Exam 15: Partial Derivatives

Estimate the error in the quadratic approximation of the given function at the origin over the given region. - $f ( x , y ) = \ln ( 1 + 3 x + 4 y ) , \quad | x | \leq 0.1 , | y | \leq 0.1$

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$

Solve the problem. -Find an equation for the level curve of the function $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ that passes through the point $( 3,4 )$ .

Find the derivative of the function at P0 in the direction of u. - $f ( x , y , z ) = \ln \left( x ^ { 2 } + 9 y ^ { 2 } - 7 z ^ { 2 } \right) , \quad P _ { 0 } ( 9,9,9 ) , \quad \mathbf { u } = 3 \mathbf { i } + 4 \mathbf { j }$

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

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = x ^ { 3 } + y ^ { 3 } - 243 x - 48 y - 5$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial z } \text { at the point } ( 6 , - 5,5 ) : f ( x , y , z ) = x y z - 10 y ^ { 2 } - 7 z$

At what points is the given function continuous? - $f ( x , y , z ) = \frac { 1 } { | x - 3 | + | y - 4 | + | z - 5 | }$

Estimate the error in the quadratic approximation of the given function at the origin over the given region. - $f ( x , y ) = \sin 5 x \sin 2 y , | x | \leq 0.1 , | y | \leq 0.1$