Question 1

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 2

$$\text { Use the definition of partial derivatives as limits to find } f _ { x } ( x , y ) \text { if } f ( x , y ) = 4 x ^ { 2 } - 8 x y + 2 y ^ { 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Use the definition of partial...

Question 3

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

Accepted Answer

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

Question 4

$$\text { Find the domain and range of the function } g ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + 3$$

Accepted Answer

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

Question 5

Evaluate the limit.
$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

A)  2 
B)  the limit does not exist 
C)  7 
D)  1 
E)  0 
A)  2 
B)  the limit does not exist 
C)  7 
D)  1 
E)  0

Question 6

Find three positive numbers whose sum is 291 and whose product is a maximum.

Accepted Answer

A)  $x = y = z = 97$ 
B)  $\quad x = 109 , y = 99 , z = 89$ 
C)  $x = 100 , y = 108 , z = 89$ 
D)  $x = 99 , y = z = - 99$ 
E)  $x = 129 , y = 79 , z = 89$ 
A)  $x = y = z = 97$ 
B)  $\quad x = 109 , y = 99 , z = 89$ 
C)  $x = 100 , y = 108 , z = 89$ 
D)  $x = 99 , y = z = - 99$ 
E)  $x = 129 , y = 79 , z = 89$

Question 7

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

Accepted Answer

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

Question 8

Use polar coordinates to find the limit.
$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 3 } + y ^ { 3 } } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

The answer of Use polar coordinates to find the limit.
\[\lim...

Question 9

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 2,2 )$.
Select the correct answer.

Accepted Answer

A)  64 
B)  78 
C)  35 
D)  28 
E)  48 
A)  64 
B)  78 
C)  35 
D)  28 
E)  48

Question 10

Use the Chain Rule to find $\frac { d w } { d t }$. $w = 3 x ^ { 6 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 2 t , \quad z = t \sin t$

Accepted Answer

A)  $3 x ^ { 5 } y ^ { 2 } ( 36 y z - 6 x z \sin 2 t + x y ( \sin t + t \cos t ) )$ 
B)  $3 x ^ { 6 } y ^ { 3 } ( 36 y z - 6 x z \sin 2 t + x y ( \sin t + t \cos t ) )$ 
C)  $648 x ^ { 5 } y ^ { 2 } \sin 2 t ( \sin t + t \cos t )$ 
D)  $648 x ^ { 6 } y ^ { 3 } z \sin 2 t ( \sin t + t \cos t )$ 
A)  $3 x ^ { 5 } y ^ { 2 } ( 36 y z - 6 x z \sin 2 t + x y ( \sin t + t \cos t ) )$ 
B)  $3 x ^ { 6 } y ^ { 3 } ( 36 y z - 6 x z \sin 2 t + x y ( \sin t + t \cos t ) )$ 
C)  $648 x ^ { 5 } y ^ { 2 } \sin 2 t ( \sin t + t \cos t )$ 
D)  $648 x ^ { 6 } y ^ { 3 } z \sin 2 t ( \sin t + t \cos t )$

Question 11

Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 7 x y + 2 y ^ { 2 }$.
Select the correct answer.

Accepted Answer

A)  $5 y - 7 x$ 
B)  $10 x - 7 x y$ 
C)  $10 x - 7 y$ 
D)  $10 x - 7$ 
E)  $5 x - 7 y$ 
A)  $5 y - 7 x$ 
B)  $10 x - 7 x y$ 
C)  $10 x - 7 y$ 
D)  $10 x - 7$ 
E)  $5 x - 7 y$

Question 12

A boundary stripe 2 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.

Accepted Answer

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

Question 13

Use implicit differentiation to find $\frac { \partial z } { \partial x }$.
$$\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 2 x ^ { 2 } = 5$$

Accepted Answer

The answer of Use implicit differentiation to find \(\frac {...

Question 14

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

Accepted Answer

A) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$$ 
B) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$$ 
C) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$$
$\mathrm { d } .$
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$$ 
A) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$$ 
B) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$$ 
C) 
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$$
$\mathrm { d } .$
$$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$$

Question 15

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 $\mathbf { v } = 7 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$.
Select the correct answer.

Accepted Answer

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

Question 16

Determine where the function $f ( x , y ) = \frac { 6 x y } { 9 x + 8 y - 1 }$
is continuous.

Accepted Answer

The answer of Determine where the function \(f ( x...

Question 17

$$\text { Find the limit } \lim _ { ( x , y , z ) \rightarrow ( 1,2,3 ) } \frac { x y + y z + x z } { x y z - 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Find the limit } \lim...

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

Question 19

$$\text { Find } f _ { y x } \text { for the function } f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Find } f _ {...

Question 20

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

A) 
$$\frac { d y } { d x } = \frac { \sin ( x - 8 y ) + e ^ { 4 y } } { \sin ( x - 8 y ) - x e ^ { 4 y } }$$ 
B) 
$$\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - y ) - x e ^ { y } }$$ 
C) 
$$\frac { d y } { d x } = \frac { 8 \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - 8 y ) - 8 x e ^ { y } }$$ 
D) 
$$\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { 8 \sin ( x - 8 y ) - x e ^ { 4 y } }$$ 
E) 
$$\frac { d y } { d x } = \frac { \sin ( x - 8 y ) + e ^ { 4 y } } { 8 \sin ( x - 8 y ) - 4 x e ^ { 4 y } }$$ 
A) 
$$\frac { d y } { d x } = \frac { \sin ( x - 8 y ) + e ^ { 4 y } } { \sin ( x - 8 y ) - x e ^ { 4 y } }$$ 
B) 
$$\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - y ) - x e ^ { y } }$$ 
C) 
$$\frac { d y } { d x } = \frac { 8 \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - 8 y ) - 8 x e ^ { y } }$$ 
D) 
$$\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { 8 \sin ( x - 8 y ) - x e ^ { 4 y } }$$ 
E) 
$$\frac { d y } { d x } = \frac { \sin ( x - 8 y ) + e ^ { 4 y } } { 8 \sin ( x - 8 y ) - 4 x e ^ { 4 y } }$$

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.

$\text { Use the definition of partial derivatives as limits to find } f _ { x } ( x , y ) \text { if } f ( x , y ) = 4 x ^ { 2 } - 8 x y + 2 y ^ { 2 } \text {. }$

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

$\text { Find the domain and range of the function } g ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + 3$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Find three positive numbers whose sum is 291 and whose product is a maximum.

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

Use polar coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 3 } + y ^ { 3 } } { x ^ { 2 } + y ^ { 2 } }$

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 2,2 )$ . Select the correct answer.

Use the Chain Rule to find $\frac { d w } { d t }$ . $w = 3 x ^ { 6 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 2 t , \quad z = t \sin t$

Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 7 x y + 2 y ^ { 2 }$ . Select the correct answer.

A boundary stripe 2 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.

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ . $\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 2 x ^ { 2 } = 5$

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

Determine where the function $f ( x , y ) = \frac { 6 x y } { 9 x + 8 y - 1 }$ is continuous.

$\text { Find the limit } \lim _ { ( x , y , z ) \rightarrow ( 1,2,3 ) } \frac { x y + y z + x z } { x y z - 2 } \text {. }$

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.

$\text { Find } f _ { y x } \text { for the function } f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 } \text {. }$

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 $\frac { d y } { d x }$ . $\cos ( x - 8 y ) = x e ^ { 4 y }$

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

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Exam 14: Partial Derivatives

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 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​.

Find the domain and range of the function g(x,y)=x2+2y2+3\text { Find the domain and range of the function } g ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + 3 Find the domain and range of the function g(x,y)=x2+2y2+3

Evaluate the limit. lim⁡(x,y)→(0,0)7xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y27xy2​

Find three positive numbers whose sum is 291 and whose product is a maximum.

Find the gradient of the function f(x,y,z)=z2e2xy. \text { Find the gradient of the function } f ( x , y , z ) = z ^ { 2 } e ^ { 2 x } \sqrt { y } \text {. } Find the gradient of the function f(x,y,z)=z2e2xy​.

Use polar coordinates to find the limit. lim⁡(x,y)→(0,0)x3+y3x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 3 } + y ^ { 3 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x3+y3​

A contour map for a function fff is shown. Use it to estimate the value of f(−2,2)f ( - 2,2 )f(−2,2) . Select the correct answer.

Use the Chain Rule to find dwdt\frac { d w } { d t }dtdw​ . w=3x6y3z,x=6t,y=cos⁡2t,z=tsin⁡tw = 3 x ^ { 6 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 2 t , \quad z = t \sin tw=3x6y3z,x=6t,y=cos2t,z=tsint

Use the definition of partial derivatives as limits to find fx(x,y)f _ { x } ( x , y )fx​(x,y) if f(x,y)=5x2−7xy+2y2f ( x , y ) = 5 x ^ { 2 } - 7 x y + 2 y ^ { 2 }f(x,y)=5x2−7xy+2y2 . Select the correct answer.

A boundary stripe 2 in. 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.

Use implicit differentiation to find ∂z∂x\frac { \partial z } { \partial x }∂x∂z​ . ln⁡(x2+z2)+yz3+2x2=5\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 2 x ^ { 2 } = 5ln(x2+z2)+yz3+2x2=5

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

Determine where the function f(x,y)=6xy9x+8y−1f ( x , y ) = \frac { 6 x y } { 9 x + 8 y - 1 }f(x,y)=9x+8y−16xy​ is continuous.

Find the limit lim⁡(x,y,z)→(1,2,3)xy+yz+xzxyz−2. \text { Find the limit } \lim _ { ( x , y , z ) \rightarrow ( 1,2,3 ) } \frac { x y + y z + x z } { x y z - 2 } \text {. } Find the limit lim(x,y,z)→(1,2,3)​xyz−2xy+yz+xz​.

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.

Find fyx for the function f(x,y)=4x3y−7xy2. \text { Find } f _ { y x } \text { for the function } f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 } \text {. } Find fyx​ for the function f(x,y)=4x3y−7xy2.

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 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.

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

$\text { Find the domain and range of the function } g ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + 3$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

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

Use polar coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 3 } + y ^ { 3 } } { x ^ { 2 } + y ^ { 2 } }$

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 2,2 )$ . Select the correct answer.

Use the Chain Rule to find $\frac { d w } { d t }$ . $w = 3 x ^ { 6 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 2 t , \quad z = t \sin t$

Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 7 x y + 2 y ^ { 2 }$ . Select the correct answer.

A boundary stripe 2 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.

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ . $\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 2 x ^ { 2 } = 5$

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

Determine where the function $f ( x , y ) = \frac { 6 x y } { 9 x + 8 y - 1 }$ is continuous.

$\text { Find the limit } \lim _ { ( x , y , z ) \rightarrow ( 1,2,3 ) } \frac { x y + y z + x z } { x y z - 2 } \text {. }$

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.

$\text { Find } f _ { y x } \text { for the function } f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 } \text {. }$