Exam 14: Partial Derivatives

A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are $100 \mathrm { ft }$ by $240 \mathrm { ft }$ . Use differentials to approximate the number of square feet of paint in the stripe.

Find the indicated partial derivative. $u = x ^ { 2 } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$

Find $h _ { z x y } ( x , y , z )$ for the function $h ( x , y , z ) = e ^ { 9 x } \cos ( y + 7 z )$ . Select the correct answer.

Find the shortest distance from the point $( 2,0 , - 3 )$ to the plane $x + y + z = 1$ . Select the correct answer.

Suppose $( 1,1 )$ is a critical point of a function $f$ with continuous second derivatives. In the case of $f _ { x x } ( 1,1 ) = 8 , f _ { x y } ( 1,1 ) = 8 , f _ { y y } ( 1,1 ) = 10$ what can you say about $f$ ? Select the correct answer.

A boundary stripe $2 \mathrm { in }$ . wide is painted around a rectangle whose dimensions are $100 \mathrm { ft }$ by $240 \mathrm { ft }$ . Use differentials to approximate the number of square feet of paint in the stripe.

Use Lagrange multipliers to find the maximum and the minimum of $f$ subject to the given constraint(s). $f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 6 x ^ { 2 } + 6 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }$ . Hint: If $x = r \cos \theta$ and $y = r \sin \theta$ , then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$ . Select the correct answer.

The length $l$ , width $w$ and height $h$ of a box change with time. At a certain instant the dimensions are $l = 3$ and $w = h = 4$ , and $l$ and $w$ are increasing at a rate of $10 \mathrm {~m} / \mathrm { s }$ while $h$ is decreasing at a rate of 1 $\mathrm { m } / \mathrm { s }$ . At that instant find the rates at which the surface area is changing.

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$ .

The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is $T$ and the relative humidity is $h$ , so we can write $I = f ( T , h )$ . The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic and Atmospheric Administration. For what value of $T$ is $f ( T , 50 ) = 107$ ? T \downarrow \rightarrow 20 30 40 50 60 70 80 74 76 78 82 83 86 85 81 82 84 86 90 94 90 86 90 93 96 101 106 95 94 94 98 107 111 125 100 99 101 109 122 129 138

Find all the second partial derivatives. $f ( x , y ) = x ^ { 4 } - 4 x ^ { 2 } y ^ { 3 }$

Find the dimensions of the rectangular box with largest volume if the total surface area is given as $384 \mathrm {~cm} ^ { 2 }$ .

Use the Chain Rule to find $\frac { \partial u } { \partial p }$ . u= x=p+5r+7t,y=p-5r+7t,z=p+5r-7t

Find the shortest distance from the point $( 3,9,8 )$ to the plane $3 x + 9 y + 4 z = 16$ Select the correct answer.

Find the linearization $L ( x , y )$ of the function at the given point. $f ( x , y ) = x \sqrt { y } , ( - 5,25 )$ Round the answers to the nearest hundredth.

Find the directional derivative of the function $f ( x , y ) = ( x + 6 ) e ^ { y }$ at the point $P ( 7,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive $x$ -axis. Select the correct answer.

$\text { Use the Chain Rule to find } \frac { d w } { d t }$ $w = 3 x ^ { 6 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 2 t , \quad z = t \sin t$