f(x,y)=3−10x+12y−5x2−6y2f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }f(x,y)=3−10x+12y−5x2−6y2

A boundary stripe 2in2 \mathrm { in }2in . wide is painted around a rectangle whose dimensions are 100ft100 \mathrm { ft }100ft by 240ft240 \mathrm { ft }240ft . Use differentials to approximate the number of square feet of paint in the stripe. Select the correct answer.

Use implicit differentiation to find ∂z∂x\frac { \partial z } { \partial x }∂x∂z​ . x4y+xz+yz2=7x ^ { 4 } y + x z + y z ^ { 2 } = 7x4y+xz+yz2=7

Find the maximum rate of change of f(x,y)=xy2+yf ( x , y ) = x y ^ { 2 } + \sqrt { y }f(x,y)=xy2+y​ at the point (2,1)( 2,1 )(2,1) . In what direction does it occur?

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 } Find the differential of the function z=x−6yx+5y​

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

Find all the saddle points of the function. $f ( x , y ) = x \sin \frac { y } { 5 }$

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

Determine the largest set on which the function is continuous. $f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$

$\text { Find the domain and range of the function } h ( x , y ) = \sqrt { 2 x - 9 y } \text {. }$

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

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 $f ( x , y , z ) = 5 x - y - 5 z \quad$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

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

Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point $( 1,3 )$ in the direction toward the point $( 3,1 )$ . Select the correct answer.

Find $h _ { z y y } ( x , y , z )$ for the function $h ( x , y , z ) = e ^ { 6 x } \cos ( y + 3 z )$ Select the correct answer.

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

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$ . Find $f ( 3 h , 4 k )$ .

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

A boundary stripe $2 \mathrm { in }$ . wide is painted around a rectangle whose dimensions are $100 \mathrm { ft }$ by $240 \mathrm { ft }$ . Use differentials to approximate the number of square feet of paint in the stripe. Select the correct answer.

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ . $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 2,1 )$ . In what direction does it occur?

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