Exam 14: Partial Derivatives

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 } , 8 x ^ { 2 } + 4 y ^ { 2 } = 8$

Find the differential of the function $z = 3 x ^ { 3 } y ^ { 6 }$ . 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 .

Use the equation $\frac { d y } { d x } = - \frac { \partial x } { \frac { \partial F } { \partial y } }$$= - \frac { \frac { \partial F } { \partial x } } { \frac { \partial F } { \partial y } } = - \frac { F _ { x } } { F _ { y } }$ to find $\frac { d y } { d x }$ . $\cos ( x - 6 y ) = x e ^ { 4 y }$

Find the equation of the normal line to the given surface at the specified point. $2 x ^ { 2 } + 8 y ^ { 2 } + 3 z ^ { 2 } = 235 , ( 6,6,7 )$

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$ Select the correct answer.

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints. $f ( x , y , z , t ) = 7 x + 7 y + 7 z + 7 t , \quad 7 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 } \right) = 8$

Use a table of numerical values of $f ( x , y )$ for $( x , y )$ near the origin to make a conjecture about the value of the limit of $f ( x , y )$ as $( x , y ) \rightarrow ( 0,0 )$ . $f ( x , y ) = \frac { x ^ { 2 } y ^ { 3 } + x ^ { 3 } y ^ { 2 } - 7 } { 1 - x y }$

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

The height of a hill (in feet) is given by $h ( x , y ) = 30 \left( 14 - 5 x ^ { 2 } - 2 y ^ { 2 } + 3 x y + 30 x - 18 y \right)$ where $x$ is the distance (in miles) east and $y$ is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?

Let $g ( r , s , t ) = r e ^ { s / t }$ . Find $g \left( 6 , \ln 2 , \frac { 1 } { 2 } \right)$ . 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 108 .

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 $P ( 2,1,1 )$ .

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 9 x ^ { 2 } + 9 y ^ { 2 } \right) } { 6 x ^ { 2 } + 6 y ^ { 2 } }$ . Hint: If $x = r \cos \theta$ and $y = r \sin \theta$ , then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$ .

If $f ( x , y ) = x ^ { 2 } + 9 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$ . Select the correct answer.

Suppose that over a certain region of space the electrical potential $V$ is given by $V ( x , y , z ) = 8 x ^ { 2 } - 7 x y + 7 x y z$ . Find the rate of change of the potential at $( - 1,1 , - 1 )$ in the direction of the vector $\mathbf { v } = 7 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$ . Select the correct answer.

Find an equation of the tangent plane to the given surface at the specified point. $z = \sqrt { 53 - x ^ { 2 } - 2 y ^ { 2 } } , ( 3,2,6 )$

Use implicit differentiation to find $\partial z / \partial x$ . $8 x ^ { 2 } + 8 y ^ { 2 } - 4 z ^ { 2 } = 2 x ( y + z )$