**Problem 1. Graphing**

Consider the function .

a) Find the partition points of *f* and make a sign chart for *f*.

, so the partition points are 0,
and
. The function is positive for
and for
. It is negative for

and for
.

b) Find the critical points of *f* and make a sign chart for *f*'.

, so the critical points are -1, 0 and 1. *f*' is negative for *x<*-1,
positive for -1 < *x* < 0, negative for 0<*x<*1 and
positive for *x>*1.

c) Find the inflection points of *f* and make a sign chart for
*f*''.

, so the inflection points are at
. *f*'' is positive for
and for
, and negative for
.

d) On the back of this page, sketch the curve *y*=*f*(*x*).
Mark all important points CLEARLY.

I can't draw this on screen, but the graph looks like a ``Mexican hat''.
There are local minima at (-1,-1) and (1,-1), and a local maximum at (0,0).
The graph is positive, decreasing and curving up for
, then negative, decreasing and curving up for
, hitting a local minimum at (-1, -1). It is then negative, increasing,
and curving up for
, and is negative, increasing, and curving down for
, hitting a local maximum at (0,0). The remainder of the graph is a mirror
image of the first half, since *f*(-*x*)=*f*(*x*).

**Problem 2. Max-min**

Consider the function .

a) Find all critical points of this function. For each one, say whether it is a local maximum, a local minimum, or neither.

. Setting this equal to zero we have

But , so dividing our equation by gives

so , so . This is the ONLY critical point. Now , and , so . Now , so , so is a local minimum.

b) Find the global maximum and minimum of *f*(*x*) in the
interval [-5,5].

The candidates are the critical points and the endpoints.
is a huge positive number, and
is a tiny negative number, so the maximum is at *x*=5 and the minimum
is at
.

**Problem 3. Marginal analysis**

The demand *x* for widgets is related to the price *p* by
the demand equation
*x* = 3000 - 100 *p*. The cost function is
.

a) Find the marginal cost, the marginal revenue, and the marginal profit
at a production level of *x*=1200.

Solving for price in terms of demand gives *p*=30 - *x*/100,
so
. Since
, we have a profit
. Taking derivatives we get our marginal quantities:

*R*'(*x*) = 30 - *x*/50, *C*'(*x*) = 2-=*x*/100,
*P*'(*x*) = 10-*x*/100.

Finally, plugging in *x*=1200 gives *R*'(1200)=6, *C*'(1200)=8
and
*P*'(1200)=-2. In other words, each additional widget costs us
$ 8, and only brings us $ 6 in additional revenue, and so decreases our
profit by $ 2.

b) What is the production level that maximizes revenue? What production level maximizes profit?

To maximize revenue, set *R*'=0. This gives *x*=1500.

To maximize profit, set *P*'=0. This gives *x*=1000.

**Problem 4. Exponential growth**

An investor invests $1000 at 7% interest, compounded continuously.

a) How much money will he have in 20 years? Express your answer as an exact expression (e.g. something like $ - no, that's not the right answer), and then approximate it numerically (e.g., $16,000).

. Now (the Law of 70, remember?), so and . Thus .

b) When will there be $10,000 in his account? Express your answer as an exact expression. You do NOT need to approximate it numerically.

, so , so , so . (This is a little under 33 years).

**Problem 5. Rates of change**

A quantity *y* is changing at a rate

When *x*=0, *y*=5. What does *y* equal when *x*=2?

Plugging in *y*(0)=5 we have
, so *C*=3. Thus:

**Problem 6. Volume**

A silo-shaped region is obtained by taking the region between the curve
and the *x*-axis and rotating it about the *y*-axis. [See figure].
We compute the volume of this region by slicing it into a stack of disks.

Note: This is a straightforward application of the ``slice and dice principle'', but doesn't directly correspond to any formulas in the book. I'm quite disappointed that only 4 or 5 people in the entire class attempted this problem.

a) Find the (approximate) radius and (approximate) volume of a disk
at height *y* and thickness
.

The radius of the disk is just the value of *x* on the curve. Since
, we have
, so our radius is
. The area is
, and the volume is (area
thickness) =
.

b) By summing the answer to (a) and taking a limit, express the volume of the silo as a definite integral.

The sum of volumes is of the form
, where
, and *y* ranges from 0 to 4. Taking the limit as the number of slices
goes to infinity gives the definite integral
.

c) Evaluate this integral to get the total volume.

The integral evaluates to .

**Part II:**

Evaluate the following. The limits and integrals should be simplified as much as possible, but you don't have to simplify the derivatives:

a)

b) . Set .

c)

.

d) .

e) *f*'(*x*), where
. Apply the chain rule twice to get

f) . This comes from the substitution .

g) *dy*/*dx*, where
is

h) . (Either apply L'Hopital 3 times, or divide top and bottom by and take a limit).

i) Note that and that .

j)
. This sum is of the form
, where
and
. Taking the limit gives
, which works out to
.