Exam 12: Multivariable Functions

Solve the exact equation $y \cos ( x y ) + x \cos ( x y ) \frac { d y } { d x } = 0$

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

Find the directional derivative of the given function at the specified point $P$ in the direction $\vec { u }$ $f ( x , y ) = \cos ( x y ) , P = \left( 2 , \frac { \pi } { 6 } \right) , \vec { u } = \left\langle \frac { 3 } { \sqrt { 10 } } , \frac { 1 } { \sqrt { 10 } } \right\rangle$

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

Find the boundary of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $1 \leq x < 2$ and $1 < y \leq 2$

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

Evaluate $f ( x , y , z ) = e ^ { x } \sin ( y ) \cos ( z )$ at $\left( 0 , \frac { \pi } { 2 } , 0 \right)$

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

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

Evaluate $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ at $( 2 , - 2 )$

Find the discriminant of the given function $f ( x , y ) = x ^ { 2 } y ^ { 3 }$

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

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