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AP Calculus Featured Question: April 2005
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by Ben Klein
Davidson College
Davidson, North Carolina
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The Piecewise-Defined Functions of 2003 AB5 and Some of Their Relatives
This Featured Question is the third in a series devoted to free-response problems from the 2003 AP Calculus Examinations, which were released in early 2005. This Featured Question builds on the last problem on the 2003 examination, a problem that used piecewise-defined functions to test student understanding of continuity and differentiability. In three separate parts, the problem asked students to explore two different piecewise-defined functions. We ask one more question about each of these original functions and then explore several related functions.
The original problem raised some interesting mathematical issues that are addressed in the article "Some Thoughts on 2003 Calculus AB Question 6" by Jim Hartman and Larry Riddle.
 Some Thoughts on 2003 Calculus AB Question 6
In the interest of completeness, we will reproduce the "official" solutions for the three parts of the original problem along with the solutions for the six new parts.
AB6
Let f be the function defined by
.
(A) Is f continuous at x = 3? Explain why or why not.
(B) Find the average value of f(x) on the closed interval
.
(C) Suppose the function g is defined by
where k and m are constants. If g is differentiable at x = 3, what are the values of k and m?
The next two parts involve the given functions f and g.
(D) Is the function f differentiable at x = 3? Explain why or why not.
(E) Evaluate 
.
We now introduce two functions that are related to the original functions f and g.
(F) Suppose the function h is defined by
where k and m are constants. (Note: The only difference between g and h is that the constant k appears outside the radical in g and inside the radical in h.) If h is differentiable at x = 3, find all possible pairs of values of k and m. For each pair of values k and m, evaluate 
.
(G) Show that if the definition of h is changed to the following, there is only one pair of values of k and m for which h is differentiable at x = 3. (Hint: The real question here is which of the values you found in part (F) will work in this part.)
The next two parts involve the function s defined by
.
Note that the domain of s consists of two disjoint intervals [0,3] and [4,7].
The issue in the next two parts is to find a function t defined on the interval (3,4) such that the extension of the function s defined by
is differentiable on the interval [0,7]. Note that S is the extension of the function s whose domain is the two disjoint intervals [0,3] and [4,7] to a function whose domain is the single interval [0,7].
Note first that the function t cannot possibly be linear since the only linear function that would have the same value and derivative at x = 4 as 5 - x is 5 - x. But then we have seen in parts (A) and (D) that the function we are now calling S is continuous but not differentiable at x = 3.
(H) Show that there is no quadratic function t that makes S differentiable on the interval [0,7], i.e., it is not possible to find constants a, b, and c with
such that the function S is differentiable on the interval [0,7].
(I) Find constants a, b, c, and d such that if
on the interval (3,4), the function S is differentiable on the interval [0,7].
Complete the question before viewing the answers and explanation!
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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