Exam 15: Partial Derivatives

Solve the problem. - $\text { Find the equation for the tangent plane to the surface } x ^ { 2 } - 10 x y z + y ^ { 2 } = 12 z ^ { 2 } \text { at the point } ( - 1 , - 1 \text {, } - 1 ) \text {. }$

Find all the second order partial derivatives of the given function. - $f ( x , y ) = x \ln ( y - x )$

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). - $f ( x , y ) = \cos \left( \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } \right)$

Write a chain rule formula for the following derivative. - $\frac { \partial \mathrm { w } } { \partial \mathrm { t } }$ for $\mathrm { w } = \mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) ; \mathrm { x } = \mathrm { g } ( \mathrm { s } , \mathrm { t } ) , \mathrm { y } = \mathrm { h } ( \mathrm { s } , \mathrm { t } ) , \mathrm { z } = \mathrm { k } ( \mathrm { s } )$

Sketch the surface z = f(x,y). - $f ( x , y ) = - \sqrt { x ^ { 2 } + y ^ { 2 } }$

Solve the problem. -Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (8, 16, 24) in the direction in which the function increases most rapidly.

Solve the problem. -Find an equation for the level curve of the function $f ( x , y ) = \int _ { x } ^ { y } t d t$ that passes through the point $( 6,9 )$ .

Find the equation for the level surface of the function through the given point. - $f ( x , y , z ) = \frac { x ^ { 2 } y } { x z + y ^ { 2 } } , ( 2,7,8 )$

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = 2 - x ^ { 4 } y ^ { 4 }$

Find the domain and range and describe the level curves for the function f(x,y). - $f ( x , y ) = ( - 4 x - 3 y ) ^ { 3 }$

Compute the gradient of the function at the given point. -f(x, y) = ln(-5x + 3y), (-9, -4)

Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) ≈ L(x, y) at the given point over the given region R. - $f ( x , y ) = 4 x + 7 y - 5 \text { at } ( 10 , - 9 ) ; R : | x - 10 | \leq 0.1 , | y + 9 | \leq 0.1$

Solve the problem. -What points of the surface $x y - z ^ { 2 } - 6 y + 36 = 0$ are closest to the origin?

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } e ^ { - x ^ { 2 } - y ^ { 2 } }$

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

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

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). - $f ( x , y ) = \frac { x + y } { x ^ { 2 } + y + y ^ { 2 } }$

Provide an appropriate response. -Find any local extrema (maxima, minima, or saddle points) of $\mathrm { f } ( \mathrm { x } , \mathrm { y } )$ given that $\mathrm { f } _ { \mathrm { x } } = 5 \mathrm { x } + 5 \mathrm { y }$ and $\mathrm { f } _ { \mathrm { y } } = 3 \mathrm { x } - 8 \mathrm { y }$ .

Find the limit. -

Give an appropriate answer. - $f ( x , y ) = \frac { x } { x + y }$