Solutions to the
1998 AP Calculus BC Exam
Free Response Questions
Louis A. Talman
Department of Mathematical & Computer Sciences
Metropolitan State College of Denver
The area of the region is
The volume of the solid generated by revolving about the -axis is
The value of is given by
, to three digits beyond the decimal.
. L'Hôpital's Rule is applicable to the latter expression. Thus, .
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 .
By the observations we have made in part b. above, the range of is .
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.
The third-degree Taylor polynomial for about is or . Thus, is approximately
We can obtain the third-degree Taylor polynomial for about by substituting for in the Taylor polynomial for and then truncating. This gives .
We can obtain the third-degree Taylor polynomial for by integrating that of term by term and truncating. We obtain .
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
Euler's Method is given by
The first Euler step with gives
The second gives
Thus, is approximately .
If is the solution to for which , then , or . Hence, , and . This gives, for ,
Average temperature is :
To the nearest degree, this is 87 degrees.
The air conditioner ran when
The approximate total cost is , or
To the nearest cent, this is .
If , with , then , or . Thus, , so that .
, so .
When , and . Also, , while . Hence, when , speed is . At time , the particle is at with speed .
Created by Mathematica (May 5, 2009)