AP CALCULUS AB 2007 SCORING GUIDELINES (Form B)

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1 AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and two line segments, as shown above. (a) For 5 < x < 5, find all values x at which f has a relative maximum. Justify your answer. (b) For 5 < x < 5, find all values x at which the graph of f has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of f ( x ) over the closed interval 5 x 5. Explain your reasoning. (a) f ( x) = 0 at x = 3, 1, 4 f changes from positive to negative at 3 and 4. Thus, f has a relative maximum at x = 3 and at x = 4. 2 : { 1 : x -values 1 : justification (b) f changes from increasing to decreasing, or vice versa, at x = 4, 1, and 2. Thus, the graph of f has points of inflection when x = 4, 1, and 2. 2 : { 1 : x -values 1 : justification (c) The graph of f is concave up with positive slope where f is increasing and positive: 5 < x < 4 and 1 < x < 2. 2 : { 1 : intervals 1 : explanation (d) Candidates for the absolute minimum are where f changes from negative to positive (at x = 1 ) and at the endpoints ( x = 5, 5 ). 5 π f( 5) = 3 + f ( x) dx = 3 + 2π > f ( 1) = f( 5) = 3 + f ( x) dx = 3 + > The absolute minimum value of f on [ 5, 5] is f ( 1) = 3. 3 : 1 : identifies x = 1 as a candidate 1 : considers endpoints 1 : value and explanation 2007 The College Board. All rights reserved.

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8 AP CALCULUS AB 2007 SCORING COMMENTARY (Form B) Question 4 Sample: 4A Score: 9 The student earned all 9 points. Sample: 4B Score: 6 The student earned 6 points: 2 points in part (a), no points in part (b), 2 points in part (c), and 2 points in part (d). Correct work is presented in parts (a) and (c). In part (b) the student only finds two of the three values, so the first point was not earned. The justification point was not earned because it is not true that f changes from increasing to decreasing at x = 1. In part (d) the student earned the first 2 points since x = 1 is identified as a candidate and the endpoints are considered. Since the student never concludes that the absolute minimum is 3, the third point was not earned. Sample: 4C Score: 4 The student earned 4 points: 2 points in part (a), no points in part (b), 1 point in part (c), and 1 point in part (d). Correct work is presented in part (a). In part (b) the student gives two additional, incorrect values, so the first point was not earned. No justification is included. In part (c) the first point is earned because of the correct intervals. The student s sign chart alone did not earn the explanation point. It was necessary to explain the reasoning from the sign chart. In part (d) the student earned the first point since x = 1 is identified as a candidate. The student does not consider both endpoints and does not give a correct answer, so the last 2 points were not earned The College Board. All rights reserved.

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