Exam 15: Partial Derivatives

Provide an appropriate response. -Find the direction in which the function is increasing most rapidly at the point $\mathrm { P } _ { 0 }$ . $f ( x , y , z ) = x y - \ln ( z ) , P _ { 0 } ( 1,2,2 )$

Determine whether the given function satisfies a Laplace equation. - $f ( x , y ) = e ^ { 10 y } \sin 10 x$

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } \right)$

Solve the problem. -Find the derivative of the function $f ( x , y ) = \tan ^ { - 1 } \frac { y } { x }$ at the point $( - 4,4 )$ in the direction in which the function decreases most rapidly.

Determine whether the given function satisfies a Laplace equation. - $f ( x , y , z ) = \cos ( 6 x ) \sin ( - 5 y ) e ^ { ( \sqrt { 61 } z ) }$

Find the limit. -

Solve the problem. -A rectangular box is to be inscribed inside the ellipsoid $2 x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 12$ . Find the largest possible volume for the box.

Solve the problem. - $\text { Write an equation for the tangent line to the curve } x ^ { 2 } - 10 x y + y ^ { 2 } = 12 \text { at the point } ( - 1,1 ) \text {. }$

Find all the second order partial derivatives of the given function. - $f ( x , y ) = x ^ { 2 } + y - e ^ { x + y }$

Compute the gradient of the function at the given point. - $f ( x , y , z ) = \tan ^ { - 1 } \frac { 7 x } { 9 y + 3 z } , ( 9,0,0 )$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial y } \text { at the point } ( - 3 , - 2 , - 8 ) : f ( x , y , z ) = x y z - 7 y ^ { 2 } - 7 z$

Find all the local maxima, local minima, and saddle points of the function. -f(x, y) = 10xy(x + y) + 4

Show that the function is a solution of the wave equation. - $w ( x , t ) = e ^ { 2 c t } \cos 2 x$

Find the linearization of the function at the given point. - $f ( x , y ) = 3 \sin x - 8 \cos y$ at $\left( 0 , \frac { \pi } { 2 } \right)$

Find the derivative of the function at P0 in the direction of u. - $f ( x , y , z ) = 2 x y ^ { 3 } z ^ { 2 } , \quad P _ { 0 } ( 2,8,4 ) , \quad \mathbf { u } = - 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$

Solve the problem. - $\text { Evaluate } \frac { \partial \mathrm { z } } { \partial \mathrm { u } } \text { at } ( \mathrm { u } , \mathrm { v } ) = ( 4,5 ) \text { for the function } \mathrm { z } = \mathrm { xy } - \mathrm { y } ^ { 2 } ; \mathrm { x } = \mathrm { u } - \mathrm { v } , \mathrm { y } = \mathrm { uv }$

Find all the second order partial derivatives of the given function. - $f ( x , y ) = x y e ^ { - y ^ { 2 } }$

Provide an appropriate answer. -Find $\frac { \partial \mathrm { w } } { \partial \mathrm { r } }$ when $\mathrm { r } = 5$ and $\mathrm { s } = - 2$ if $\mathrm { w } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { xz } + \mathrm { y } ^ { \wedge } 2 , \mathrm { x } = 3 \mathrm { r } + 7 , \mathrm { y } = \mathrm { r } + \mathrm { s }$ , and $\mathrm { z } = \mathrm { r } - \mathrm { s }$

Compute the gradient of the function at the given point. - $f ( x , y ) = \tan ^ { - 1 } \frac { - 9 x } { y }$

Provide an appropriate response. -Find any local extrema (maxima, minima, or saddle points) of $f ( x , y )$ given that $\mathrm { f } _ { \mathrm { x } } = - 4 \mathrm { x } + 4 \mathrm { y }$ and $\mathrm { f } _ { \mathrm { y } } = 7 \mathrm { x } - 8 \mathrm { y }$ .