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AP Calculus Question of the Month: December
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by Ben Klein
Davidson College
Davidson, North Carolina
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Differential Equations and Infinite Series: BC6 from the 2003 Examination
Problem 6 on the 2003 BC Examination introduced a function defined by a convergent power series. (Incidentally, it would be worth your while to check that the series really does converge for all real numbers x. Doing so is an easy application of the ratio test.) The last part of the problem asked students to verify that the function satisfied a certain differential equation. One solution technique for this last part (C) revealed the given function's "hidden identity."
In this Question of the Month, we explore the differential equation a little more deeply and then explore a similar differential equation. The second differential equation is similar to the first but behaves very differently in at least one way.
We use some techniques from the BC syllabus and some topics from the AB syllabus for both differential equations. So, even though most of this Question of the Month involves BC topics, some topics are directly relevant for AB students as well. In particular, parts (F), (G), and (I) use topics from the AB syllabus.
As in the past, even though you can download the original problem and its solution for yourself, we will reproduce both of them here.
BC6
The function f is defined by the power series:
for all real numbers x.
(A) Find
. Determine whether f has a local maximum, a local minimum, or neither at x = 0. Give a reason for your answer.
(B) Show that1-1/3! approximates 
with error less than 1/100.
(C) Show that 
is a solution to the differential equation 
.
Part (C) was the final part of the original problem. Each of these first three parts involves BC-only topics, i.e., manipulation of power series.
(D) You can see from the second solution to part (C), or you may already have noticed, that:
.
Use this characterization of f to verify the claim in part (B). [You can also graph 
to verify your answer to part (A).]
(E) Suppose that the power series
has a positive radius of convergence. If 
is the sum of this series on its interval of convergence and 
satisfies the differential equation 
on the interval of convergence of the series, show that 
. Thus, 
is the only solution of the form 
to the given differential equation on an interval containing x = 0.
Note: Part (E) shows that 
is the only solution of a particular nature that satisfies the given differential equation. The next part will show that the given function 
is the only solution of any type.
(F) Use the fact that
to show that if 
is differentiable on an open interval containing x = 0 and solves the differential equation 
, then 
on that interval.
Note: We now introduce another differential equation and one of its solutions.
(G) Show that 
solves the differential equation 
.
(H) Use the method of part (E) to solve the differential equation 
. That is, suppose that the power series 
has a positive radius of convergence, that 
is the sum of this series on its interval of convergence, and that 
satisfies the differential equation 
on the interval of convergence of the series. Find the coefficients 
and identify the function 
. [This time you should find that 
is not the only solution to the differential equation.]
Note: You should do part (H) and check your solution before doing part (I).
(I) Use the fact that the differential equation 
is equivalent to the differential equation 
if x is nonzero, to show that the functions discovered in part (H) are the only solutions to the differential equation 
.
Click here to view the answers and commentary!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. In 2003 he completed a four-year term on the AP Calculus Development Committee.
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