Exam 12: Multivariable Functions

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

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

Find $\nabla f$ for $f ( x , y ) = x ^ { 2 } y$

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

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).

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

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

Find the domain and range of $f ( x , y ) = \frac { x } { y + 2 }$

Evaluate the limit if it exists: $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } x ^ { 2 } - x y$

Evaluate $f ( x , y , z ) = x e ^ { x } \cos ( z )$ at $( 2,1,0 )$

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

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

Evaluate the limit if it exists: $\lim _ { ( x , y ) \rightarrow ( 0.0 ) } ( y + 1 ) ^ { 2 } \cos ( x )$

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

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

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = 2 x y ^ { 3 }$ and $\frac { \partial f } { \partial y } = 3 x ^ { 2 } y ^ { 2 }$

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

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 $2 \leq x \leq 12$ and $\pi \leq y \leq 7$

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = \cos ( x y z )$

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 $1 \leq x < 2$ and $1 < y \leq 2$