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AP Calculus Question of the Month: May
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by Ben Klein
Davidson College
Davidson, North Carolina
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An Alternative Solution Method for 1988 AB6
Question AB6 from the 1988 examination asks students to find a formula for a cubic function given certain information about the function. We will present the problem as it appeared on the examination and then present the solution as given on AP Central. (Free-response question collections dating back to 1978 are available for purchase by following the link in "More" at right.)
We will then state a useful theorem that we will use here to find a simpler solution to the original problem and a slight variation of it. (The theorem is useful in many other contexts as well.) Next, we will present a problem that has a simple solution when solved using the theorem, and finally we will ask for a proof of the theorem itself.
It is worth noting that if a problem like 1988 AB6 were asked on an examination now, it would definitely appear on the non-calculator section of the examination. Students whose calculators have a computer algebra capability would be at a huge advantage, although the advantage would be smaller if they used the theorem presented below.
AB6
Let f be a differentiable function, defined for all real numbers x, with the following properties:
Find 
. Show your work.
Here is a paraphrased version of the "official" solution, but you may want to work the problem yourself before reading the solution.
From (ii) then, 
It follows that a = 12 and b = -6 and thus
. Antidifferentiating, we obtain
. Now from (iii), we have
.
Thus, finally,
.
Now consider the following theorem, the proof of which will be the last part of this Question of the Month. As you consider the theorem, remember that 0! = 1.
Theorem: If
is a polynomial of degree n, then
Now that you have this tool to help, here are some problems for you to solve.
(A) Use the theorem above to solve 1988 AB6.
(B) Use the theorem above to solve 1988 AB6 if (iii) is replaced by
.
(C) Find
, if 
is a cubic polynomial with the following properties:
(i) 
and (ii) 
(D) Prove the theorem given above.
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. He just completed a four-year term on the AP Calculus Development Committee.
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