Exam 15: Partial Derivatives

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { \partial y } { \partial x }$ at the point $\left( 1,4 , e ^ { 7 } \right)$ for $\ln ( x z ) y + 6 y ^ { 3 } = 0$ .

Find the equation for the level surface of the function through the given point. - $f ( x , y , z ) = \int _ { y } ^ { z } ( \ln \theta + 1 ) d \theta + \int _ { 0 } ^ { x } t e ^ { t } d t , \left( 4 , e ^ { 3 } , e ^ { - 5 } \right)$

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

Give an appropriate answer. -Given the function $f ( x , y , z )$ and the positive number $\varepsilon$ as in the formal definition of a limit, find a positive number $\delta$ as in the definition that insures $| f ( x , y , z ) - f ( 0,0,0 ) | < \varepsilon$ . $f ( x , y , z ) = \frac { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } { x + 1 } ; \varepsilon = 0.03$

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

Solve the problem. -

Solve the problem. -Find the derivative of the function $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 }$ at the point $( 3,4 )$ in the direction in which the function decreases most rapidly.

Find the limit. - $\lim _ { P \rightarrow ( 1,1,8 ) } \sec ^ { 2 } x - \tan ^ { 2 } y + z$

Sketch the surface z = f(x,y). - $f ( x , y ) = 3 - x ^ { 2 }$

Solve the problem. -Find an equation for the level surface of the function $f ( x , y , z ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + y ) ^ { 2 n } } { ( 2 n ) ! z ^ { 2 n } }$ that passes through the point $( \pi , \pi , 1 )$ .

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. - $f ( x , y , z ) = e ^ { 9 x + 7 y + 5 z } \text { at } ( 0,0,0 ) ; R : | x | \leq 0.1 , | y | \leq 0.1 , | z | \leq 0.1$

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $\text { Show that } f ( x , y , z ) = x ^ { 2 } y ^ { 5 } z ^ { 5 } \text { is continuous at every point } \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right) \text {. }$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Quadratic approximation to f(x, y) = ln(1 + 10x + y)

Solve the problem. -Find the point on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that is farthest from the point $( 3,1 , - 1 )$ .

Compute the gradient of the function at the given point. - $f ( x , y , z ) = - 7 x - 7 y - 6 z , \quad ( 3 , - 8,5 )$

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { d y } { d x }$ at the point $( 1,0 )$ for $\cos x y + y e ^ { x } = 0$ .

Show that the function is a solution of the wave equation. -w(x, t) = cos ( ct) sin ( x)

Find the limit. -

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

Solve the problem. -If the length, width, and height of a rectangular solid are measured to be 5, 2, and 9 inches respectively and each measurement is accurate to within 0.1 inch, estimate the maximum possible error in computing the volume of The solid.