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AP Calculus Question of the Month: October
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by Ben Klein
Davidson College
Davidson, North Carolina
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What's My Cubic? -- AB5 from the 2001 Examination
Problem 5 on the 2001 AB examination asks students to determine the coefficients of a cubic polynomial given certain information about the behavior of the cubic. We will present the original problem, the "official" and an "alternative" solution, and then a sequence of related problems. Even though you can download the original problem and its solution for yourself, we will reproduce both of them here.
AB5
A cubic polynomial function f is defined by:
f(x) = 4x3 + ax2 + bk + k
where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.
(A) Find the values of a and b.
(B) If
, what is the value of k?
The original problem consisted of the two parts above. The first of the parts below asks for an alternative solution to the original problem. The remaining parts contain questions similar in spirit to the original problem.
(C) Recall the FACTS: If a cubic polynomial function has a local minimum (respectively a local maximum), then the function has a local maximum (respectively a local minimum). Moreover, the point of inflection of the cubic is at the midpoint of the segment whose endpoints are the two local extreme points. (Click here to access a proof of these facts.) Use these facts to find the values of a and b.
(D) Suppose now that f has a local minimum at x = -1 and a local maximum at x = -5. Find the values of a and b.
This next question is a trick. Be sure to think carefully before answering it.
(E) Suppose now that f has a local minimum at x = -1 and a local maximum at x = 5. Find the values of a and b.
(F) Suppose now that all we know about f is the following:
.
Find the values of a, b, and k.
The question above suggests that whenever the values of three integrals involving f are given, we have enough information to discover the values of a, b, and k. However, as the following question shows, things are not quite that simple.
(G) Suppose now that all we know about f is the following:
.
Show that we can find the values of a and k but cannot find the value of b. Conclude that we can find the x-coordinate of the point of inflection but cannot even determine whether f has local extreme points, much less determine their locations.
(H) Suppose that f is as in part (G). Convince yourself that if we know the value of f at even one value of x, with one exception, we can determine the value of b in addition to the values of a and k that we already know. What is the exceptional value of x?
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 recently served a four-year term on the AP Calculus Development Committee.
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