Question 1

$\text { Determine where the function } f ( x , y , z ) = \frac { x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 } \text { is continuous. }$

Accepted Answer

The answer of \(\text { Determine where the function }...

Question 2

Let $g ( r , s , t ) = r e ^ { s / t }$. Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$.

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  36 
D)  6 
A)  $\frac { 2 } { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  36 
D)  6

Question 3

Find the equation of the normal line to the given surface at the specified point. $$z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$$

Accepted Answer

The answer of Find the equation of the normal line...

Question 4

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

Accepted Answer

A)  
$D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\}$ 
$R = \{ z \mid 0 < z \}$ 
B) 
$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
C) 
$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$ 
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)  
$D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\}$ 
$R = \{ z \mid 0 < z \}$ 
B) 
$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
C) 
$\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$ 
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 5

$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {, }$

Accepted Answer

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

Question 6

$\text { Find three positive numbers whose sum is } 291 \text { and whose product is a maximum. }$

Accepted Answer

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

Question 7

Use the Chain Rule to find $\frac { \partial z } { \partial s }$.
$$z = e ^ { \gamma } \cos ( \theta ) , r = 6 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$$

Accepted Answer

A) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( 6 t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
B) 
$\frac { \partial z } { \partial s } = \left( 6 t \cos ( \theta ) + \frac { s e ^ { \gamma } \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
C) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t } } \right)$ 
D) 
$\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 6 t \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
E) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
A) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( 6 t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
B) 
$\frac { \partial z } { \partial s } = \left( 6 t \cos ( \theta ) + \frac { s e ^ { \gamma } \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
C) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t } } \right)$ 
D) 
$\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 6 t \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
E) 
$\frac { \partial z } { \partial s } = e ^ { y } \left( \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$

Question 8

Find all the second partial derivatives. $$f ( x , y ) = x ^ { 4 } - 4 x ^ { 2 } y ^ { 3 }$$

Accepted Answer

The answer of Find all the second partial derivatives. \[f...

Question 9

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 3 x ^ { 2 } + 4 y ^ { 2 } \right)$.

Accepted Answer

A)  52 
B)  16 
C)  73 
D)  19 
A)  52 
B)  16 
C)  73 
D)  19

Question 10

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

A)  $\frac { 85 } { 2 } , \left( 2 , \frac { 7 } { 2 } \right)$ 
B)  $\frac { 29 } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$ 
C)  $\frac { \sqrt { 29 } } { 2 } , \left( 1 , \frac { 5 } { 2 } \right)$ 
D)  $\frac { \sqrt { 85 } } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$ 
E)  none of these 
A)  $\frac { 85 } { 2 } , \left( 2 , \frac { 7 } { 2 } \right)$ 
B)  $\frac { 29 } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$ 
C)  $\frac { \sqrt { 29 } } { 2 } , \left( 1 , \frac { 5 } { 2 } \right)$ 
D)  $\frac { \sqrt { 85 } } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$ 
E)  none of these

Question 11

$\text { Find the gradient of } f ( x , y , z ) = x ^ { 4 } e ^ { y z } \text { at the point } P ( - 2,0 , - 9 ) \text {. }$

Accepted Answer

The answer of \(\text { Find the gradient of }...

Question 12

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 } .$

Accepted Answer

A)  
None 
B) 
Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  
Saddle point $( 0,0,0 )$ 
D) 
Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$ 
A)  
None 
B) 
Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  
Saddle point $( 0,0,0 )$ 
D) 
Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$

Question 13

Find the differential of the function. $$y = x ^ { 9 } + 2 x$$

Accepted Answer

The answer of Find the differential of the function. \[y...

Question 14

Find the local maximum, and minimum value and saddle points of the function. $$f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 9 x + 6 y + 9$$

Accepted Answer

The answer of Find the local maximum, and minimum value...

Question 15

$\text { Sketch the graph of the function } g ( x , y ) = x ^ { 2 } \text {. }$

Accepted Answer

The answer of \(\text { Sketch the graph of the...

Question 16

Use implicit differentiation to find $\partial z / \partial x$.
$$8 x ^ { 2 } + 8 y ^ { 2 } - 4 z ^ { 2 } = 2 x ( y + z )$$

Accepted Answer

The answer of Use implicit differentiation to find \(\partial z...

Question 17

Find the first partial derivatives of the function. $$c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$$

Accepted Answer

The answer of Find the first partial derivatives of the...

Question 18

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 3,3 )$.

Accepted Answer

A)  78 
B)  48 
C)  56 
D)  35 
E)  28 
A)  78 
B)  48 
C)  56 
D)  35 
E)  28

Question 19

Find the indicated partial derivative. $$f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y , f _ { m x }$$

Accepted Answer

A)  $f _ { xx x } = 12 x ^ { 2 } y$ 
B)  $f _ { xx x } = - 36 x y$ 
C)  $f _ { xx x } = 3 x y$ 
D)  $f _ { xxx } = - 72 x y$ 
E)  $f _ { xx x } = - 3 x y$ 
A)  $f _ { xx x } = 12 x ^ { 2 } y$ 
B)  $f _ { xx x } = - 36 x y$ 
C)  $f _ { xx x } = 3 x y$ 
D)  $f _ { xxx } = - 72 x y$ 
E)  $f _ { xx x } = - 3 x y$

$\text { Determine where the function } f ( x , y , z ) = \frac { x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 } \text { is continuous. }$

Let $g ( r , s , t ) = r e ^ { s / t }$ . Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$ .

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

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

$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {, }$

$\text { Find three positive numbers whose sum is } 291 \text { and whose product is a maximum. }$

Use the Chain Rule to find $\frac { \partial z } { \partial s }$ . $z = e ^ { \gamma } \cos ( \theta ) , r = 6 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$

Find all the second partial derivatives. $f ( x , y ) = x ^ { 4 } - 4 x ^ { 2 } y ^ { 3 }$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 3 x ^ { 2 } + 4 y ^ { 2 } \right)$ .

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?

$\text { Find the gradient of } f ( x , y , z ) = x ^ { 4 } e ^ { y z } \text { at the point } P ( - 2,0 , - 9 ) \text {. }$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 } .$

Find the differential of the function. $y = x ^ { 9 } + 2 x$

Find the local maximum, and minimum value and saddle points of the function. $f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 9 x + 6 y + 9$

$\text { Sketch the graph of the function } g ( x , y ) = x ^ { 2 } \text {. }$

Use implicit differentiation to find $\partial z / \partial x$ . $8 x ^ { 2 } + 8 y ^ { 2 } - 4 z ^ { 2 } = 2 x ( y + z )$

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 3,3 )$ .

Find the indicated partial derivative. $f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y , f _ { m x }$

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

Determine where the function f(x,y,z)=xyzx2+y2+z2−2 is continuous. \text { Determine where the function } f ( x , y , z ) = \frac { x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 } \text { is continuous. } Determine where the function f(x,y,z)=x2+y2+z2−2xyz​ is continuous.

Let g(r,s,t)=res/tg ( r , s , t ) = r e ^ { s / t }g(r,s,t)=res/t . Find g(4,ln⁡3,12)g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)g(4,ln3,21​) .

Find the equation of the normal line to the given surface at the specified point. z+2=xeycos⁡z,(4,0,0)z + 2 = x e ^ { y } \cos z , ( 4,0,0 )z+2=xeycosz,(4,0,0)

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​ .

Find the differential of the function z=e3xsin⁡5y, \text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {, } Find the differential of the function z=e3xsin5y,

Find three positive numbers whose sum is 291 and whose product is a maximum. \text { Find three positive numbers whose sum is } 291 \text { and whose product is a maximum. } Find three positive numbers whose sum is 291 and whose product is a maximum.

Use the Chain Rule to find ∂z∂s\frac { \partial z } { \partial s }∂s∂z​ . z=eγcos⁡(θ),r=6st,θ=s2+t2z = e ^ { \gamma } \cos ( \theta ) , r = 6 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }z=eγcos(θ),r=6st,θ=s2+t2​

Find all the second partial derivatives. f(x,y)=x4−4x2y3f ( x , y ) = x ^ { 4 } - 4 x ^ { 2 } y ^ { 3 }f(x,y)=x4−4x2y3

Find the limit lim⁡(x,y)→(1,2)(3x2+4y2)\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 3 x ^ { 2 } + 4 y ^ { 2 } \right)lim(x,y)→(1,2)​(3x2+4y2) .

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 gradient of f(x,y,z)=x4eyz at the point P(−2,0,−9). \text { Find the gradient of } f ( x , y , z ) = x ^ { 4 } e ^ { y z } \text { at the point } P ( - 2,0 , - 9 ) \text {. } Find the gradient of f(x,y,z)=x4eyz at the point P(−2,0,−9).

Find and classify the relative extrema and saddle points of the function f(x,y)=e−2xsin⁡4yf ( x , y ) = e ^ { - 2 x } \sin 4 yf(x,y)=e−2xsin4y for x≥0x \geq 0x≥0 and 0≤y≤π2.0 \leq y \leq \frac { \pi } { 2 } .0≤y≤2π​.

Find the differential of the function. y=x9+2xy = x ^ { 9 } + 2 xy=x9+2x

Find the local maximum, and minimum value and saddle points of the function. f(x,y)=x2−xy+y2−9x+6y+9f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 9 x + 6 y + 9f(x,y)=x2−xy+y2−9x+6y+9

Sketch the graph of the function g(x,y)=x2. \text { Sketch the graph of the function } g ( x , y ) = x ^ { 2 } \text {. } Sketch the graph of the function g(x,y)=x2.

Use implicit differentiation to find ∂z/∂x\partial z / \partial x∂z/∂x . 8x2+8y2−4z2=2x(y+z)8 x ^ { 2 } + 8 y ^ { 2 } - 4 z ^ { 2 } = 2 x ( y + z )8x2+8y2−4z2=2x(y+z)

Find the first partial derivatives of the function. c=ln⁡(a+a4+b4)c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)c=ln(a+a4+b4​)

A contour map for a function fff is shown. Use it to estimate the value of f(−3,3)f ( - 3,3 )f(−3,3) .

Find the indicated partial derivative. f(x,y)=x2y4−3x4y,fmxf ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y , f _ { m x }f(x,y)=x2y4−3x4y,fmx​

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

$\text { Determine where the function } f ( x , y , z ) = \frac { x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 } \text { is continuous. }$

Let $g ( r , s , t ) = r e ^ { s / t }$ . Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$ .

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

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

$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {, }$

$\text { Find three positive numbers whose sum is } 291 \text { and whose product is a maximum. }$

Use the Chain Rule to find $\frac { \partial z } { \partial s }$ . $z = e ^ { \gamma } \cos ( \theta ) , r = 6 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$

Find all the second partial derivatives. $f ( x , y ) = x ^ { 4 } - 4 x ^ { 2 } y ^ { 3 }$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 3 x ^ { 2 } + 4 y ^ { 2 } \right)$ .

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?

$\text { Find the gradient of } f ( x , y , z ) = x ^ { 4 } e ^ { y z } \text { at the point } P ( - 2,0 , - 9 ) \text {. }$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 } .$

Find the differential of the function. $y = x ^ { 9 } + 2 x$

Find the local maximum, and minimum value and saddle points of the function. $f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 9 x + 6 y + 9$

$\text { Sketch the graph of the function } g ( x , y ) = x ^ { 2 } \text {. }$

Use implicit differentiation to find $\partial z / \partial x$ . $8 x ^ { 2 } + 8 y ^ { 2 } - 4 z ^ { 2 } = 2 x ( y + z )$

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

A contour map for a function $f$ is shown. Use it to estimate the value of $f ( - 3,3 )$ .

Find the indicated partial derivative. $f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y , f _ { m x }$