AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 5. Scoring Guideline.

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1 017 AP Calculus BC Sample Student Responses and Scoring Commentary Inside: RR Free Response Question RR Scoring Guideline RR Student Samples RR Scoring Commentary 017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home for the AP Program: apcentral.collegeoard.org

2 AP CALCULUS BC 017 SCORING GUIDELINES Question (a) f ( x) = ( x 7) ( x 7x + ) ( )( ) 1 f ( ) = ( 18 1 ) = + ( x 7) 7 () f ( x) = = 0 x = x 7x + ( ) The only critical point in the interval 1 < x <. has x-coordinate 7. f changes sign from positive to negative at Therefore, f has a relative maximum at x = x = : f ( ) 1 : x-coordinate : 1 : relative maximum with justification 1 (c) ( ) = lim = lim ( ) x 7x + x x 1 x = lim ln ( x ) ln ( x 1) lim ln = ( x 1 ) 8 = ln ( ) ln ( ) = ( ) = ( ) f x dx dx dx 1 : antiderivative : 1 : limit expression 1 : answer lim ln ln ln 1 (d) f is continuous, positive, and decreasing on [, ). : answer with conditions The series converges y the integral test since converges. + x 7x dx OR n 7n + > 0 and 1 n > 0 for n. Since lim n 7n + 1 = and the series n 1 n= n converges, n the series converges y the limit comparison test. n= n 7n The College Board.

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9 AP CALCULUS BC 017 SCORING COMMENTARY Question Overview In this prolem the function f is defined y f( x) =. In part (a) students were asked to find the x 7x + slope of the line tangent to the graph of f at x =. Students needed to differentiate f and find the slope of the line tangent to the graph of f at x = y evaluating f (. ) [LO.B/EK.B1] In part () students were asked to find the x-coordinate of each critical point of f in the interval 1 < x <. and to classify each critical point as the location of a relative minimum, a relative maximum, or neither, justifying these classifications. Students should have oserved that f is differentiale on 1 < x <. and found that f ( x) = 0 has just one solution, 7 x =, in this interval. Then students needed to determine that f has a relative maximum at that f changes sign from positive to negative at the critical point 7 x = y noting 7 x =. [LO.A/EK.A1] In part (c) students were given the partial fraction decomposition for f( x ) and asked to evaluate f ( x) dx or to show that the integral diverges. Students should have expressed the given improper integral as a limit of proper, and used the partial fraction decomposition for ( ) integrals, lim f ( x) dx f x to find that f ( x) dx = ln ( ) ln ( 1) ( ln ln ) = ln ( ) ln ( ). 1 needed to take the limit of this expression as to find that the improper integral converges to 8 ( ) Applying limit theorems, students ln. [LO 1.1C/EK 1.1C1-1.1C, LO.D/EK.D1-.D] In part (d) students were asked to determine whether the series f( n) converges or diverges, stating the conditions of the test used for this determination. Students n= needed to comine the results of part (c) with the integral test or use a limit comparison test to the convergent 1 p-series to find that the series f( n) converges. For either test, students should have oserved that the n= n n= necessary conditions hold, namely that f is continuous, positive, and decreasing on [, ). [LO.1A/EK.1A6] This prolem incorporates the following Mathematical Practices for AP Calculus (MPACs): reasoning with definitions and theorems, connecting concepts, implementing algeraic/computational processes, uilding notational fluency, and communicating. Sample: A Score: 9 The response earned all 9 points: points in part (a), points in part (), points in part (c), and points in part (d). In part (a) the student earned oth points y presenting a correct expression for f ( x) and evaluating it 1 correctly at x =, arriving at the value of. The student does not need to simplify the numerical expression ( ( ) 7) ( ( ) 7 ( ) + ) to earn the points. The student chooses to simplify and does so correctly. In part () the student earned oth points with the sentence at the ottom. In this sentence the x-coordinate is correctly identified, 017 The College Board.

10 AP CALCULUS BC 017 SCORING COMMENTARY Question (continued) the critical point is correctly classified as a relative maximum, and the student s justification is correct. In part (c) the student earned the antiderivative point with the correct expression presented on the right side of the second line. The limit expression point was earned with the expression on the left side of the second line. The student earned the answer point at the end of the last line. The expression ln ( ) ln ( ) does not need to e simplified. In part (d) the student earned oth points for using the integral test to conclude that the given series converges and for stating the conditions for the integral test on the left side of the page. Sample: B Score: 6 The response earned 6 points: points in part (a), points in part (), points in part (c), and no points in part (d). In part (a) the student earned oth points y presenting a correct expression for f ( x) and evaluating it correctly 1 at x =, arriving at the value of. The student does not need to simplify the numerical expression ( 1 7) ( ) to earn the points. The student chooses to simplify and does so correctly. In part () the student earned oth points with the oxed sentence in the lower left corner. In this sentence the x-coordinate is correctly identified, the critical point (which is identified in the work in the upper right corner) is correctly classified as a relative maximum, and the student s justification is correct. In part (c) the student earned the antiderivative point in the second line. The limit expression point was earned in the first line. The student does not determine the value of the limit correctly and did not earn the answer point. In part (d) the student does not draw any conclusion aout the convergence of the given series, so no points were earned. Sample: C Score: The response earned points: points in part (a), 1 point in part (), no points in part (c), and no points in part (d). In part (a) the student earned oth points y presenting a correct expression for f ( x) and evaluating it 1 correctly at x =, arriving at the value of. The student does not need to simplify the numerical expression ( 1 7) ( ) to earn the points. The student chooses to simplify and does so correctly. In part () the 1 student earned the first point for identifying x = as the x-coordinate of the critical point. The student does not 1 provide sufficient justification for the classification of a relative maximum. The justification discusses the ehavior of f as increasing to the left and decreasing to the right rather than focusing on the sign change of f 1 at x =. Thus, the second point was not earned. In part (c) the student does not evaluate the integral, so no 1 points were earned. In part (d) the student makes an incorrect claim aout the convergence of the series without any support for the conclusion. No test is used, and no conditions are stated. The student did not earn any points. 017 The College Board.

AP CALCULUS BC 2007 SCORING GUIDELINES

AP CALCULUS BC 2007 SCORING GUIDELINES Question 4 Let f be the function defined for x > 0, with f( e ) = 2 and f, the first derivative of f, given by f ( x) = x 2 ln x. (a) Write an equation for the line