Exam 11: Partial Derivatives

Let $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ .(a) Sketch the intersection of $z = f ( x , y )$ and $z = 1$ in the $x y$ -plane.(b) Sketch the intersection of $z = f ( x , y )$ and $x = 0$ in the $y z$ -plane.(c) Sketch the intersection of $z = f ( x , y )$ and $y = 0$ in the $x z$ -plane.(d) Sketch the graph of $z = f ( x , y )$ in $\mathbb { R } ^ { 3 }$ .

Use differentials to approximate $\frac { \sqrt [ 3 ] { 28 } } { \sqrt { 10 } }$ .

Let $f ( x , y , z ) = x ^ { 2 } y + y ^ { 3 } z + x z ^ { 3 }$ and let $P ( 2,1 , - 1 )$ .(a) Find the directional derivative at $P$ in the direction of $( 1,2,3)$ .(b) In what direction does $f$ increase most rapidly? (c) What is the maximum rate of change of $f$ at the point $P$ ?

Describe the vertical traces $x = k$ and $y = k$ and the horizontal traces $z = k$ for the function $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ .

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + x z$ . Find the directional derivative of $f$ at $( 1,2,0 )$ in the direction of the vector $\mathbf { v } = \{ 1 , - 1,1 \}$ .

Describe the level curves of the function $f ( x , y ) = \sqrt { 1 - x ^ { 2 } - 2 y ^ { 2 } }$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - 3 x y + y ^ { 2 } } { x ^ { 2 } + 2 y ^ { 2 } }$

Find the range of the function $f ( x , y ) = \ln \left( x - y ^ { 2 } \right)$ .

Show that $T ( t , x ) = \cos ( 2 x ) e ^ { - 4 t }$ satisfies the heat equation $\frac { \delta T } { \delta t } = \frac { \delta ^ { 2 } T } { \delta x ^ { 2 } }$

How many third-order partial derivatives does a function $f ( x , y )$ have?

If $f$ is a function of $x$ and $y$ , and $y$ is a function of $x$ , then indirectly $f$ depends only on $x$ : $g ( x ) = f ( x , y ( x ) )$ . Use the Chain Rule to write an expression for $\frac { d g } { d x }$ in terms of $\frac { \delta f } { \delta x }$ and $\frac { \delta f } { \delta y }$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 2 x ^ { 3 } + 4 y ^ { 3 } + 3 x ^ { 2 } - 12 x - 192 y + 5$ .

Use the level curves of $f ( x , y )$ shown below to estimate the critical points of $f$ . Indicate whether $f$ has a saddle point or a local maximum or minimum at each of those points.

Consider $f ( x , y , z ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ . Where $f$ is continuous?

Given $f ( x , y ) = x ^ { 2 } \sin ( x y )$ , find $f _ { x }$ , $f _ { y }$ , and $f _ { x y }$ .

Find an equation of the tangent plane to the parametric surface $x = u ^ { 2 } , y = u - v ^ { 2 } , z = v ^ { 2 }$ at the point $( 1,0,1 )$ .

Suppose that $z = x - y , x = 4 \left( t ^ { 3 } - 1 \right)$ , and $y = \ln t$ . Find $\frac { d z } { d t }$ .

If $g$ is a differentiable function and $f ( x , y ) = g \left( x ^ { 2 } + y ^ { 2 } \right)$ , show that $y f _ { x } - x f _ { y } = 0$

Let $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 4 } + y ^ { 4 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ Does this function have a limit at the origin? If so, prove it. If not, demonstrate why not.

If $f ( x , y ) = y e ^ { x y }$ , find the values $x _ { 0 }$ for which $f \left( x _ { 0 } , 5 \right) = 5$ , and then find an equation of the plane tangent to the graph of $f$ at $\left( x _ { 0 } , 5,5 \right)$ .