Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function $P(L)=200 L+10 L^2-L^3$ gives the output of a system as a function of the number of laborers $L$. The average product $A(L)$ is the average output per laborer when $L$ laborers are working; that is $A(L)=P(L) / L$. The marginal product $M(L)$ is the approximate change in output when one additional laborer is added to $L$. laborers; that is, $M(L)=\frac{d P}{d L}$. (a). For the production function given here, compute and graph $P, A$, and $M$. (b). Suppose the peak of the average product curve occurs at $L=L_0$, so that $A^{\prime}\left(L_0\right)=0$. Show that for a general production function, $M\left(L_0\right)=A\left(L_0\right)$

Calculate y'(x) for the following relations. y=

$$\frac { e ^ { y } } { 1 + \sin x }$$

An angler hooks a trout and reels in his line at 4in/s. Assume the tip of the fishing rod is 12ft above the water and directly above the angler, and the fish is pulled horizontally directly toward the angler. Find the horizontal speed of the fish when it is 20ft from the angler.

Find

$$\frac { d y } { d x }$$

.

$$y = x ^ { 5 / 4 }$$

(a). Find the derivative function $f^{\prime}$ for the following function $f$. (b). Find an equation of the line tangent to the graph of $f$ at $(a, f(a))$ for the given value of $a$. $(c)$. Graph $f$ and the tangent line.

$$f(x)=3 x^2+2 x-10 ; a=1$$

$$y=e^x ; a=\ln 3$$

(a). Find an equation of the tangent line at $x=a$. (b). Use a graphing utility to graph the curve and the tangent line on the same set of axes.

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find $f ^ { - 1 }$. f(x)=3x+4; (16, 4)

Use the General Power Rule where appropriate to find the derivative of the following functions.

$$f(x)=x^e$$

Find the derivative of the following functions.

$$f ( x ) = 3 x ^ { - 9 }$$

Find the derivative of the following functions. $y = \frac { \cot x } { 1 + \csc x }$

Evaluate and simplify the following derivatives.

$$\frac { d } { d x } \left. \left( \tan ^ { - 1 } e ^ { - x } \right) \right| _ { x = 0 }$$

$$y=\frac{e^x}{x} ; a=1$$

(a). Find an equation of the line tangent to the given curve at $a$. (b). Use a graphing utility to graph the curve and the tangent line on the same set of axes.

Use the General Power Rule where appropriate to find the derivative of the following functions. $f ( x ) = 2 x ^ { \sqrt { 2 } }$

Find

$$\frac { d ^ { 2 } y } { d x ^ { 2 } }$$

sin x+

$$x^2y$$

=10