Sample Student Responses and Scoring Commentary

2017 AP Calculus AB Sample Student Responses and Scoring Commentary Inside: • Free Response Question 1 • Scoring Guideline • Student Samples • Scoring Commentary AP Central is the official online home for the AP Program: apcentral.collegeboard.org AP® CALCULUS AB/CALCULUS BC 2017 SCORING GUIDELINES Question 1 1 : units in parts (a), (c), and (d) (a) Volume = 10 ∫0 A( h ) dh ≈ ( 2 − 0 ) ⋅ A( 0 ) + ( 5 − 2 ) ⋅ A( 2 ) + (10 − 5 ) ⋅ A( 5 ) = 2 ⋅ 50.3 + 3 ⋅14.4 + 5 ⋅ 6.5 = 176.3 cubic feet (b) The approximation in part (a) is an overestimate because a left Riemann sum is used and A is decreasing. (c) 10 ∫0 f ( h ) dh = 101.325338 2: { 1 : left Riemann sum 1 : approximation 1 : overestimate with reason 2: { 1 : integral 1 : answer The volume is 101.325 cubic feet. (d) Using the model, V ( h ) = dV h ∫0 f ( x ) dx. dV dh  =  ⋅  2 : dV 3:  dt  1 : answer  dh dt  h 5 dt h 5= = dh =  f (h) ⋅   dt  h=5 = f ( 5 ) ⋅ 0.26 = 1.694419 When h = 5, the volume of water is changing at a rate of 1.694 cubic feet per minute. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. AP® CALCULUS AB/CALCULUS BC 2017 SCORING COMMENTARY Question 1 Overview In this problem students were presented with a tank that has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given by a continuous and decreasing function A, where A( h ) is measured in square feet. Values of A( h ) for heights h = 0, 2, 5, and 10 are supplied in a table. In part (a) students were asked to approximate the volume of the tank using a left Riemann sum and indicate the units of measure. Students needed to respond by incorporating data from the table in a left Riemann sum expression approximating 10 ∫0 A( h ) dh using the subintervals [ 0, 2] , [ 2, 5] , and [5, 10]. [LO 3.2B/EK 3.2B2] In part (b) students needed to explain that a left Riemann sum approximation for the definite integral of a continuous, decreasing function 50.3 overestimates the value of the integral. [LO 3.2B/EK 3.2B2] In part (c) the function f given by f ( h ) = 0.2h e +h is presented as a model for the area, in square feet, of the horizontal cross section at height h feet. Students were asked to find the volume of the tank using this model, again indicating units of measure. Using the model f for cross-sectional areas of the tank, students needed to express the volume of the tank as 10 ∫0 f ( h ) dh and use the graphing calculator to produce a numeric value for this integral. [LO 3.4D/EK 3.4D2] In part (d) water is pumped into the tank so that the water’s height is increasing at the rate of 0.26 foot per minute at the instant when the height of the water is 5 feet. Students were asked to use the model from part (c) to find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet, again indicating units of measure. Students needed to realize that the volume of water in the tank, as a function of its height h, is given by V (h) = h ∫0 f ( x ) dx and then use the Fundamental Theorem of Calculus to find that the rate of change of the volume of water with respect to its height is given by V ′( h ) = f ( h ) . Then, using the chain rule for derivatives, students needed to relate the rates of change of volume with respect to time and height and the rate of change of the water’s height with respect to time. Information in the problem suffices to be able to find these rates when the water’s height is 5 feet. [LO 2.3C/EK 2.3C2, LO 3.3A/EK 3.3A2] This problem incorporates the following Mathematical Practices for AP Calculus (MPACs): reasoning with definitions and theorems, connecting concepts, implementing algebraic/computational processes, connecting multiple representations, building notational fluency, and communicating. Sample: 1A Score: 9 The response earned all 9 points: the units point, 2 points in part (a), 1 point in part (b), 2 points in part (c), and 3 points in part (d). The units point was earned because the units of ft3 in parts (a) and (c) and ft3/min in part (d) are all correct. In part (a) the left Riemann sum point was earned by the numerical expression in the first line. This expression would have also earned the approximation point without simplification. The student chooses to simplify, does so correctly, and thus earned the approximation point. In part (b) the statement “overestimate because A is a decreasing function” earned the point. In part (c) the definite integral earned the integral point, and dV 101.325 earned the answer point. In part (d) the second line on the right earned the 2 points for . The third dt line on the right would have earned the answer point without simplification. The student chooses to give a final answer of 1.694 that is computed correctly and earned the answer point. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org. AP® CALCULUS AB/CALCULUS BC 2017 SCORING COMMENTARY Question 1 (continued) Sample: 1B Score: 6 The response earned 6 points: the units point, 2 points in part (a), 1 point in part (b), 1 point in part (c), and 1 point in part (d). The units point was earned because the units of ft3 in parts (a) and (c) and ft3/minute in part (d) are all correct. In part (a) the left Riemann sum point was earned by the symbolic expression in the first line. The approximation point was earned by the second line. The numerical expression in the second line would have earned the approximation point without simplification to 176.3. The student chooses to simplify, does so correctly, and thus earned the approximation point. In part (b) the response of “overestimated” with the reason that “ A( h ) is continuously decreasing” earned the point. In part (c) the expression 10 ∫0 f ( h ) dh earned the integral point. The answer of 101 did not earn the answer point because the result is not accurate to three places dV dh dV after the decimal point. In part (d) the equation= [ f ( 5 ) − f ( 0 )] points for using earned 1 of the 2 dt dt dt the chain rule. The student has made an error in the application of the Fundamental Theorem of Calculus. The dV is not eligible to earn the answer point. numerical evaluation of the student’s expression for dt Sample: 1C Score: 3 The response earned 3 points: no units point, no points in part (a), 1 point in part (b), 2 points in part (c), and no points in part (d). The units of ft/min2 in part (d) are incorrect, so the student did not earn the units point. Because the student uses a right Riemann sum, no points were earned in part (a). The confusion of left with right has implications in part (b). Because the setup in part (a) for a right Riemann sum is accurate, the student is eligible to earn the point in part (b) if the response is consistent for a right Riemann sum. In part (b) the implication that A is decreasing leads to an “under-estimate” is consistent with a right Riemann sum. Thus, the student earned the point. In part (c) ⌠  10 50.3 ⌡0 e.2h + h dh earned the integral point, and 101.325 earned the answer point. In part (d) no points were earned because the statement of dV = f ( h ) is incorrect, and the remaining work only implies, dt dV incorrectly, that = f= ( 5 ) 6.517. The answer point was not earned because the student’s expression for dV dt dt is not eligible to earn the answer point. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.