AP Calculus BC - NBISD

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2017AP Calculus BCFree-Response Questions 2017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logoare registered trademarks of the College Board. Visit the College Board on the Web: www.collegeboard.org.

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONSCALCULUS BCSECTION II, Part ATime—30 minutesNumber of questions—2A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.1. A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given bythe function A, where A(h) is measured in square feet. The function A is continuous and decreases as hincreases. Selected values for A(h) are given in the table above.(a) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate thevolume of the tank. Indicate units of measure.(b) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain yourreasoning.(c) The area, in square feet, of the horizontal cross section at height h feet is modeled by the function f givenby f (h)50.3e0.2hh. Based on this model, find the volume of the tank. Indicate units of measure.(d) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rateof 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water ischanging with respect to time when the height of the water is 5 feet. Indicate units of measure. 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.GO ON TO THE NEXT PAGE.-2-

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONS2. The figure above shows the polar curves r0qf (q )1sin q cos (2 q ) and rp. Let R be the region in the first quadrant bounded by the curve r2the region in the first quadrant bounded by the curve rf (q ), the curve rg(q )2 cos q forf (q ) and the x-axis. Let S beg(q ), and the x-axis.(a) Find the area of R.(b) The ray qk , where 0kp, divides S into two regions of equal area. Write, but do not solve, an2equation involving one or more integrals whose solution gives the value of k.p, let w(q) be the distance between the points with polar coordinates ( f (q ), q) and2p( g(q ), q ). Write an expression for w(q). Find wA, the average value of w(q) over the interval 0 q.2(c) For each q , 0q(d) Using the information from part (c), find the value of q for which w(q ) wA. Is the function w(q )increasing or decreasing at that value of q ? Give a reason for your answer.END OF PART A OF SECTION II 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.GO ON TO THE NEXT PAGE.-3-

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONSCALCULUS BCSECTION II, Part BTime—1 hourNumber of questions—4NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.3. The function f is differentiable on the closed interval 6, 5 and satisfies f ( 2) 7. The graph of f , thederivative of f, consists of a semicircle and three line segments, as shown in the figure above.(a) Find the values of f ( 6) and f (5).(b) On what intervals is f increasing? Justify your answer.(c) Find the absolute minimum value of f on the closed interval 6, 5 . Justify your answer.(d) For each of f ( 5) and f (3), find the value or explain why it does not exist. 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.GO ON TO THE NEXT PAGE.-4-

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONS4. At time t0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internaltemperature of the potato is 91 degrees Celsius ( C) at time tis greater than 27 C for all times t0, and the internal temperature of the potato0. The internal temperature of the potato at time t minutes can bemodeled by the function H that satisfies the differential equationmeasured in degrees Celsius and H(0)d 2Hdt 20. Use this equation to approximate theto determine whether your answer in part (a) is an underestimate or an overestimate of theinternal temperature of the potato at time t(c) For t1 (H 27), where H (t) is491.(a) Write an equation for the line tangent to the graph of H at tinternal temperature of the potato at time t 3.(b) UsedHdt3.10, an alternate model for the internal temperature of the potato at time t minutes is the functionG that satisfies the differential equationand G(0)dGdt (G 27)2 / 3, where G(t) is measured in degrees Celsius91. Find an expression for G(t ). Based on this model, what is the internal temperature of thepotato at time t3? 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.GO ON TO THE NEXT PAGE.-5-

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONS5. Let f be the function defined by f (x)322x 7x5.(a) Find the slope of the line tangent to the graph of f at x3.(b) Find the x-coordinate of each critical point of f in the interval 1 x 2.5. Classify each critical point asthe location of a relative minimum, a relative maximum, or neither. Justify your answers.(c) Using the identity that322x 7x521 , evaluate2x 5x 15f (x) dx or show that the integraldiverges.3(d) Determine whether the seriesn 522n 7n5converges or diverges. State the conditions of the testused for determining convergence or divergence. 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.GO ON TO THE NEXT PAGE.-6-

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONSf (nf (0)f (0 )011) n f (n) (0) for all n(0 )6. A function f has derivatives of all orders for 1 xThe Maclaurin series for f converges to f (x) for x11. The derivatives of f satisfy the conditions above.1.x2(a) Show that the first four nonzero terms of the Maclaurin series for f are x 2x3x4 , and write the34general term of the Maclaurin series for f.(b) Determine whether the Maclaurin series described in part (a) converges absolutely, convergesconditionally, or diverges at x 1. Explain your reasoning.(c) Write the first four nonzero terms and the general term of the Maclaurin series for g(x) 1 (d) Let Pn represent the nth-degree Taylor polynomial for g about x 2 0 evaluated at xthe function defined in part (c). Use the alternating series error bound to show that 1 1 P4 g 2 2 1.500STOPEND OF EXAM 2017 The College Board.Visit the College Board on the Web: www.collegeboard.org.-7-x0f (t ) d t .1, where g is2

2017 AP CALCULUS BC FREE-RESPONSE QUESTIONS 3 fx 2 2x 7x 5 3 2 1 2x2 7x 5 2x 5 x 1 2 3 n 5 2n 7n 5 5. Let f be the function defined by . (a) Find the slope of the line tangent to the graph of f at x 3

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