AP calculus BC student samples and commentary from the 2019 AP exam administration: free response question 5

AP Calculus BC Student Samples and Commentary from the 2019 AP Exam Administration Free Response Question 5 2019 AP ® Calculus BC Sample Student Responses and Scoring Commentary © 2019 The College Boa[.]

2019 AP Calculus BC ® Sample Student Responses and Scoring Commentary Inside: Free Response Question R Scoring Guideline R Student Samples R Scoring Commentary © 2019 The College Board College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of the College Board Visit the College Board on the web: collegeboard.org AP Central is the official online home for the AP Program: apcentral.collegeboard.org AP® CALCULUS BC 2019 SCORING GUIDELINES Question (a) f  x   f    (b)    2x  2 x  2x  k   : denominator of f  x   :  : f  x   : answer 2 1   k2   k  3 k 1 A B    x x  2 x x      x  2x   : partial fraction  decomposition 3:   : antiderivatives  : answer   A x  2  B  x  4 1  A , B 6 0 1    f  x  dx    dx    0  x  x   x 1 1   ln x   ln x     x  1 1 ln  ln  ln  ln   ln  6 6     2 1 1 dx   dx   dx   dx    2 0 x  x  0  x  1 0  x  1 1  x  12  (c)   lim    b 1  Because lim   b 1 b 1 dx  lim  dx  b 1 b  x  1 0  x  12   x b  x2   lim    lim     b 1  x  x   b 1  x  x  b  1  lim    lim 1    b b  1 b 1 b 1      does not exist, the integral diverges b 1 © 2019 The College Board Visit the College Board on the web: collegeboard.org  : improper integral  :  : antiderivative  : answer with reason © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org AP® CALCULUS BC 2019 SCORING COMMENTARY Question Note: Student samples are quoted verbatim and may contain spelling and grammatical errors Overview This problem deals with a family of functions f  x   , where k is a constant x  2x  k In part (a) students were asked to find the positive value of k such that the slope of the line tangent to the graph of f at x  equals A response should demonstrate differentiation rules to find f  x  and then identify f   as the slope of the line tangent to the graph of f at x  0, so the k can be found by solving f    In part (b) students were asked to evaluate 0 f  x  dx in the case where k  8 A response should demonstrate 1   Then that, with k  8, f  x  can be expressed using partial fractions as f  x   x4 x2 x  2x  0 f  x  dx can be evaluated using antidifferentiation and the Fundamental Theorem of Calculus In part (c) students were asked to evaluate 0 f  x  dx or show that it diverges, in the case where k  A response should note that x  x    x  12 , so the graph of f has a vertical asymptote at x  and improper integral Thus 2 0 f  x  dx is the sum 0 f  x  dx  1 0 f  x  dx is an f  x  dx, providing each of the summands converges Expressing either summand as a one-sided limit of a proper integral, a response should demonstrate that the summand diverges and conclude that 0 f  x  dx diverges For part (a) see LO FUN-3.C/EK FUN-3.C.1, LO CHA-2.C/EK CHA-2.C.1 For part (b) see LO FUN-6.F.b/EK FUN-6.F.1, LO FUN-6.C/EK FUN-6.C.2, LO FUN-6.B/EK FUN-6.B.3 For part (c) see LO LIM-6.A/EK LIM6.A.1, LO FUN-6.C/EK FUN-6.C.2, LO LIM-6.A/EK LIM-6.A.2 This problem incorporates the following Mathematical Practices: Practice 1: Implementing Mathematical Processes, Practice 2: Connecting Representations, and Practice 4: Communication and Notation Sample: 5A Score: The response earned points: points in part (a), points in part (b), and points in part (c) In part (a) the  response earned the first point in line with the term x  x  k  f  x    x  x  k  2  2 in the presented derivative expression  x   The second point was also earned in line with this presented expression for and the consistent work leading to this value In part (b) the response earned the first point in line on the right with the equation A B 1   and the declaration in line on the right that A  and B   The 6  x    x    x   x   the derivative f  x  The third point was earned in line with the declaration that k  © 2019 The College Board Visit the College Board on the web: collegeboard.org AP® CALCULUS BC 2019 SCORING COMMENTARY Question (continued) 1 ln x   ln x  The 6 3 4  ln and no response would have earned the third point in line on the left with the expression ln 6 numerical simplification The response simplifies the expression correctly, and the third point was earned with  ln In part (c) the response earned the first point in line with the expression response earned the second point in line on the left with the antiderivatives R 1   lim  dx  lim  dx The response earned the second point in line with the 2   R 1 0  x  1 R 1 R  x  1 R 1  lim  The response earned the third point in in the expression lim    x  R 1 x  R x 1 R 1 line with the boxed conclusion “[t]he integral diverges” and with the correct work that includes both one-sided limits evaluated in line as      antiderivative  Sample: 5B Score: The response earned points: points in part (a), points in part (b), and point in part (c) In part (a) the  response earned the first point in line with the denominator x  x  k expression  in the presented derivative x  x  k     1 x    f  x   The second point was also earned in line with this presented  x2  x  k  expression for the derivative f  x  The response did not earn the third point, as the value k   in line does not utilize the given information that k  In part (b) the response earned the first point in line on the A B   and the declaration in lines and on the right that right with the equation  x   x   x  x  1 A  and B   The response earned the second point in line on the left with the antiderivatives 6 1 1 ln x   ln x   The response would have earned the third point in line on the left with the 0 6 expression  ln  ln  ln  ln 2 and no numerical simplification The response simplifies this expression 1 correctly to the boxed ln in line on the left, and the third point was earned with this final expression In part (c) the response did not earn the first point because there is no indication that the integral   dx is 0  x  12 improper The response earned the second point by letting u  x  in line and with antiderivative  The response did not earn the third point because there is no conclusion of divergence © 2019 The College Board Visit the College Board on the web: collegeboard.org in line u AP® CALCULUS BC 2019 SCORING COMMENTARY Question (continued) Sample: 5C Score: The response earned points: point in part (a), point in part (b), and point in part (c) In part (a) the response  earned the first point in line with the correct denominator x  x  k f  x    in the presented derivative expression  x2  x  k    x   The response did not earn the second point because the presented derivative  x2  x  k  for f  x  in line is incorrect The response did not earn the third point because the circled answer k  on the right is incorrect In part (b) the response earned the first point in line with the equation 1 1 A B   dx   dx and with the declaration in the last line on the right that B   0 x  x  0  x   x   The missing parentheses in the integrand on the right side of the equation not impact earning the point The response did not earn the second point because there is no antiderivative presented The response did 1 1 1 1 ln  ln  ln  ln not earn the third point due to the undefined expression in line on the left 6 ln Note that even if the circled answer had been correct, this response would not have been eligible for the third point because of the undefined, incorrect expression In part (c) the response did not earn the first point and A      because there is no indication that the integral  dx is improper The response earned the second point  0  x  12 with the antiderivative divergence 1 The response did not earn the third point because there is no conclusion of x 1 © 2019 The College Board Visit the College Board on the web: collegeboard.org  ... with the declaration that k  © 2019 The College Board Visit the College Board on the web: collegeboard.org AP? ? CALCULUS BC 2019 SCORING COMMENTARY Question (continued) 1 ln x   ln x  The. .. Visit the College Board on the web: collegeboard.org AP? ? CALCULUS BC 2019 SCORING COMMENTARY Question Note: Student samples are quoted verbatim and may contain spelling and grammatical errors Overview... © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the College Board on the web: collegeboard.org © 2019 The College Board Visit the