Exam 12: Multivariable Functions

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

Find the domain and range of $f ( x , y , z ) = \frac { 1 } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 1 }$

Find the domain and range of $f ( x , y , z ) = \frac { x } { y ^ { 2 } - z ^ { 2 } - 3 }$

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

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

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

Find the equation of the plane tangent to the surface given by the function at the specified point $P$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) ,$

Find $\nabla f$ for $f ( x , y ) = \frac { \tan ( x ) } { y }$ .

Find a function of two variables with the given gradient, $\nabla f = \left\langle 2 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 2 } \right\rangle$

Find the range of $f ( x , y , z ) = \frac { 2 } { x ^ { 4 } + y ^ { 2 } + z ^ { 6 } + 4 }$

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

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

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

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

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

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

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

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

Find the discriminant of the given function $f ( x , y ) = 3 x ^ { 2 } + 4 x y + 9 y ^ { 2 }$

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