Question 1

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

Accepted Answer

A) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid 0 \leq x < \infty , 2 \leq y < \infty \} \
R = \{ z \mid 0 \leq z \}
\end{array}$$ 
B) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid - \infty < x < \infty , - \infty < y < \infty \} \
R = \{ z \mid 3 \leq z \}
\end{array}$$ 
C) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid - \infty < x < \infty , 2 \leq y < \infty \} \
R = \{ z \mid - \infty < z < \infty \}
\end{array}$$ 
D) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid 0 \leq x < \infty , 0 \leq y < \infty \} \
R = \{ z \mid 2 \leq z \}
\end{array}$$ 
A) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid 0 \leq x < \infty , 2 \leq y < \infty \} \
R = \{ z \mid 0 \leq z \}
\end{array}$$ 
B) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid - \infty < x < \infty , - \infty < y < \infty \} \
R = \{ z \mid 3 \leq z \}
\end{array}$$ 
C) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid - \infty < x < \infty , 2 \leq y < \infty \} \
R = \{ z \mid - \infty < z < \infty \}
\end{array}$$ 
D) 
$$\begin{array} { l } 
D = \{ ( x , y ) \mid 0 \leq x < \infty , 0 \leq y < \infty \} \
R = \{ z \mid 2 \leq z \}
\end{array}$$

Question 2

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

Accepted Answer

The answer of \[\text { Find three positive numbers whose...

Question 3

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

Accepted Answer

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

Question 4

$$\text { Find } h ( x , y ) = g ( f ( x , y ) ) \text {, if } g ( t ) = t ^ { 2 } + \sqrt { t } \text { and } f ( x , y ) = 7 x + 6 y - 6$$

Accepted Answer

The answer of \[\text { Find } h ( x...

Question 5

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

The answer of Find and classify the relative extrema and...

Question 6

$$\text { Use the definition of partial derivatives as limits to find } f _ { x } ( x , y ) \text { if } f ( x , y ) = 5 x ^ { 2 } - 9 x y + 2 y ^ { 2 } \text {. }$$

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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 9

$$\text { Find } \frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } \text { for } z = y \tan 7 x$$

Accepted Answer

The answer of \[\text { Find } \frac { \partial...

Question 10

$$\text { Find the differential of the function } z = 3 x ^ { 3 } y ^ { 6 } \text {. }$$

Accepted Answer

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

Question 11

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)  $x = 1 + t , y = 2 , z = 4 t - 2$

Question 12

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

Accepted Answer

The answer of \[\text { Find three positive numbers whose...

Question 13

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$, use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$.

Accepted Answer

A)  $100 x + 36 y = 124$ 
B)  $100 x - 124 y = 12$ 
C)  $10 x + 12 y = 124$ 
D)  $10 x + 12 y = 12$ 
E)  $100 x - 12 y = 12$ 
A)  $100 x + 36 y = 124$ 
B)  $100 x - 124 y = 12$ 
C)  $10 x + 12 y = 124$ 
D)  $10 x + 12 y = 12$ 
E)  $100 x - 12 y = 12$

Question 14

Use differentials to estimate the amount of metal in a closed cylindrical can that is $12 \mathrm {~cm}$ high and 8 $\mathrm { cm }$ in diameter if the metal in the top and bottom is $0.09 \mathrm {~cm}$ thick and the metal in the sides is $0.01$ $\mathrm { cm }$ 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 15

Evaluate the limit.
$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

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

Question 16

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$.
$$w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$$

Accepted Answer

The answer of Use the Chain Rule to find \(\frac...

Question 17

Let $g ( r , s , t ) = r e ^ { s / t }$. Find $g \left( 9 , \ln 5 , \frac { 1 } { 2 } \right)$.
Select the correct answer.

Accepted Answer

A)  $\frac { 9 } { 10 }$ 
B)  $\frac { 45 } { 2 }$ 
C)  225 
D)  $9 \sqrt { 5 }$ 
A)  $\frac { 9 } { 10 }$ 
B)  $\frac { 45 } { 2 }$ 
C)  225 
D)  $9 \sqrt { 5 }$

Question 18

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

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

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

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

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

$\text { Find } h ( x , y ) = g ( f ( x , y ) ) \text {, if } g ( t ) = t ^ { 2 } + \sqrt { t } \text { and } f ( x , y ) = 7 x + 6 y - 6$

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 $0 \leq y \leq \frac { \pi } { 2 } .$

$\text { Use the definition of partial derivatives as limits to find } f _ { x } ( x , y ) \text { if } f ( x , y ) = 5 x ^ { 2 } - 9 x y + 2 y ^ { 2 } \text {. }$

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

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

$\text { Find } \frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } \text { for } z = y \tan 7 x$

$\text { Find the differential of the function } z = 3 x ^ { 3 } y ^ { 6 } \text {. }$

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 $( 1,2,2 )$ .

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

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$ .

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$ . $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$

Let $g ( r , s , t ) = r e ^ { s / t }$ . Find $g \left( 9 , \ln 5 , \frac { 1 } { 2 } \right)$ . Select the correct answer.

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

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

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.

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 h(x,y)=g(f(x,y)), if g(t)=t2+t and f(x,y)=7x+6y−6\text { Find } h ( x , y ) = g ( f ( x , y ) ) \text {, if } g ( t ) = t ^ { 2 } + \sqrt { t } \text { and } f ( x , y ) = 7 x + 6 y - 6 Find h(x,y)=g(f(x,y)), if g(t)=t2+t​ and f(x,y)=7x+6y−6

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 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 ) Find the gradient of f(x,y,z)=x4eyz at the point P(−2,0,−9)

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 ∂2z∂x2 for z=ytan⁡7x\text { Find } \frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } \text { for } z = y \tan 7 x Find ∂x2∂2z​ for z=ytan7x

Find the differential of the function z=3x3y6. \text { Find the differential of the function } z = 3 x ^ { 3 } y ^ { 6 } \text {. } Find the differential of the function z=3x3y6.

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 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.

If f(x,y)=x2+6y2f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }f(x,y)=x2+6y2 , use the gradient vector ∇f(10,2)\nabla f ( 10,2 )∇f(10,2) to find the tangent line to the level curve f(x,y)=124f ( x , y ) = 124f(x,y)=124 at the point (10,2)( 10,2 )(10,2) .

Evaluate the limit. lim⁡(x,y)→(0,0)4xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y24xy2​

Use the Chain Rule to find ∂w∂s\frac { \partial w } { \partial s }∂s∂w​ where s=4,t=0s = 4 , t = 0s=4,t=0 . w=x2+y2+z2,x=st,y=scos⁡t,z=ssin⁡tw = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin tw=x2+y2+z2,x=st,y=scost,z=ssint

Let g(r,s,t)=res/tg ( r , s , t ) = r e ^ { s / t }g(r,s,t)=res/t . Find g(9,ln⁡5,12)g \left( 9 , \ln 5 , \frac { 1 } { 2 } \right)g(9,ln5,21​) . Select the correct answer.

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​) . Select the correct answer.

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

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

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

$\text { Find } h ( x , y ) = g ( f ( x , y ) ) \text {, if } g ( t ) = t ^ { 2 } + \sqrt { t } \text { and } f ( x , y ) = 7 x + 6 y - 6$

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 $0 \leq y \leq \frac { \pi } { 2 } .$

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

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

$\text { Find } \frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } \text { for } z = y \tan 7 x$

$\text { Find the differential of the function } z = 3 x ^ { 3 } y ^ { 6 } \text {. }$

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 $( 1,2,2 )$ .

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

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$ .

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$ . $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$

Let $g ( r , s , t ) = r e ^ { s / t }$ . Find $g \left( 9 , \ln 5 , \frac { 1 } { 2 } \right)$ . Select the correct answer.

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