Exam 14: Partial Derivatives

Find the absolute minimum value of the function $f ( x , y ) = 6 + 3 x y - 2 x - 4 y$ on the set $D . D$ is the region bounded by the parabola $y = x ^ { 2 }$ and the line $y = 4$ Select the correct answer.

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is 84 . Select the correct answer.

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { 5 x ^ { 2 } y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } } }$

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

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 3 x ^ { 2 } + 4 y ^ { 2 } \right)$

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$ . Select the correct answer.

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

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

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ . $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

Use the Chain Rule to find $\frac { \partial u } { \partial p }$ . $u = \frac { x + y } { y + z }$ $x = p + 3 r + 4 t , y = p - 3 r + 4 t , z = p + 3 r - 4 t$

At what point is the following function a local minimum? $f ( x , y ) = 8 x ^ { 2 } + 5 y ^ { 2 }$ Select the correct answer.

Find $f _ { y } ( - 18,6 )$ for $f ( x , y ) = \sin ( 3 x + 9 y )$ . Select the correct answer.

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

Find the equation of the normal line to the given surface at the specified point. $2 x ^ { 2 } + 8 y ^ { 2 } + 3 z ^ { 2 } = 235 , ( 4,4,5 )$

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 $30 \mathrm { kmh }$ . T 10 20 30 40 50 \downarrow v \rightarrow 20 18 17 14 13 12 16 15 12 10 7 8 12 8 6 2 2 -1 8 5 1 -2 -6 -7

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