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A Graphically Presented Derivative: AB/BC4 from 2003
This Featured Question is the fifth in a series devoted to free-response problems from the 2003 AP Calculus Examinations, which were released in early 2005. This month we extend AB/BC4, a problem that asks students to discover information about a function given a graphical presentation of its derivative.
All but the last of the new parts are very much in the spirit of the original question and are at the same level of difficulty. In particular, something like parts (E) through (H) could well appear in a similar question on a subsequent AP Calculus Examination.
Instructors should note that the official commentary about AB/BC4 states, "A small percentage of students attempted to solve part (D) analytically. Few were successful." The last of the new parts below asks students to carry out this analytic procedure, but now we expect that students will have a computer algebra system (CAS) to help with the computations.
In the interest of completeness, we reproduce here the "official" solutions for the four parts of the original problem.
AB/BC4
Let f be a function defined on the closed interval 
. The graph of f', the derivative of f, consists of one line segment and a semicircle, as shown in the accompanying figure.
(A) On what intervals, if any, is f increasing? Justify your answer.
(B) Find the x-coordinate of each point of inflection of the graph of f on the open interval
-3 < x < 4 . Justify your answer.
(C) Find an equation for the line tangent to the graph of f at the point (0,3).
(D) Find f(-3) and f(4). Show the work that leads to your answers.
The original problem consisted of the four parts (A) through (D) above. The following five parts are new. The first new part is really an extension of part (A), with an assist from part (D).
(E) Find the absolute minimum and the absolute maximum of f on the closed interval [ -3,4]. Use your answer to find the zeros of f on this interval, i.e., the values of x that satisfy f(x) = 0 and 
.
(F) Describe the behavior of the graph of f at x = 2.
(G) If f(0) = k, find the number of zeros of f as a function of k. [In the original problem, k = 3.]
In the following parts, assume that f(0)=3, as in the original problem.
(H) Extend the definition of f' , and hence f, to the interval
by replacing the line segment from (0,-2) to (-3,1) in the graph of f' with a ray that passes through (-3,1) and has endpoint (0,-2). Like the original line segment, this ray has slope -1. The graph of f' now consists of a ray and a semicircle. Find f(-4) and f(-6) and then find all zeros of f on the interval
.
(I) Find explicit formulas for the functions f, f' and f'', on the interval [-3, 4].
Note: It makes sense to find a formula for f' first, then handle f'' , and finally handle f. It also makes sense to use a CAS to find the formula for f, since at least part of this formula is fairly complicated. It will be instructive to use your formula for f to plot the graph of f so that you can confirm your answers to parts (A), (B), (D), (E), and (F). You can also use the formula to verify your answer to part (H). The formula you obtain for f on [-3, 4] should apply to
as well.
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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