1) (10 points)

Find the equation of the tangent line to the graph of
at *x*=3.

2) (30 points)

Compute the following limits:

3) (30 points)

Compute the derivative of the following functions. You don't need to
simplify your answer:

4) (10 points)

The total profit (in dollars) from the sale of *x* espresso machines
is

Use the marginal profit to approximate the profit from the sale of the espresso machine.

5) (20 points)

Let .

a. Find the domain of *f*(*x*).

b. Compute *f*'(*x*) and find the critical points of *f*.

NOTE: You may choose to do a single chart for parts c and d.

c. Give the intervals where *f* is increasing and those where *f*
is decreasing. Find all local maxima and all local minima.

d. Compute *f*''(*x*). Give the intervals where *f* is
concave upward and those where *f* is concave downward. Is there any
inflection point for *f*?

e. If *f*(*x*) for
represents the cost function of producing
*x* hundred espresso machines,
how many machines would you produce in order to minimize your cost?