Exam 15: Partial Derivatives

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

Find the extreme values of the function subject to the given constraint. -f(x, y) = 12x + 3y, xy = 4, x > 0, y > 0

Find the specific function value. - $\text { Find } f ( 0,1 , - 1 ) \text { when } f ( x , y , z ) = 3 ^ { x } - 7 y z + 2 x \text {. }$

Sketch the surface z = f(x,y). - $f ( x , y ) = 1 - x - 2 y$

Solve the problem. -Find an equation for the level curve of the function $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \sqrt { \mathrm { y } ^ { 2 } - 16 }$ that passes through the point $( 0,4 )$ .

Solve the problem. - $\text { Evaluate } \frac { \partial \mathrm { u } } { \partial \mathrm { x } } \text { at } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = ( 4,5,1 ) \text { for the function } \mathrm { u } = \mathrm { p } ^ { 2 } - \mathrm { q } ^ { 2 } - \mathrm { r } ; \mathrm { p } = \mathrm { xy } , \mathrm { q } = \mathrm { y } ^ { 2 } , \mathrm { r } = \mathrm { xz }$

Solve the problem. -Evaluate $\frac { \partial \mathrm { z } } { \partial \mathrm { v } }$ at $( \mathrm { u } , \mathrm { v } ) = ( 2,3 )$ for the function $\mathrm { z } = \mathrm { xy } ^ { 2 } - \ln \mathrm { x } ; \mathrm { x } = \mathrm { e } ^ { \mathrm { u } + \mathrm { v } } , \mathrm { y } = \mathrm { uv }$ .

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { \partial y } { \partial x }$ at the point $( 2,3,2 )$ for $3 x ^ { 2 } + 6 \ln x z - 4 y z ^ { 2 } + 4 e ^ { z } = 0$ .

Provide an appropriate response. -Find $\frac { d w } { d t }$ at $t = 0$ for the function $w = \sin ( x ) \cos ( y ) \ln ( z )$ where $x = - 9 t ^ { 2 } + 6 t + \frac { \pi } { 2 } , y = t , z = e ^ { - 8 t }$ .

Solve the problem. -Find the point on the line x - 3y = 6 that is closest to the origin.

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = x e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $f ( x , y ) = \frac { x ^ { 3 } + y ^ { 6 } } { x ^ { 3 } }$

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = e^{ ( y z + \sin x )}$

Solve the problem. -Find an equation for the level curve of the function $f ( x , y ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { y } \right) ^ { n }$ that passes through the point $( 1,5 )$ .

Solve the problem. -Evaluate $\frac { \mathrm { dw } } { \mathrm { dt } }$ at $\mathrm { t } = \frac { 5 } { 2 } \pi$ for the function $\mathrm { w } = \frac { \mathrm { xy } } { \mathrm { z } } ; \mathrm { x } = \sin \mathrm { t } , \mathrm { y } = \cos \mathrm { t } , \mathrm { z } = \mathrm { t } ^ { 2 }$ .

Find the requested partial derivative. - $( \partial w / \partial x ) _ { y , z }$ if $w = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } + 9 x y z$

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = x + 2 y - 2 z , \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$

Sketch a typical level surface for the function. - $f ( x , y , z ) = \cos \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$

Write a chain rule formula for the following derivative. - $\frac { \partial \mathrm { u } } { \partial \mathrm { t } }$ for $\mathrm { u } = \mathrm { f } ( \mathrm { v } ) ; \mathrm { v } = \mathrm { h } ( \mathrm { s } , \mathrm { t } )$

Find the specific function value. - $\text { Find } f ( - 8 , - 5 ) \text { when } f ( x , y ) = 2 y ^ { 2 } - 9 x y$