Exam 14: Partial Derivatives

Find the directional derivative of the function $f ( x , y ) = ( x + 5 ) e ^ { y }$ at the point $P ( 6,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive x-axis.

Use implicit differentiation to find $\partial z / \partial x$ . $7 x ^ { 2 } + 8 y ^ { 2 } - 3 z ^ { 2 } = 2 x ( y + z )$

Find $h_{m y}(x, y, z)$ for the function $h ( x , y , z ) = e ^ { 9 x } \cos ( y + 7 z )$

Find $f _ { m w } ( x , y )$ for the function $f ( x , y ) = x ^ { 4 } - 9 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } + 6 y ^ { 4 }$

Find the differential of the function. $y = x ^ { 7 } + 3 x$

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ $\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 4 x ^ { 2 } = 6$

Find the differential of the function. $u = e ^ { 6 t } \sin 3 x$

Use the Chain Rule to find $\frac { \partial w } { \partial r }$ and $\frac { \partial w } { \partial t }$ if $r = 5$ $s = 2$ and $t = 0$ $w = \frac { x ^ { 2 } y } { z ^ { 2 } } , \quad x = r e ^ { s t } , \quad y = s e ^ { n t } , \quad z = e ^ { r s t }$

Find the limit if $g ( x ) = x ^ { 4 }$ . $\lim _ { x \rightarrow 2 } \frac { g ( x ) - g ( 2 ) } { x - 2 }$

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 h \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

Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function g whose graph is a paraboloid. Which is the contour map of a cone?

Which of the given points are the points on the hyperboloid $x ^ { 2 } - y ^ { 2 } + 4 z ^ { 2 } = 4$ where the normal line is parallel to the line that joins the points $( - 1,1,3 )$ and $( 0,2,5 )$ . Select all that apply.

The length l, width w and height h of a box change with time. At a certain instant the dimensions are $l = 4$ and $w = h = 6$ , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing.

Find the local maximum, and minimum value and saddle points of the function. $f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 } - 9 x + 6 y + 13$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 3 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 2 y ^ { 2 } } }$

Use the linearization L(x, y) of the function. $f ( x , y ) = \sqrt { 9 - x ^ { 2 } - 2 y ^ { 2 } }$ at $( - 1,1 )$ to approximate $f ( - 0.93,0.75 )$ .

Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 9 x y + 2 y ^ { 2 }$ .

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

The wind-chill index I is the perceived temperature when the actual temperature is T and the wind speed is v so we can write $I = f ( T , v )$ . The following table of values is an excerpt from a table compiled by the National Atmospheric and Oceanic Administration. Use the table to find a linear approximation $L ( T , v )$ to the wind chill index function when T is near $16 ^ { \circ } \mathrm { C }$ and v is near 30 kmh.

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