Exam 15: Partial Derivatives

Solve the problem. -The resistance $R$ produced by wiring resistors of $R _ { 1 }$ and $R _ { 2 }$ ohms in parallel can be calculated from the formu: $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } } .$ If $R _ { 1 }$ and $R _ { 2 }$ are measured to be 10 ohms and 5 ohms respectively and if these measurements are accurate to within $0.05$ ohms, estimate the maximum percentage error in computing $R$ .

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

Provide an appropriate response. -Find the value(s) of $t$ corresponding to the extrema of $f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )$ subject to the constraints $x ^ { 2 } +$ $y ^ { 2 } = 6 t , 0 \leq t \leq \pi$ , and $z = \frac { \pi } { 6 }$ . Classify each extremum as a minimum or maximum. (Hint: $w = f ( x , y , z )$ is a differentiable function of t.

Solve the problem. -Write an equation for the tangent line to the curve $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 100 } = 1$ at the point $\left( \frac { 7 } { \sqrt { 2 } } , \frac { 10 } { \sqrt { 2 } } \right)$ .

Show that the function is a solution of the wave equation. - $w ( x , t ) = e ^ { x - c t }$

Find the limit. - $\lim _ { P \rightarrow ( 1 , - 1,0 ) } \frac { - 7 x z + 6 x y } { x ^ { 2 } + y ^ { 2 } - z ^ { 2 } }$

Provide an appropriate response. -Which order of differentiation will calculate $f _ { x y }$ faster, $x$ first or y first? $f ( x , y ) = x ^ { 2 } y + \sqrt { y ^ { 2 } + 1 }$

Find the absolute maxima and minima of the function on the given domain. - $f ( x , y ) = 8 x ^ { 2 } + 9 y ^ { 2 } \text { on the closed triangular region bounded by the lines } y = x , y = 2 x \text {, and } x + y = 6$

At what points is the given function continuous? - $f ( x , y ) = \sqrt { 2 x + 7 y }$

Solve the problem. -Find the extreme values of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to $3 x - y + z = 6$ and $x + 2 y + 2 z = 2$ .

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). - $f ( x , y ) = \cos ^ { - 1 } \left( \frac { x ^ { 3 } - x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } \right)$

Compute the gradient of the function at the given point. -f(x, y) = 3x - 10y, (3, -4)

Find the absolute maxima and minima of the function on the given domain. -f(x, y) = 6x + 7y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1)

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = \cos x \sin ^ { 2 } ( y z )$

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. - $f ( x , y ) = e ^ { 7 x + 10 y }$ at $( 0,0 ) ; R : | x | \leq 0.1 , | y | \leq 0.1$

Provide an appropriate response. -Find any local extrema (maxima, minima, or saddle points) of $f ( x , y )$ given that $f _ { x } = 8 x - 24$ and $f _ { y } = 2 y - 4$ .

Solve the problem. -Find the extreme values of $f ( x , y , z ) = x y z$ subject to $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and $x + y = 2$ .

Solve the problem. -Find parametric equations for the normal line to the surface -6x - 7y - 6z = -11 at the point (1, -1, 2).

At what points is the given function continuous? - $f ( x , y ) = \frac { x - y } { 2 x ^ { 2 } + x - 6 }$

Find the absolute maximum and minimum values of the function on the given curve. -Function: $f ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } ;$ curve: $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1 , x \geq 0$ , $y \geq 0$ . (Use the parametric equations $x = 3$ cos $t$ , $y = 4 \sin t$ .)