Question 1

For $f ( x , y ) = x ^ { 2 } y$ find a line in the x direction tangent to the surface defined by f at (1,2).

Accepted Answer

A)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,4 \rangle$ 
B)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,2 \rangle$ 
C)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,1 \rangle$ 
D)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 4,1,0 \rangle$ 
A)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,4 \rangle$ 
B)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,2 \rangle$ 
C)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,1 \rangle$ 
D)  $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 4,1,0 \rangle$

Question 2

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = x ^ { 2 } + x y z + y ^ { 3 }$

Accepted Answer

The answer of Find \(\frac { \partial f } {...

Question 3

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 1$ , $y  0$

Accepted Answer

A)  Open 
B)  Closed 
C)  Both open and closed 
D)  Neither open nor closed 
A)  Open 
B)  Closed 
C)  Both open and closed 
D)  Neither open nor closed

Question 4

Find $\frac { d z } { d t }$ for $z = \sin \left( x ^ { 2 } y \right) , x = \frac { s } { t } , y = t ^ { 2 } e ^ { s t }$

Accepted Answer

The answer of Find \(\frac { d z } {...

Question 5

Find the maximum and the minimum values of the function $f ( x , y , z ) = x + 2 y + 3 z$ subject to $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$

Accepted Answer

Maximum is

Question 6

Find the compliment of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x \geq 0$

Accepted Answer

All points

Question 7

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$

Accepted Answer

A)  $10$ 
B)  $10 \sqrt { 3 }$ 
C)  $\frac { 10 } { 3 }$ 
D)  $\frac { 10 } { \sqrt { 3 } }$ 
A)  $10$ 
B)  $10 \sqrt { 3 }$ 
C)  $\frac { 10 } { 3 }$ 
D)  $\frac { 10 } { \sqrt { 3 } }$

Question 8

Find a function of two variables for the surface of revolution formed when the given function on the specified interval is revolved around the y-axis. $f ( x ) = x ^ { 2 }$ on $[ 0,2 ]$

Accepted Answer

The answer of Find a function of two variables for...

Question 9

Evaluate $f ( x , y , z ) = x ^ { 2 } y z$ at $( 1,2 , - 1 )$

Accepted Answer

The answer of Evaluate \(f ( x , y ,...

Question 10

Find the line tangent to the surface given by the function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Accepted Answer

The answer of Find the line tangent to the surface...

Question 11

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \langle 1 , \sqrt { 3 } \rangle$

Accepted Answer

The answer of Find the directional derivative to the given...

Question 12

Find the domain of $f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }$

Accepted Answer

A)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq y \right\}$ 
B)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq - y \right\}$ 
C)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : | x | \neq | y | \right\}$ 
D)  $\operatorname { Dom } ( f ) = R ^ { 2 }$ 
A)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq y \right\}$ 
B)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq - y \right\}$ 
C)  $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : | x | \neq | y | \right\}$ 
D)  $\operatorname { Dom } ( f ) = R ^ { 2 }$

Question 13

Find and classify the critical points for $f ( x , y ) = 3 x ^ { 4 } - 2 x y + 9 y ^ { 4 }$

Accepted Answer

@#LAT-DLM& is a sadd

Question 14

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + 4 y$ subject to $x ^ { 2 } + y ^ { 2 } = 1$

Accepted Answer

A)  4 
B)  -4 
C)  1 
D)  -1 
A)  4 
B)  -4 
C)  1 
D)  -1

Question 15

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) , \vec { u } = \langle \sqrt { 2 } , \sqrt { 2 } \rangle$

Accepted Answer

The answer of Find the directional derivative to the given...

Question 16

Find $\nabla f$ for $f ( x , y ) = e ^ { x \sin ( y ) }$

Accepted Answer

The answer of Find $\nabla f$ for \(f ( x...

Question 17

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x y z ^ { 2 } , P = ( 1,2,1 ) , \vec { u } = \langle 1 , \sqrt { 3 } , 0 \rangle$

Accepted Answer

The answer of Find the directional derivative to the given...

Question 18

Find the direction in which the given function increases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Accepted Answer

The answer of Find the direction in which the given...

Question 19

Find the compliment of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $x > 0$ and $y < 0$

Accepted Answer

The answer of Find the compliment of the given subset...

Question 20

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 0$

Accepted Answer

The answer of Determine if the given subset of \(R...

For $f ( x , y ) = x ^ { 2 } y$ find a line in the x direction tangent to the surface defined by f at (1,2).

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = x ^ { 2 } + x y z + y ^ { 3 }$

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 1$ , $y < 0$ and $z > 0$

Find $\frac { d z } { d t }$ for $z = \sin \left( x ^ { 2 } y \right) , x = \frac { s } { t } , y = t ^ { 2 } e ^ { s t }$

Find the maximum and the minimum values of the function $f ( x , y , z ) = x + 2 y + 3 z$ subject to $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$

Find the compliment of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x \geq 0$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$

Find a function of two variables for the surface of revolution formed when the given function on the specified interval is revolved around the y-axis. $f ( x ) = x ^ { 2 }$ on $[ 0,2 ]$

Evaluate $f ( x , y , z ) = x ^ { 2 } y z$ at $( 1,2 , - 1 )$

Find the line tangent to the surface given by the function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \langle 1 , \sqrt { 3 } \rangle$

Find the domain of $f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }$

Find and classify the critical points for $f ( x , y ) = 3 x ^ { 4 } - 2 x y + 9 y ^ { 4 }$

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + 4 y$ subject to $x ^ { 2 } + y ^ { 2 } = 1$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) , \vec { u } = \langle \sqrt { 2 } , \sqrt { 2 } \rangle$

Find $\nabla f$ for $f ( x , y ) = e ^ { x \sin ( y ) }$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x y z ^ { 2 } , P = ( 1,2,1 ) , \vec { u } = \langle 1 , \sqrt { 3 } , 0 \rangle$

Find the direction in which the given function increases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Find the compliment of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $x > 0$ and $y < 0$

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 0$

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 12: Multivariable Functions

For f(x,y)=x2yf ( x , y ) = x ^ { 2 } yf(x,y)=x2y find a line in the x direction tangent to the surface defined by f at (1,2).

Find ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ for f(x,y,z)=x2+xyz+y3f ( x , y , z ) = x ^ { 2 } + x y z + y ^ { 3 }f(x,y,z)=x2+xyz+y3

Determine if the given subset of R3R ^ { 3 }R3 is open, closed, both open and closed, or neither open nor closed. All points (x,y,z)( x , y , z )(x,y,z) so that x≥1x \geq 1x≥1 , y<0y < 0y<0 and z>0z > 0z>0

Find dzdt\frac { d z } { d t }dtdz​ for z=sin⁡(x2y),x=st,y=t2estz = \sin \left( x ^ { 2 } y \right) , x = \frac { s } { t } , y = t ^ { 2 } e ^ { s t }z=sin(x2y),x=ts​,y=t2est

Find the maximum and the minimum values of the function f(x,y,z)=x+2y+3zf ( x , y , z ) = x + 2 y + 3 zf(x,y,z)=x+2y+3z subject to x2+y2+z2=1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1x2+y2+z2=1

Find the compliment of the set in R3R ^ { 3 }R3 : all points (x,y,z)( x , y , z )(x,y,z) so that x≥0x \geq 0x≥0

Find the directional derivative to the given function at the specified point PPP in the direction of the vector u⃗\vec { u }u f(x,y,z)=2zxy,P=(2,2,3),u⃗=⟨1,1,1⟩f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \ranglef(x,y,z)=2zxy​,P=(2,2,3),u=⟨1,1,1⟩

Find a function of two variables for the surface of revolution formed when the given function on the specified interval is revolved around the y-axis. f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 on [0,2][ 0,2 ][0,2]

Evaluate f(x,y,z)=x2yzf ( x , y , z ) = x ^ { 2 } y zf(x,y,z)=x2yz at (1,2,−1)( 1,2 , - 1 )(1,2,−1)

Find the directional derivative to the given function at the specified point PPP in the direction of the vector u⃗\vec { u }u f(x,y)=x2+xy,P=(1,2),u⃗=⟨1,3⟩f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \langle 1 , \sqrt { 3 } \ranglef(x,y)=x2+xy,P=(1,2),u=⟨1,3​⟩

Find the domain of f(x,y)=xx2−y2f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }f(x,y)=x2−y2x​

Find and classify the critical points for f(x,y)=3x4−2xy+9y4f ( x , y ) = 3 x ^ { 4 } - 2 x y + 9 y ^ { 4 }f(x,y)=3x4−2xy+9y4

Find the minimum value of the function f(x,y)=x2+4yf ( x , y ) = x ^ { 2 } + 4 yf(x,y)=x2+4y subject to x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1

Find ∇f\nabla f∇f for f(x,y)=exsin⁡(y)f ( x , y ) = e ^ { x \sin ( y ) }f(x,y)=exsin(y)

Find the direction in which the given function increases most rapidly at the given point. f(x,y,z)=xez−yex,P=(0,2,0)f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )f(x,y,z)=xez−yex,P=(0,2,0)

Find the compliment of the given subset of R2R ^ { 2 }R2 : all points (x,y)( x , y )(x,y) so that x>0x > 0x>0 and y<0y < 0y<0

Determine if the given subset of R3R ^ { 3 }R3 is open, closed, both open and closed, or neither open nor closed. All points (x,y,z)( x , y , z )(x,y,z) so that x≥0x \geq 0x≥0

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

For $f ( x , y ) = x ^ { 2 } y$ find a line in the x direction tangent to the surface defined by f at (1,2).

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = x ^ { 2 } + x y z + y ^ { 3 }$

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 1$ , $y < 0$ and $z > 0$

Find $\frac { d z } { d t }$ for $z = \sin \left( x ^ { 2 } y \right) , x = \frac { s } { t } , y = t ^ { 2 } e ^ { s t }$

Find the maximum and the minimum values of the function $f ( x , y , z ) = x + 2 y + 3 z$ subject to $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$

Find the compliment of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x \geq 0$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$

Find a function of two variables for the surface of revolution formed when the given function on the specified interval is revolved around the y-axis. $f ( x ) = x ^ { 2 }$ on $[ 0,2 ]$

Evaluate $f ( x , y , z ) = x ^ { 2 } y z$ at $( 1,2 , - 1 )$

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \langle 1 , \sqrt { 3 } \rangle$

Find the domain of $f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }$

Find and classify the critical points for $f ( x , y ) = 3 x ^ { 4 } - 2 x y + 9 y ^ { 4 }$

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + 4 y$ subject to $x ^ { 2 } + y ^ { 2 } = 1$

Find $\nabla f$ for $f ( x , y ) = e ^ { x \sin ( y ) }$

Find the direction in which the given function increases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Find the compliment of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $x > 0$ and $y < 0$

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 0$