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2002 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)2. The number of gallons, P(t ), of a pollutant in a lake changes at the rate P (t ) 1 - 3e - 0.2 t gallons per day,where t is measured in days. There are 50 gallons of the pollutant in the lake at time t 0. The lake isconsidered to be safe when it contains 40 gallons or less of pollutant.(a) Is the amount of pollutant increasing at time t 9 ? Why or why not?(b) For what value of t will the number of gallons of pollutant be at its minimum? Justify your answer.(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.05(d) An investigator uses the tangent line approximation to P t at t 0 as a model for the amount of pollutantin the lake. At what time t does this model predict that the lake becomes safe?3. Let R be the region in the first quadrant bounded by the y-axis and the graphs of y 4 x - x 3 1 and3y x.4(a) Find the area of R.(b) Find the volume of the solid generated when R is revolved about the x-axis.(c) Write an expression involving one or more integrals that gives the perimeter of R. Do not evaluate.END OF PART A OF SECTION IICopyright 2002 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.3

2002 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)CALCULUS BCSECTION II, Part BTime—45 minutesNumber of problems—3No calculator is allowed for these problems.4. The graph of a differentiable function f on the closed interval [ -3, 15] is shown in the figure above. The graphof f has a horizontal tangent line at x 6. Let g ( x ) 5 05 0505Ix605f t dt for -3 x 15.(a) Find g 6 , g ′ 6 , and g ′′ 6 .(b) On what intervals is g decreasing? Justify your answer.(c) On what intervals is the graph of g concave down? Justify your answer.(d) Find a trapezoidal approximation of5. Consider the differential equation05I15-305f t dt using six subintervals of lengthDt 3.dy3- x .dxy(a) Let y f x be the particular solution to the given differential equation for 1 x 5 such that the liney -2 is tangent to the graph of f. Find the x-coordinate of the point of tangency, and determine whetherf has a local maximum, local minimum, or neither at this point. Justify your answer.05(b) Let y g x be the particular solution to the given differential equation for -2 x 8, with the initial05condition g (6) - 4. Find y g x .Copyright 2002 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.4GO ON TO THE NEXT PAGE.

2002 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B) 1 -1 x is xn with interval of convergence 1 x 1.1(a) Find the Maclaurin series for ln and determine the interval of convergence. n6. The Maclaurin series for lnn 11 3x (b) Find the value of( -1) nÊ n .n 1(c) Give a value of p such thatis correct. Ên 1 (d) Give a value of p such thatcorrect.( -1) nnp converges, butÊ n2 p1diverges. Give reasons why your value of pn 1 11Ê n p diverges, but Ê n 2 p converges. Give reasons why your value of p isn 1n 1END OF EXAMINATIONCopyright 2002 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.5

AP Calculus BC2003 Free-Response QuestionsThe materials included in these files are intended for use by AP teachersfor course and exam preparation; permission for any other use must besought from the Advanced Placement Program . Teachers may reproduce them, inwhole or in part, in limited quantities for noncommercial, face-to-face teachingpurposes. This permission does not apply to any third-party copyrights containedherein. This material may not be mass distributed, electronically or otherwise.These materials and any copies made of them may not be resold, and thecopyright notices must be retained as they appear here.These materials were produced by Educational Testing Service (ETS ), which develops and administers the examinations of the Advanced PlacementProgram for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and theirprograms, services, and employment policies are guided by that principle.The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college and opportunity.Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, theCollege Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges through major programs and services incollege admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT , thePSAT/NMSQT , and the Advanced Placement Program (AP ). The College Board is committed to the principles of equity andexcellence, and that commitment is embodied in all of its programs, services, activities, and concerns.For further information, visit www.collegeboard.comCopyright 2003 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams,APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board.AP Central is a trademark owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned by theCollege Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks ofEducational Testing Service. Other products and services may be trademarks of their respective owners.For the College Board’s online home for AP professionals, visit AP Central at apcentral.collegeboard.com.

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONSCALCULUS BCSECTION II, Part ATime—45 minutesNumber of problems—3A graphing calculator is required for some problems or parts of problems.1. Let R be the shaded region bounded by the graphs of y as shown in the figure above.x and y e 3x and the vertical line x 1,(a) Find the area of R.(b) Find the volume of the solid generated when R is revolved about the horizontal line y 1.(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis isa rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.Copyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.GO ON TO THE NEXT PAGE.2

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS2. A particle starts at point A on the positive x-axis at time t 0 and travels along the curve from A to B to C05to D, as shown above. The coordinates of the particle’s position x (t ), y(t ) are differentiable functions of t,dyptp t 1dxwhere x (t ) sinand y (t ) is not explicitly given. At time t 9, the - 9cosdt62dtparticle reaches its final position at point D on the positive x-axis. (a) At point C, isdydxpositive? At point C, ispositive? Give a reason for each answer.dtdt(b) The slope of the curve is undefined at point B. At what time t is the particle at point B ?05(c) The line tangent to the curve at the point x (8), y(8) has equation y 5x - 2. Find the velocity vector9and the speed of the particle at this point.(d) How far apart are points A and D, the initial and final positions, respectively, of the particle?Copyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.GO ON TO THE NEXT PAGE.3

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS5y and the curve C given by x 1 y 2 . Let S be the3shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P.3. The figure above shows the graphs of the line x (a) Find the coordinates of point P and the value ofdxfor curve C at point P.dy(b) Set up and evaluate an integral expression with respect to y that gives the area of S.(c) Curve C is a part of the curve x 2 - y 2 1. Show that x 2 - y 2 1 can be written as the polar equation1.r2 2cos q - sin 2 q(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle q thatrepresents the area of S.END OF PART A OF SECTION IICopyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.4

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONSCALCULUS BCSECTION II, Part BTime—45 minutesNumber of problems—3No calculator is allowed for these problems.054. Let f be a function defined on the closed interval -3 x 4 with f 0 3. The graph of f ′, the derivativeof f, consists of one line segment and a semicircle, as shown above.(a) On what intervals, if any, is f increasing? Justify your answer.(b) Find the x-coordinate of each point of inflection of the graph of f on the open interval -3 x 4. Justifyyour answer.0 5(c) Find an equation for the line tangent to the graph of f at the point 0, 3 .05(d) Find f ( -3) and f 4 . Show the work that leads to your answers.Copyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.GO ON TO THE NEXT PAGE.5

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS5. A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth ofthe coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume Vof coffee in the pot is changing at the rate of -5p h cubic inches per second. (The volume V of a cylinder withradius r and height h is V pr 2 h. )(a) Show thatdhh .dt5(b) Given that h 17 at time t 0 , solve the differential equationdhh for h as a function of t.dt5(c) At what time t is the coffeepot empty?6. The function f is defined by the power seriesf ( x) for all real numbers x.05 ( -1) n x 2 nÊ (2n 1) !n 0 1-( -1) n x 2 nx2x4x6 L L3!7!5!(2n 1) !05(a) Find f 0 and f 0 . Determine whether f has a local maximum, a local minimum, or neither at x 0.Give a reason for your answer.(b) Show that 1 -11.approximates f (1) with error less than3!10005(c) Show that y f x is a solution to the differential equation xy y cos x.END OF EXAMINATIONCopyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.6

AP Calculus BC2003 Free-Response QuestionsForm BThe materials included in these files are intended for use by AP teachersfor course and exam preparation; permission for any other use must besought from the Advanced Placement Program . Teachers may reproduce them, inwhole or in part, in limited quantities for noncommercial, face-to-face teachingpurposes. This permission does not apply to any third-party copyrights containedherein. This material may not be mass distributed, electronically or otherwise.These materials and any copies made of them may not be resold, and thecopyright notices must be retained as they appear here.These materials were produced by Educational Testing Service (ETS ), which develops and administers the examinations of the Advanced PlacementProgram for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and theirprograms, services, and employment policies are guided by that principle.The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college and opportunity.Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, theCollege Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges through major programs and services incollege admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT , thePSAT/NMSQT , and the Advanced Placement Program (AP ). The College Board is committed to the principles of equity andexcellence, and that commitment is embodied in all of its programs, services, activities, and concerns.For further information, visit www.collegeboard.comCopyright 2003 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams,APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board.AP Central is a trademark owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned by theCollege Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks ofEducational Testing Service. Other products and services may be trademarks of their respective owners.For the College Board’s online home for AP professionals, visit AP Central at apcentral.collegeboard.com.

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)CALCULUS BCSECTION II, Part ATime—45 minutesNumber of problems—3A graphing calculator is required for some problems or parts of problems.1. Let f be the function given by f ( x ) 4 x 2 - x 3 , and let l be the line y 18 - 3 x, where l is tangent to thegraph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded bythe graph of f, the line l, and the x-axis, as shown above.(a) Show that l is tangent to the graph of y f ( x ) at the point x 3.(b) Find the area of S.(c) Find the volume of the solid generated when R is revolved about the x-axis.Copyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.GO ON TO THE NEXT PAGE.2

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)052. The figure above shows the graphs of the circles x 2 y 2 2 and x - 10 50 52 y 2 1. The graphs intersect atthe points 1, 1 and 1, -1 . Let R be the shaded region in the first quadrant bounded by the two circles and thex-axis.(a) Set up an expression involving one or more integrals with respect to x that represents the area of R.(b) Set up an expression involving one or more integrals with respect to y that represents the area of R.(c) The polar equations of the circles are r 2 and r 2 cos q , respectively. Set up an expressioninvolving one or more integrals with respect to the polar angle q that represents the area of R.Distancex(mm)060120180240300360DiameterB( x )(mm)243028302624263. A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table abovegives the measurements of the diameter of the blood vessel at selected points along the length of the bloodvessel, where x represents the distance from one end of the blood vessel and B( x ) is a twice-differentiablefunction that represents the diameter at that point.(a) Write an integral expression in terms of B( x ) that represents the average radius, in mm, of the blood vesselbetween x 0 and x 360.(b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemannsum with three subintervals of equal length. Show the computations that lead to your answer.(c) Using correct units, explain the meaning of pI B( x) 22751252dx in terms of the blood vessel.(d) Explain why there must be at least one value x, for 0 x 360, such that B ( x ) 0.END OF PART A OF SECTION IICopyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.3

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)CALCULUS BCSECTION II, Part BTime—45 minutesNumber of problems—3No calculator is allowed for these problems.4. A particle moves in the xy-plane so that the position of the particle at any time t is given byx (t ) 2e 3t e -7t and y(t ) 3e 3t - e -2 t .(a) Find the velocity vector for the particle in terms of t, and find the speed of the particle at time t 0.(b) Finddydyin terms of t, and find lim.dxt dx(c) Find each value t at which the line tangent to the path of the particle is horizontal, or explain why noneexists.(d) Find each value t at which the line tangent to the path of the particle is vertical, or explain why none exists.I5. Let f be a function defined on the closed interval 0, 7 . The graph of f, consisting of four line segments,is shown above. Let g be the function given by g( x ) (a) Find g(3), g (3), and g (3).x2f (t ) dt.(b) Find the average rate of change of g on the interval 0 x 3.(c) For how many values c, where 0 c 3, is g (c) equal to the average rate found in part (b) ? Explainyour reasoning.(d) Find the x-coordinate of each point of inflection of the graph of g on the interval 0 x 7. Justify youranswer.Copyright 2003 by College Entrance Examination Board. All rights reserved.Available to AP professionals at apcentral.collegeboard.com and tostudents and parents at www.collegeboard.com/apstudents.GO ON TO THE NEXT PAGE.4

2003 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)6. The function f has a Taylor series about x 2 that converges to f ( x ) for all x in the interval of convergence.(n 1)!for n 1, and f (2 ) 1.The nth derivative of f at x 2 is given by f (n ) (2) 3n(a) Write the first four terms and the general term of the Taylor se

2. The rate at which people enter an amusement park on a given day is modeled by the function E defined by Et tt 15600 2 24 160. The rate at which people leave the same amusement park on the same day is modeled by the function L defined by Lt tt 9890 2 38 370. Both Et and Lt are measured in people per hour a