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AP Calculus BC Exam CiECTION II: Free Response 00 NOT OPEN THIS BOOKl.Er OR BEGIN PART B UNTIL YOU ARE TOLD TO DO SO. I I 0 At a Glance Total Tine 1 hour, 30 minutes Number of Que5tions 0 Percent of Total Score Writing Instrument Either pencil or pen with black or dark blue ink Weight The questions are weighted equally, but the parts of a question are not necessarily given equal weight. Part A Number of Questions 2 Time 30 minutes Electromc Device Graphing calculator required Percent of Section II Score 33,3% Part B Number of Questions 4 Time 60 minutes Electronic Device None allowed Percent of Section II Score 66.6% s-.,....,4-c. PORTAr!TId.ntIflcatIoainformaUon; PLEASE PRINT WITh PEN: - I 1. First two ettars of your last name 4 Unless I check the box below, grant the First letter of your first name College Board the unlimited right to use, reproduce, and publish my free-response materials, both written and oral, for 2. Date of birth educational research and instructional purposes My name and the name of my school will not be used iii any way in Mnnth Day ñ ar connection with my free-response 3 Six-digit school code materials l understand that I am free to _... mark Na with no effect on my score or Instructions f ] No, I do not grant the College Board these rights. The questions for Section II are printed in this booklet. Do nut begin Part B until you are told to do so. Write your solution 10 each pact of each question in the space provided. Write clearly and lcgibh. Cross out any errors you make erased or crossed-out work will not be scored. Manage your time careh.iily. C)uring the timed portion for Part A, work only on the questions in Part A. You are permitted to use your calculator to solve an equation. lind the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation. ftmcnon, or integral you are using. if you use other bt.iilt-in leatures or programs. you must show the mathematical steps necessary to produce your results. During the timed portion for Part B, you may continue to work on the questions in Part A without the use of a calculator. Für each part of Section II, you may wish to look over the questions beinre starting to work on them. It is not expected that everyone will he able to complete all parts of all Show all of your work, even though a question may not explicitly remind mrn [C) do Sn. Clearly label am functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and tltat you verily the ieedcd conditions under which relevant theorems, properties. definitions, or tests are applied. Your work will he scored on the correctness and completeness of your methods as well as your answers, Answers without Supporting work will usually not receive credit. Your work trit]st be expressed in standard mathematical notation rather than Lalculator syntx. For example. J r dx may not he written as fnlntti. X, 1. 5). Unless otherwise specified. answers (numeric or allgebraic need not he simplified. If you use decimal approximations in..alculations, your work will be scored on accuracy. Unless otherwise specifled. your final answers should be accurate to three places atter the decimal point. Unless otherwise specified. the domain of a function j is assumed to be the set of all real numbers x for which flx) is a real number. PP C::,j:..:; Prt. L.;f -:y n-n

-, = 2009 AP CALCULUS BC FREE-RESPONSE QUESTIONS. x(e) Note: figure not drawn to scale. ) 3. A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time r seconds after she leaps, the horizontal distance from the front edge of the platform to the diver s shoulders is given by xft), and the vertical distance from the water surface to her shoulders is given by y(r), where x(t) and y(t) are measured in meters. Suppose that the diver s shoulders are I t.4 meters above the water when she makes her leap and that dx = 0.8 and dt dy 3.6 98t, dt for 0 t A, where A is the time that the diver s shoulders enter the water. (a) find the maximum vertical distance from the water surface to the diver s shoulders. (b) Find A, the time that the diver s shoulders enter the water. (c) Find the total distance traveled by the diver s shoulders from the time she leaps from the platform until the time her shoulders enter the water. (d) Find the angle 0, 0 < 0 < shoulders enter the water. between the path of the diver and the water at the instant the diver s WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF PART A OF SECTION II 2009 The College Board. All rights resei ed. Visit the College Board on the Web: www.collegeboardcom. -4-

xy. 2009 APE CALCUlUS BC FREERESPONSE QUESTIONS CALCULUS BC ) SECTION II, Part B Time 45 minutes Number of problems 3 No calculator is allowed for these problems. dy. 4. Consider the differential equation = 6x Let y = f(x) be a particular solution to this differential equation with the initial condition f( l) = 2. (a) Use Euler s method with two steps of equal size, starting at x = I, to approximate f(o). Show the work that leads to your answer. fb) At the point ( 1,2), the value of is 12. Find the second-degree Taylor polynomial for f about x = 1. (c) find the particular solution y = f(x) to the given differential equation with the initial condition f( I) = 2. WRITE ALL WORK IN THE PINK EXAM BOOKLET. ) 2009 The College Boar& All nghts reserve& Visit the College Board on the Web: ww.collegeboar±com. GO ON TO THE NEXT PAGE.

2010 AP CALCULUS BC FREE-RESPONSE QUESTIONS 3. A particle ts moving along a ccirve so that its posttion at time t is (x(t), y(r)), where x(t) = 4r + $ and / v(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known dv t 3 hat =rte 1. dt (a) Find the speed of the particle at time t 3 seconds. (b) Find the total distance traveled by the particle for 0 4 seconds. (c) find the time t, 0 t 4, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer. fd) There is a point with xcoordinate 5 through which the particle passes twice. find each of the following. (i) (ii) The two values of t when that occurs The slopes of the tines tangent to the particle s path at that point (iii) The y-coordinate of that point, given y(?) 3 + ± WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF PART A OF SECTION II ) 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. -4-

(a) Use Euler s method, starting at x = I with two steps of equal size, to approximate f(0). Show the work fb) find lirn Show the work that leads to your answer. equatton with the initial condition f(i) = 0. For this particular solution, f(x) < I for all values of x. that leads to your answer. 5. Consider the ditferential equation = I p. Let v = 1(x) be the particular solution to this differential -6- Visit the College Board on the Web: wwwcollegeboar&com. 2010 The College Board. END OF EXAM WRITE ALL WORK IN THE PINK EXAM BOOKLET. series to write the first three nonzero terms and the general term of the Taylor series for f relative minimum, or neither at x = 0. Give a reason for your answer. value of g(l). Explain why this estimate differs from the actual value of g(1) by less than about x = 0. decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about x = 0 to estimate the (a) Write the first three nonzero terms and the general term of the Taylor series for cos x about x = 0. Use this (h) Use the Taylor series for f about x 0 found in part (a) to determine whether f has a relative maximcim, (c) Write the fifth-degree Taylor polynomial for g about x = 0. (d) The Taylor series for g about x = 0, evaluated at x = 1, is an alternating series with individual terms that g(x)= 1 +f(r)dt. 6. The function f, defined above, has derivatives of all orders. Let g be the function defined by 7 f(x)= for x 0 for x = 0 1(1) = U. (c) Find the particular SOlUtIOn v = ;(x) to the differential eqciation = I with the initial condition 2010 AP CALCULUS BC FREE-RESPONSE QUESTIONS

CALCuLUS BC SECTION II, Part A Time 30 minutes Number of problems 2. (minutes) H(t) 66 60 52 44 43 2. As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 t 10, where (b) Using correct units, explain the meaning of 1J H(t) dr in the context of this problem. Use a trapezoidal problem. the tea? WRITE ALL WORK IN THE EXAM BOOKLET. A graphing calculator is required for these problems. 1. At time t, a particle moving in the xy-plane is at position (xq), y(t)), where x(t) and y(t) are not explicitly given. fort 0, = 4t+1 and = sin(t2). Attime t = 0, x(0)= 0 and y(o)= 4. (a) Find the speed of the particle at time t 3, and find the acceleration vector of the particle at time r = 3. (b) Find the slope of the line tangent to the path of the particle at time t = 3. (c) find the position of the particle at time t = 3. (d) find the total distance traveled by the particle over the time interval 0 t 3. 2011 AP CALCULUS BC FREE-RESPONSE QUESTIONS 0 2 5 9 10 (degrees Celsius) time t is measured in minutes and temperature H(t) is measured in degrees Celsius. Values of 1-1(r) at selected values of time t are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer. sum with the four subintervals indicated by the table to estimate j.jh(t) dt. fc) Evaluate JH (r) cit. Using correct units, explain the meaning of the expression in the context of this (d) At time t = 0, biscuits with temperature 1000C were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that B (r) = l3.84e017. Using the given models, at time t = 10, how much cooler are the biscuits than

y No calculator is allowed for these problems. Number of problems 4 Time 60 minutes SECTION II, Part B CALCULUS BC -4- GO ON TO THE NEXT PAGE. Visit the College Board on the Web: www.collegeboard.org. 2011 The College Board. WRITE ALL WORK IN ThE EXAM BOOKLET. (c) The volume V, found in part (b), changes as k changes. If -L- =.-, determine when k = (a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of R in terms of k. (b) The region R is rotated about the x-axis to form a solid. Find the volume, V, of the solid in terms of k. the vertical line x = k, where k > 0. The region R is shown in the figure above. 3. Let 1(x) = e2x. Let R be the region in the first quadrant bounded by the graph off, the coordinate axes, and -á 2k ( e uii. it 13C FREE-RESPONSE QUESTIONS

the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential 2 an overestimate of the amount of solid waste that the landfill contains at time r = (c) Find the particular solution W = W(t) to the differential equation = -1z-(W 300) with initial 2011 AP CALCULUS BC FREE-RESPONSE QUESTIONS 0 y = jf(5)tx)( shown above, show that P4(-) f(-) WRITE ALL WORK N THE EXAM BOOKLET. 5. At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models equation = 300) for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010. (a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time t = 4). d2w. d2w in terms of W. Use, to determine whether your answer in part (a) is an underestimate or (b) Find dt dt condition W(0) = 1400. l20. Graph of y = 6. Let 1(x) sin(x2) + cos x. The graph of y = f(x)) is shown above. (a) Write the first four nonzero terms of the Taylor series for sin x about x = 0, and write the first four nonzero terms of the Taylor series for sin (x2) about x = 0. (b) Write the first four nonzero terms of the Taylor series for ms x about x = 0. Use this series and the series for sin(x2), found in part (a), to write the first four nonzero terms of the Taylor series for f about x = 0. (d) Let P4 (x) be the fourth-degree Taylor polynomial for f about x = 0. Using information from the graph of (c) Find the value of ft6)(0).

A graphing calculator is required for these problems. Number of problems 2 Time 30 minutes 2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS ) (c) Find the speed of the particle at time t = 4. find the acceleration vector of the particle at time t = 4. (d) Find the distance traveled by the particle from time t = 2 to t 4. (b) Find the x-coordinate of the particle s position at time r = 4. particle is at position (1, ). It is known that -- e Find the slope of the path of the particle at time t 2. - dx IE dy 2 and = sin t. (a) Is the horizontal movement of the particle to the left or to the right at time t = 2? Explain your answer. 2. for t 0, a particle is moving along a curve so that its position at time t is (x(r), y(t)). At time r 2, the 2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS W Q) = 0.4 Iicos(0.06t). Based on the model, what is the temperature of the water at time t = 25? (d) For 20 t 25, the function W that models the water temperature has first derivative given by approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning. 4.,.J 0 with the four subintervals indicated by the data in the table to approximate j WQ) Ut. Does this (c) For 0 t 20, the average temperature of the water in the tub is -J W(t) di. Use a left Riemann sum (b) Use the data in the table to evaluate I w (t) Ut. Using correct units, interpret the meaning of I w tt) Ut in the context of this problem. units, interpret the meaning of your answer in the context of this problem. (a) Use the data in the table to estimate W (12). Show the computations that lead to your answer, Using correct the water is 55 f. The water is heated for 30 minutes, beginning at time t = 0. Values of WQ) at selected where W(t) is measured in degrees Fahrenheit and t is measured in minutes. At time : 0, the temperature of times t for the first 20 minutes are given in the table above. 1. The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W, W(t) (degrees Fahrenheit) 57.1 61.8 67.9 71.0 t (minutes) 0 4 9 15 20 SECTION II, Part A CALCULUS BC

2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part B Time 60 minutes Number of problems 4 No calculator is allowed for these problems. /2.3 (-4, I) Graph off (3. 1) 3. Let f be the continuous function defined on [ 4, 3] whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let g be the function given by g(x) = ;xf(t) Ut. (a) Find the values of g(2) and g( 2). (b) For each of g ( 3) and g ( 3), find the value or state that it does not exist. (c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. for each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (U) For 4 < x < 3, find all values of x for which the graph of g has a paint of inflection, Explain your reasoning. 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. -4- GO ON TO THE NEXT PAGE.

2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS x 1 1.1 1.2 1.3 1.4 f (x) 8 10 12 13 14.5 4. The function f is twice differentiable for x > 0 with f(1) = 15 and f (l) = 20. Values of f, the derivative of f, are given for selected values of x in the table above. (a) Write an equation for the line tangent to the graph of f at x = 1. Use this line to approximate f(i.4). (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate J f (x) it. Use the approximation for jf (x) to estimate the value of f(1,4). Show the computations that lead to your answer. (c) Use Euler s method, starting at x = 1 with two steps of equal size, to approximate f(l.4). Show the computations that lead to your answer. (d) Write the second-degree Taylor polynomial for f about x = 1. Use the Taylor polynomial to approximate f(l.4). 2012 The College Board. Visit the College Board an the Web: www.couegeboard.org. -5- GO ON TO THE NEXT PAGE.

2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS 5. The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t = 0, when the bird is first weighed, its weight is 20 grams. If 3(r) is the weight of the bird, in grams, at time t days after it is first weighed, then dr5 Let y = 8(t) be the solution to the differential equation above with initial condition 3(0) = 20. (a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning. d2b. d28 (b) Find i tn terms of B. Use s to explain why the graph of B cannot resemble the following graph. to f) (c) Use separation of variables to find y condition 3(0) = 20. Time (days) 3(t), the particular solution to the differential equation with initial. 2012 AP CALCULUS BC FREE-RESPONSE QUESTIONS 6. The function g has derivatives of all orders, and the Maclaurin series for g is 3 (_Vx X X 3 5 12n+33 5+7 (a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g. (b) The Maclaurin series for g evaluated at x = is an alternating series whose terms decrease in absolute value to 0. The approximation for g() using the first two nonzero terms of this series is this approximation differs from g(--) by less than. Show that. (c) Write the first three nonzero terms and the general term of the Maclaurin series for g (x).

A graphing calculator is required for these problems. Number of problems 2 Time 30 minutes SECTION II, Part A CALCULUS time t = 1.5. (c) For the particle described in part (h), find the position vector in terms of t. find the velocity vector at the interval 1 t 2 for which the x-coordinate of the particle s position is 1. (b) A particle moves along the polar curve r = 4 2sin 8 so that at time t seconds, 8 = t2. Find th time t in Find the area of S. (a) Let S be the shaded region that is inside the graph of r = 3 and also inside the graph of r = 4 2sin 6. when 8 = and U =. D.r 2. The graphs of the polar curves r = 3 and r = 4 2sin U are shown in the figure above. The curves intersect (a) Find G (S). Using correct units, interpret your answer in the context of the problem. (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t = 5 hours? Show the (d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday. work that leads to your answer. workday? Justify your answer. plant processes gravel at a constant rate of 100 tons per hour. workday ft = 0), the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 t 8, the is modeled by G(t) = 90 + 45cos] where t is measured in hours and 0 r 8. At the beginning of the 1. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant 2013 AP CALCULUS BC FREE-RESPONSE QUESTIONS

) END OF EXAM STOP 2013 A? CALCULUS BC FREE-RESPONSE QUESTIONS 5. Consider the differential equation = y2 (2x + 2). Let y = f(x) be the particular solution to the differential ) equation with initial condition f(0) = 1. Show the work that leads to your answer. (a) Find urn ft 1 o smx (b) Use Euler s method, starting at x 0 with two steps of equal size, to approximate (c) Find y = 1(x), the particular solution to the differential equation with initial condition ff0) = 1. 2013 AP CALCULUS BC FREE-RESPONSE QUESTIONS 6. A function f has derivatives of all orders at x = 0. Let P(x) denote the nth-degree Taylor polynomial ) for f about x = 0. (a) It is known that 1(0) = 4 and that 3. Show that [(0) 2. f (O) = 4 Find P3(x). and (b) It is known that f (O) (c) The function h has first derivative given by h (x) = f(2x). It is known that h(0) = 7. Find the third-degree Taylor polynomial for h about x = 0.

A graphing calculator is required for these problems. Time 30 minutes Number of problems 2 SECTION II, Part A CALCULUS BC -2- GO ON TO THE NEXT PAGE. Visit the College Board on the Web: www.collegeboard.org. 2014 The College Board. clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. (d) for t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass problem. clippings in the bin over the interval 0 t 30. (b) Find the value of A (15). Using correct units, interpret the meaning of the value in the context of the (c) find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass (a) find the average rate of change of A(t) over the interval 0 t 30. Indicate units of measure. in days. remaining ía the bin is modeled by A(t) = 6.687(0.931), where A(t) is measured in pounds and t is measured I. Grass clippings are placed in a bin, where they decompose. for 0 t 30, the amount of grass clippings 2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS ) )

. ) (b) for the curve r = 3 2sin(20), find the value of at 8 = (c) The distance between the two curves changes for 0 < 0 <-. Find the rate at which the distance between the two curves is changing with respect to 0 when 8 = (d) A particle is moving along the curve r = 3 2sin(20) so that = 3 for all times t 0. find the value of at 0 = dr di 6 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org. -3- GO ON TO THE NEXT PAGE.. 2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS y 2. The graphs of the polar curves r = 3 and r = 3 2sin(20) are shown in the figure above for 0 0 r. Find (a) Let R be the shaded region that is inside the graph of r = 3 and inside the graph of r = 3 2sin(20). the area of R. END OF PART A OF SECTION II )

2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS lx 5. Let R be the shaded region bounded by the graph of y = xex2, the line y 2x, and the vertical line x = 1, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 2. (c) Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of R. 2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS 6. The Taylor series for a function f about x = I is given by (_l) (x 1) and converges to f(x) for x I < R, where R is the radius of convergence of the Taylor series. (a) Find the value of R. (b) Find the first three nonzero terms and the general term of the Taylor series for f, the derivative off, about x = 1. (c) The Taylor series for f about x = I, found in part (b), is a geometric series. find the function f to which the series converges for Ix I < R. Use this function to determine f for jx t < R. STOP END OF EXAM

2015 A?4 CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part A Time 30 minutes Number of problems 2 A graphing calculator is required for these problems. 1. The rate at which rainwater tiows into a drainpipe is modeled by the function R, where R(t) = 20siniJ cubic tëet per hour, t is measured in hours, and 0 t 8. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by D(c) = 0.04t3 ± 0,4t2 + 0.96t cubic feet per hour, for o t 8. There are 30 cubic feet of water in the pipe at time r = 0. (a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 0 t 8? i b) Is the amount of water in the pipe increasing or decreasing at time t = 3 hours? Give a reason for your answer. (c) At what time t, 0 t 8, is the amount of water in the pipe ac a minimum? Justify your answer. Cd) The pipe can hold 50 cubic feet of water before overflowing. for t > 8, water continues to tiow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow.. 2015 AP CALCULUS BC FREE-RESPONSE QUESTIONS 2. At time t 0, a particle moving along a curve in the xv-plane has position (x(t), y(t)) with velocity vector v(t) = (cos(t2), e ). At t = I, the particle is at the point (3, 5). (a) find the x-coordinate of the position of the particle at time t = 2. (b) for 0 < t < 1, there is a point on the curve at which the line tangent to the curve has a slope of 2. At what time is the object at that point? (c) find the time at which the speed of the particle is 3. Cd) find the total distance traveled by the particle from time r = 0 to time t 1.. END OF PART A OF SECTION II

tb) Let k = 4, so that f(c) = ta) Let k = 3, so that f(x) =,, whose x-coordinate is 4. Write an equation for the line tangent to the graph off at the point ( 2 kx) k 2x 5. Consider the function 1(x), - kx 3x where k is a nonzero constant. The derivative off is given by FNfl fl FYAM STOP the third-degree Taylor polynomial for g(x) = e f(x) about x = 0. tc) Write the first four nonzero terms of the Maclaurin series for e. Use the Maclam-in series for e to write rational function for x < R. (b) Write the first four nonzero terms of the Maclaurin series for f, the derivative of f Express f as a (a) Use the ratio test to find R. converges to f(x) for ]x < R, where R is the radius of convergence of the Maclaurin series. n=t 6. The Maclaurin series for a function f is given by 3) x = x + 3x3 + x 1 + and 2015 AP CALCULUS BC FREE-RESPONSE QUESTIONS find ff(x) dx. (U) Let k 6. so that f(x) =. Find 6x the partial fraction decomposition for the function f (c) find the value of k for which f has a critical point at x 5. neither at x = 2. Justify your answer. 4x Determine whether] has a relative minimum, a relative maximum, or 2015 AP CALCULUS BC FREE-RESPONSE QUESTIONS

. 1U 16 I he Cullege 13oiid. Viii the Cuheec I3iurd uii the Web:. ww ttiieeebonrdncg. GO ON TO THE NEXT PAGE.. Al. I buj zl L Ui JE QUESTIONS CALCULUS BC SECTION II, Part A Time 3f) minutes Number of problems 2 A graphing calculator is required for these problems. 1. Water is pumped into a tank at a rate modeled by W(t) = 2000e r/20 liters per hour for 0 t 8, where t is measured in hours. Water is removed from the tank at a rate modeled by R(t) liters per hour, where 1? is differentiable and decreasing on 0 r 8. Selected values of RQ) are shown in the table above. At time t = 0, there are 50,000 liters of water in the tank. (a) Estimate R (2). Show the work that leads to your answer. Indicate units of measure. Ib) Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer. (c) Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours. (U) For 0 t 3, is there a time t when the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.

. : L 2 y(i) CO ON TO TI-IF NFXT PACF he Collec Iiad on he Web: www cr)ileeeboatdnlg. t) 6 1 e Cle tird. END OF PART A OF SECTION JI fb) Find the slope of the line tangent to the path of the particle at t = 3. (d) Find the total distance traveled by the particle from t = 0 to 2. (a) Find the position of the particle at t = 3. (c) Find the speed of the particle at t 3. At t = 0, the particle is at position (5, 1). where ± sin(3t2). The graph of y, consisting of three line segments, is shown in the figure above. 2. At time t, the position of a particle moving in the xy-plane is given by the parametric functions (x(i), y(t)),,u J 1..UIiUZ LL -E2O.MSE QUESTID\1S

2() I 6 lhe foiie Hoard. the Coileec Board on [he \Veh: wwwcoiieeho,rd.ur (O ON TO THF NE(T PA(F -,.,.. S.., - i-- C..J. d J,L. U U 11.) CALCULUS BC SECTIoN Ii, Part B Timc 6Q minutes Number of problems 4 No calculator is allowed [Or these problems. t-l 4) ( 4, 4) (. 4) (12. 4) Graph ol f 3. The figure above shows the graph otthe piecewise-linear function j: For 4 x 12, the function g is defined by g() = I2f(t) ill (a) Does,ç have a relative minimum, a relative maximum, or neither at x = 10? Justify your answer. (b) Does the graph of g have a point of intlection at x = 4? Justify your answer, (c) Find the absolute minimum value and the absolute maximum value of g on the interval 4 x [2. Justify your answers. (U) For 4 x 12, find all intervals for which g(.t) 0.

(c) Let y = g(x) be the particular solution to the given differential equation with g( l) = 2. Find point ( 2, 8) 7 Justify your answer. point ( 2, 8). Does the graph off have a relative minimum, a relative maximum, or neither at the tb) Let y = 1(x) be the particular solution to the given differential equation whose graph passes through the terms of i. and. (a) Find -- in instant, what is the rate of change of the heiuht of the liquid with respect to time? /i = 3 inches. the radius of the surfaee of the liquid is decreasing at a iute of inch per second. At this (c ) The funnel contains liquid that is draining from the bottom At the instant when the height of the liquid is tb) Find the volume of the ftrnnel. (a) Find the average value of the radius of the funnel. the radius of the funnel is given by r + h2), where 0 ii 10. The units of r and h are inches. 5. The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height It, 2016 AP CALCULUS BC FREE-RESPONSE QUESTIONS method, starting at x 0 with two steps of equal size, to approximate h(l). (U) Let y h(x) he the particular solution to the given differential equation with h(0) = 2. Use Euler s 13(x+ [)) urn I I. Show the work that [cads to your answer. c/i. 2 4. C nsidei the di) trentiaj cduati1n I

21)16 the Ca] Ieee Board. Voit the Cohlcee Hoard on the Web: wwwcoiiecehoard.oil. - tujuj J a. The function f has a Faylor series about.v = 1 (hat converges to f(x) for all x in the interval of convergence. (H I). it is known that 1(1) I, [ (I) = and the,ith derivative off at v = I is civen by f (l) = ( I) for n 2. (a) Write the first four nonzero terms and the general term of the Taylor series for f about x = 1. (b) The Taylor series for f about x = I has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer. (c) The Taylor series for f about x = 1 can be used to represent f(1.2) as an alternating series. Use the first three nonzero terms of the alternating series to approximate f(1.2), (d) Show that the approximation found in part (c) is within 0.001 of the exact value of 1(1.2). STOP END OF EXAM