Solutions to the
1998 AP Calculus BC Exam
Free Response Questions

Louis A. Talman
Department of Mathematical & Computer Sciences
Metropolitan State College of Denver

Problem 1.

a.

The area of the region is

b.

The volume of the solid generated by revolving about the -axis is

c.

The value of is given by

, to three digits beyond the decimal.

Problem 2

a.

.  L'Hôpital's Rule is applicable to the latter expression.  Thus, .

b.

If , then .  Consequently, only when .  Now (by part a, above) , while .  Consequently there are numbers and , , such that implies that and implies that .  But must have an absolute minimum in the interva l , and it cannot be located at either or .  Because is the only critical point in this interval, it must give the absolute minimum for when , and therefore for .

c.

By the observations we have made in part b. above, the range of is .

d.

Let us assume, for the moment, that .  Then, arguing as we have in parts a. and b. above, we find that   has an absolute minimum at .  This minimum value is , which is independent of .  If , we obtain the same result after the change of variables , which amounts to a reflection about the -axis.

Problem 3.

a.

The third-degree Taylor polynomial for about is or .  Thus, is approximately

b.

We can obtain the third-degree Taylor polynomial for   about by substituting for in the Taylor polynomial for and then truncating.  This gives .

c.

We can obtain the third-degree Taylor polynomial for by integrating that of term by term and truncating.  We obtain .

d.

We cannot determine from what is given.  It is possible that , in which case we would have given by

However, it is also consistent with what has been given that , and if this is the case, then would be given by

Problem 4

a.

b.

Euler's Method is given by

The first Euler step with gives

The second gives

Thus, is approximately .

c.

If is the solution to for which , then , or . Hence, , and .  This gives, for ,

Problem 5

a.

b.

Average temperature is :

To the nearest degree, this is 87 degrees.

c.

The air conditioner ran when

d.

The approximate total cost is , or

To the nearest cent, this is .

Problem 6

a.

If , with , then , or .  Thus, , so that .

b.

, so .

c.

When , and .  Also, , while .  Hence, when , speed is .  At time , the particle is at with speed .

Created by Mathematica  (May 5, 2009)