M403K Final Exam Solutions
May 10, 2002

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  .