Exam 11: Partial Derivatives

Find $\frac { \delta z } { \delta y }$ for $x ^ { 3 } - y + x z ^ { 3 } = 0$

Let $f ( x , y ) = \sqrt { 2 x + 3 y }$ .(a) In which direction does $f$ increase most rapidly at the point $( 3,1 )$ ? (b) What is the maximum rate of change of $f$ at the point $( 3,1 )$ ? (c) Find a unit vector $\mathbf { u }$ such that $D _ { u } f = 0$ at $( 3,1 )$ .

Find the linear approximation to the function $f ( x , y ) = \ln ( 3 x - y )$ at $( 1,2 )$ and use it to approximate $f ( 0.95,2.03 )$ .

The level curves of a function $f ( x , y )$ and a curve with equation $g ( x , y ) = c$ ( $C$ constant) are given below. Estimate the point where $f$ has a maximum value and the point where $f$ has a minimum value, subject to the constraint that $g ( x , y ) = c$ . Indicate your answer in the figure.

A right triangle as leg $A$ wit length 4, leg ${ B }$ with length 3, and hypotenuse with length 5. Use a total differential to approximate the length of the hypotenuse if leg $A$ had length 4.2 and leg ${ B }$ had length 2.9.

Describe the level surfaces $k = 1$ , $k = 0$ , $k = - 1$ for the function $f ( x , y , z ) = 1 - x ^ { 2 } - \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 3 } z ^ { 2 }$ .

Find the extreme values of $f ( x , y ) = x y + 2 y ^ { 2 } + x ^ { 4 } - y ^ { 4 }$ on the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Let $z = x + y$ , and let $x$ and $y$ be functions of $S$ and $t$ with $x ( 0,0 ) = 1 , y ( 0,0 ) = 2$ , $\delta x / \delta s = 3$ , and $\delta y / \delta s = 4$ at $( s , t ) = ( 0,0 )$ . Find $\delta z / \delta \text {s }$ when $( s , t ) = ( 0,0 )$ .

Use the method of Lagrange multipliers to find points on the surface of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ where the function $f ( x , y , z ) = x + y + z$ has (a) a minimum (b) a maximum.

Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the wall is 0.05 cm thick and the metal in the top and bottom is 0.1 cm thick.

If $f ( r , \theta ) = r \sin \theta + r ^ { 2 } \tan ^ { 2 } \theta$ , find the partial derivative of $f$ with respect to $r$ and the partial derivative with respect to $\theta$ , both at the point $\left( - 4 , \frac { \pi } { 6 } \right)$ .

Consider the equation $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49$ .(a) Sketch this surface. (b) Find an equation of the tangent plane to the surface at the point $( 6,2,3 )$ .(c) Find a symmetric equation of the line perpendicular to the tangent plane at the point $( 6,2,3 )$ .

A bug is crawling on the surface $z = x ^ { 2 } + x y + 2 y ^ { 2 }$ . When he reaches the point $( 2,1,8 )$ he wants to avoid vertical change. In which direction should he head? (He wants the directional derivative in the $Z$ -direction to be zero.)

If $f$ is a function of $x$ and $y$ , and $y$ is a function of $x$ , then indirectly $f$ depends only on $x : g ( x ) = f ( x , y ( x ) )$ . If $f ( x , y ) = \sin x + \sqrt { 1 - y ^ { 2 } }$ and $y ( x ) = \cos x$ , calculate $\frac { d g } { d x }$ .

Consider the surface given by $z = x y ^ { 3 } - x ^ { 2 } y$ . Find an equation for the tangent plane to the surface at the point $( 3,2,6 )$ . Also, find parametric equations for the normal line to the surface at the point $( 3,2,6 )$

Find $\frac { \delta z } { \delta x }$ for $x ^ { 3 } - y ^ { 2 } z + \sin ( x y z ) = 0$

Determine if $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 2 } + x y } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\1 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ is everywhere continuous, and if not, locate the point(s) of discontinuity.

Consider $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 4 } - y ^ { 4 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0.1 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ Where is $f$ continuous?

Find $f _ {z x }$ , $f _ { y y }$ , and $f _ { y x}$ if $f ( x , y ) = \sin x ^ { 2 } y$ .

Let $x ^ { 3 } y + y ^ { 3 } z + z ^ { 3 } x = 4$ , find $\frac { \delta z } { \delta x }$ .