AP Calculus BC 2010 Scoring Guidelines

1 AP Calculus BC 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT , the PSAT/NMSQT , the Advanced Placement Program (AP ), SpringBoard and ACCUPLACER.

2 The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. 2010 The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: Visit the College Board on the Web: AP Central is the official online home for the AP Program: AP Calculus BC 2010 Scoring Guidelines Question 1 2010 The College Board.

3 Visit the College Board on the Web: There is no snow on Janet s driveway when snow begins to fall at midnight. From midnight to 9 , snow accumulates on the driveway at a rate modeled by ()cos7tftte= cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 () The rate (),gt in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by ()0for0 6125 for 67108 for 79 .tgttt < = < (a) How many cubic feet of snow have accumulated on the driveway by 6 (b) Find the rate of change of the volume of snow on the driveway at 8 (c) Let ()ht represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight.

4 Express h as a piecewise-defined function with domain (d) How many cubic feet of snow are on the driveway at 9 (a) () dt= or cubic feet 2 : { 1 : integral 1 : answer (b) Rate of change is ()() = or cubic feet per hour. 1 : answer (c) ()00h= For 06,t< ()( )( ) s dsds=+=+= For 67,t< ()( )( )()66601251256 .tththg s dsdst=+=+= For 79,t< ()( )( )()7712510781251087 .ttgs dsdshtht=+=+=+ Thus, ()()()0for061256for 671251087for 79thttttt = < + < 3 : ()()() 1 : for 06 1 : for 67 1 : for 79htthtthtt < < (d) Amount of snow is ()( ) dt h = or cubic feet. 3 : () 1 : integral 1 : 9 1 : answerh AP Calculus BC 2010 Scoring Guidelines Question 2 2010 The College Board.}

5 Visit the College Board on the Web: t (hours) 0 2 5 7 8 ()Et (hundreds of entries) 0 4 13 21 23 A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ()0t= and 8 () The number of entries in the box t hours after noon is modeled by a differentiable function E for Values of (),Et in hundreds of entries, at various times t are shown in the table above. (a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time Show the computations that lead to your answer.

6 (b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of () Using correct units, explain the meaning of ()8018 Etdt in terms of the number of entries. (c) At 8 , volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where ()3230298976 Ptttt= + hundreds of entries per hour for According to the model, how many entries had not yet been processed by midnight ()12 ?t= (d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer. (a) ()()()756475 EEE = hundred entries per hour 1 : answer (b) ()()()()()()()()()80022557 7812 3 2 1 or 10 dtEEEEEE EE ++++ +++ = ()8018 Etdt is the average number of hundreds of entries in the box between noon and 8 3 : 1 : trapezoidal sum 1 : approximation 1 : meaning (c) ()1282323 167Pt dt = = hundred entries 2 : { 1 : integral 1 : answer (d) ()0Pt = when and () Entries are being processed most quickly at time 3 : () 1 : considers 0 1 : identifies candidates 1.}

7 Answer with justificationPt = AP Calculus BC 2010 Scoring Guidelines Question 3 2010 The College Board. Visit the College Board on the Web: A particle is moving along a curve so that its position at time t is () ()(),,xtyt where ()248xttt= + and ()yt is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that = (a) Find the speed of the particle at time 3t= seconds. (b) Find the total distance traveled by the particle for 04t seconds. (c) Find the time t, 04,t 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?

8 Give a reason for your answer. (d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle s path at that point (iii) The y-coordinate of that point, given ()123ye=+ (a) Speed ()()()() =+= meters per second 1 : answer (b) ()24xtt = Distance ()() = + = or meters 2 : {1 : integral1 : answer (c) 0dy dtdydxdx dt== when 310tte = and 240t This occurs at Since () ,x > the particle is moving toward the right at time or 3 : 1 : considers 0 1 : or.}

9 Direction of motion with reasondydxt = = (d) ()5xt= at 1t= and 3t= At time 1,t= the slope is dtdydxdx dt==== At time 3,t= the slope is dtdydxdx dt==== ()( )3211334dyyydtedt==++= 3 : 1 : 1 and 3 1 : slopes1 : -coordinatetty== AP Calculus BC 2010 Scoring Guidelines Question 4 2010 The College Board. Visit the College Board on the Web: Let R be the region in the first quadrant bounded by the graph of 2,yx= the horizontal line 6,y= and the y-axis, 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 (c) Region R is the base of a solid.

10 For each y, where 06,y the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. (a) Area ()()9932004626183xxxdxxx=== = = 3 : 1 : integrand1 : antiderivative 1 : answer (b) Volume ()()()92207276xdx = 3 : { 2 : integrand1 : limits and constant (c) Solving 2yx= for x yields Each rectangular cross section has area 16yyy = Volume 640316ydy= 3 : {2 : integrand1 : answer AP Calculus BC 2010 Scoring Guidelines Question 5 2010 The College Board.}}