Exam 15: Partial Derivatives

Solve the problem. -A simple electrical circuit consists of a resistor connected between the terminals of a battery. The voltage $V$ (in volts) is dropping as the battery wears out. At the same time, the resistance $R$ (in ohms) is increasing as the resistor heats up. The power $P$ (in watts) dissipated by the circuit is given by $P = \frac { V ^ { 2 } } { R }$ . Use the equation $\frac { \mathrm { dP } } { \mathrm { dt } } = \frac { \partial \mathrm { P } } { \partial \mathrm { V } } \frac { \mathrm { dV } } { \mathrm { dt } } + \frac { \partial \mathrm { P } } { \partial \mathrm { R } } \frac { \mathrm { dR } } { \mathrm { dt } }$ to find how much the power is changing at the instant when $R = 5$ ohms, $V = 4$ volts, $d R / d t = 1$ ohms/sec and $d V$ , $= - 0.04$ volts/sec.

Solve the problem. -Amarillo Motors manufactures an economy car called the Citrus, which is notorious for its inability to hold a respectable resale value. The average resale value of a set of 1998 Amarillo Citrus's is summarized in the table below along with the age of the car at the time of resale and the number of cars included in the average. Fit a line of the form $\ln ( \mathrm { V } ) = \mathrm { m } \cdot \mathrm { a } + \mathrm { b }$ to the data, where $\mathrm { V }$ is the resale value in thousands of dollars and a is the age of the car in years..

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

Solve the problem. -Evaluate $\frac { \partial \mathrm { u } } { \partial \mathrm { y } }$ at $( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = ( 2,3,0 )$ for the function $\mathrm { u } = \mathrm { epq } \cos ( \mathrm { r } ) ; \mathrm { p } = \frac { 1 } { \mathrm { x } } , \mathrm { q } = \mathrm { x } ^ { 2 } \ln \mathrm { y } , \mathrm { r } = \mathrm { z }$ .

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = ( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial y } \text { at the point } ( 3 , - 9,9 ) : f ( x , y , z ) = 2 x ^ { 2 } y + 2 y ^ { 2 } + 10 z$

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = x ^ { 2 } y , x ^ { 2 } + 2 y ^ { 2 } = 6$

Provide an appropriate answer. -Find $\frac { \partial z } { \partial v }$ when $u = 5$ and $v = 1$ if $z ( x ) = \frac { x } { \sqrt { x + 5 } }$ and $x = u \cdot v$ .

Provide an appropriate answer. -Find $\frac { \partial \mathrm { w } } { \partial \mathrm { u } }$ when $\mathrm { u } = 2$ and $\mathrm { v } = - 5$ if $\mathrm { w } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \frac { \mathrm { xy } } { \mathrm { z } } , \mathrm { x } = \frac { \mathrm { u } } { \mathrm { v } } , \mathrm { y } = \mathrm { u } + \mathrm { v }$ , and $\mathrm { z } = \mathrm { u } \cdot \mathrm { v }$ .

Find the domain and range and describe the level curves for the function f(x,y). - $f ( x , y ) = \frac { 9 x ^ { 2 } } { y }$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to $f ( x , y ) = \frac { 1 } { ( 1 + x + 3 y ) ^ { 2 } }$

Solve the problem. -Find the least squares line through the points (1, 18), (2, 42), and (3, 36).

Solve the problem. -Find the points on the curve $x y ^ { 2 } = 686$ that are closest to the origin.

Find the limit. - $( x , y ) - \left( 0 , - \frac { \pi } { 4 } \right) ^ { \frac { \sec x + 7 } { 10 x - \tan y } }$

Find the absolute maximum and minimum values of the function on the given curve. -Function: $f ( x , y ) = x y ;$ curve: $x ^ { 2 } + y ^ { 2 } = 49 , x \geq 0 , y \geq 0$ . (Use the parametric equations $x = 7$ cos $t , y = 7 \sin t$ .)

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. - $f ( x , y , z ) = 8 x ^ { 2 } + 7 y ^ { 2 } + 2 z ^ { 2 } \text { at } ( 1 , - 2,3 ) ; R : | x - 1 | \leq 0.1 , | y + 2 | \leq 0.1 , | z - 3 | \leq 0.1$

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

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = 7 x ^ { 2 } + 3 y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = 1$

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). -Does knowing that $3 - x ^ { 2 } y ^ { 3 } < \frac { 3 \tan ^ { - 1 } x y } { x y } < 3$ tell you anything about $( x , y ) \rightarrow ( 0,0 ) \frac { 3 \tan ^ { - 1 } x y } { x y }$ ? Give reasons for your answer.

Find the specific function value. -Find f(100, 3) when f(x, y) = y log x.