Question 1

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

Accepted Answer

$\vec { r } ( t ) = \langle 1,2 , - 3 \rangle + t \langle 0,1 , - 4 \rangle$

Question 2

Find the discriminant of the given function $f ( x , y ) = e ^ { y } \cos ( x )$

Accepted Answer

$- e ^ { 2 y }$

Question 3

Find $\frac { d z } { d t }$ for $z = x y ^ { 2 } , x = t s , y = t ^ { 2 } s$

Accepted Answer

$5 t ^ { 4 } s ^ { 3 }$

Question 4

Where is the function continuous? $$f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

A)  $R ^ { 2 }$ 
B)  $x \neq 0$ 
C)  $y \neq 0$ 
D)  $( x , y ) \neq ( 0,0 )$ 
A)  $R ^ { 2 }$ 
B)  $x \neq 0$ 
C)  $y \neq 0$ 
D)  $( x , y ) \neq ( 0,0 )$

Question 5

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

Accepted Answer

A)  $\frac { \sqrt { 3 } + 1 } { 2 }$ 
B)  $2 \sqrt { 3 } - 4$ 
C)  $\frac { - \sqrt { 3 } - 1 } { 2 }$ 
D)  $- 1$ 
A)  $\frac { \sqrt { 3 } + 1 } { 2 }$ 
B)  $2 \sqrt { 3 } - 4$ 
C)  $\frac { - \sqrt { 3 } - 1 } { 2 }$ 
D)  $- 1$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

The answer of Find the boundary of the set in...

Question 8

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = x y$ and $\frac { \partial f } { \partial y } = x y$

Accepted Answer

The answer of Find a function with the given partial...

Question 9

Evaluate $f ( x , y ) = x \cos ( y )$ at $( 1,0 )$

Accepted Answer

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

Question 10

Find $\frac { \partial f } { \partial y }$ for $f ( x , y ) = e ^ { x - y ^ { 2 } }$

Accepted Answer

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

Question 11

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

Accepted Answer

The answer of Find the equation of the plane tangent...

Question 12

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

Accepted Answer

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

Question 13

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $x > 0$ and $y < 0$

Accepted Answer

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

Question 14

Find the domain and range of $f ( x , y ) = \frac { \ln ( y ) } { \sqrt { x - 1 } }$

Accepted Answer

None

Question 15

Find the maximum and the minimum values of the function $f ( x , y ) = x + 3 y$ subject to $2 x ^ { 2 } + y ^ { 2 } = 36$

Accepted Answer

Maximum is

Question 16

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 $x > 0$ , $y < 0$ , and $z < 0$

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

Find $\frac { d z } { d t }$ for $z = x e ^ { y } , x = \cos ( t ) , y = e ^ { t }$

Accepted Answer

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

Question 19

Evaluate the limit if it exists: $$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + y ^ { 2 } }$$

Accepted Answer

The answer of Evaluate the limit if it exists: \[\lim...

Question 20

Find the equation of the plane tangent to the surface given by the given function at the specified point $P$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )$

Accepted Answer

A)  $z = - 1 - y - 2$ 
B)  $z = - x - y + 7$ 
C)  $z = - x - y + 2$ 
D)  $z = - x - y$ 
A)  $z = - 1 - y - 2$ 
B)  $z = - x - y + 7$ 
C)  $z = - x - y + 2$ 
D)  $z = - x - y$

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

Find the discriminant of the given function $f ( x , y ) = e ^ { y } \cos ( x )$

Find $\frac { d z } { d t }$ for $z = x y ^ { 2 } , x = t s , y = t ^ { 2 } s$

Where is the function continuous? $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$

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

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

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = x y$ and $\frac { \partial f } { \partial y } = x y$

Evaluate $f ( x , y ) = x \cos ( y )$ at $( 1,0 )$

Find $\frac { \partial f } { \partial y }$ for $f ( x , y ) = e ^ { x - y ^ { 2 } }$

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

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

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $x > 0$ and $y < 0$

Find the domain and range of $f ( x , y ) = \frac { \ln ( y ) } { \sqrt { x - 1 } }$

Find the maximum and the minimum values of the function $f ( x , y ) = x + 3 y$ subject to $2 x ^ { 2 } + y ^ { 2 } = 36$

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 $x > 0$ , $y < 0$ , and $z < 0$

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

Find $\frac { d z } { d t }$ for $z = x e ^ { y } , x = \cos ( t ) , y = e ^ { t }$

Evaluate the limit if it exists: $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + y ^ { 2 } }$

Find the equation of the plane tangent to the surface given by the given function at the specified point $P$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )$

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)=x2−y2f ( x , y ) = x ^ { 2 } - y ^ { 2 }f(x,y)=x2−y2 find a line in the y direction tangent to the surface defined by f at (1,2).

Find the discriminant of the given function f(x,y)=eycos⁡(x)f ( x , y ) = e ^ { y } \cos ( x )f(x,y)=eycos(x)

Find dzdt\frac { d z } { d t }dtdz​ for z=xy2,x=ts,y=t2sz = x y ^ { 2 } , x = t s , y = t ^ { 2 } sz=xy2,x=ts,y=t2s

Where is the function continuous? f(x,y)=xyx2+y2f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }f(x,y)=x2+y2xy​

Find the boundary 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 a function with the given partial derivatives ∂f∂x=xy\frac { \partial f } { \partial x } = x y∂x∂f​=xy and ∂f∂y=xy\frac { \partial f } { \partial y } = x y∂y∂f​=xy

Evaluate f(x,y)=xcos⁡(y)f ( x , y ) = x \cos ( y )f(x,y)=xcos(y) at (1,0)( 1,0 )(1,0)

Find ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ for f(x,y)=ex−y2f ( x , y ) = e ^ { x - y ^ { 2 } }f(x,y)=ex−y2

Determine if the given subset of R2R ^ { 2 }R2 is open, closed, both open and closed, or neither open nor closed. All points (x,y)( x , y )(x,y) so that x>0x > 0x>0 and y<0y < 0y<0

Find the domain and range of f(x,y)=ln⁡(y)x−1f ( x , y ) = \frac { \ln ( y ) } { \sqrt { x - 1 } }f(x,y)=x−1​ln(y)​

Find the maximum and the minimum values of the function f(x,y)=x+3yf ( x , y ) = x + 3 yf(x,y)=x+3y subject to 2x2+y2=362 x ^ { 2 } + y ^ { 2 } = 362x2+y2=36

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 > 0x>0 , y<0y < 0y<0 , and z<0z < 0z<0

Find dzdt\frac { d z } { d t }dtdz​ for z=xey,x=cos⁡(t),y=etz = x e ^ { y } , x = \cos ( t ) , y = e ^ { t }z=xey,x=cos(t),y=et

Evaluate the limit if it exists: lim⁡(x,y)→(0,0)xyx2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2xy​

Find the equation of the plane tangent to the surface given by the given function at the specified point PPP f(x,y)=x3−x2y,P=(1,2)f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )f(x,y)=x3−x2y,P=(1,2)

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

Find the discriminant of the given function $f ( x , y ) = e ^ { y } \cos ( x )$

Find $\frac { d z } { d t }$ for $z = x y ^ { 2 } , x = t s , y = t ^ { 2 } s$

Where is the function continuous? $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$

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

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = x y$ and $\frac { \partial f } { \partial y } = x y$

Evaluate $f ( x , y ) = x \cos ( y )$ at $( 1,0 )$

Find $\frac { \partial f } { \partial y }$ for $f ( x , y ) = e ^ { x - y ^ { 2 } }$

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $x > 0$ and $y < 0$

Find the domain and range of $f ( x , y ) = \frac { \ln ( y ) } { \sqrt { x - 1 } }$

Find the maximum and the minimum values of the function $f ( x , y ) = x + 3 y$ subject to $2 x ^ { 2 } + y ^ { 2 } = 36$

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 $x > 0$ , $y < 0$ , and $z < 0$

Find $\frac { d z } { d t }$ for $z = x e ^ { y } , x = \cos ( t ) , y = e ^ { t }$

Evaluate the limit if it exists: $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + y ^ { 2 } }$

Find the equation of the plane tangent to the surface given by the given function at the specified point $P$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )$