AP Calculus AB 2011 Scoring Guidelines

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1 AP Calculus AB 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 9, the College Board is composed of more than 5,7 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 3, high schools, and 3,8 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. 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. 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: AP Central is the official online home for the AP Program: apcentral.collegeboard.com.

2 SCORING GUIDELINES Question For t 6, a particle is moving along the x-axis. The particle s position, x( t, is not explicitly given. t vt = sin e +. The acceleration of the particle is given by The velocity of the particle is given by ( ( t ( t cos( at = e e and x ( =. (a Is the speed of the particle increasing or decreasing at time t = 5.5? Give a reason for your answer. (b Find the average velocity of the particle for the time period t 6. (c Find the total distance traveled by the particle from time t = to t = 6. (d For t 6, the particle changes direction exactly once. Find the position of the particle at that time. (a v ( 5.5 =.5337, a ( 5.5 =.3585 : conclusion with reason The speed is increasing at time t = 5.5, because velocity and acceleration have the same sign. 6 (b Average velocity = vt ( =.99 6 : { : integral : answer 6 : { : integral (c Distance = vt ( =.573 : answer (d vt ( = when t = Let b = vt ( changes sign from positive to negative at time t = b. b xb ( = + vt ( =.3 or.35 : considers vt ( = : integral : answer The College Board.

3 SCORING GUIDELINES Question t (minutes H ( t (degrees Celsius As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t, where time t is measured in minutes and temperature H ( t is measured in degrees Celsius. Values of H ( t 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. (b Using correct units, explain the meaning of ( H t in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate (. H t (c Evaluate H ( t. Using correct units, explain the meaning of the expression in the context of this problem. (d At time t =, biscuits with temperature C 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.73t B ( t = 3.8 e. Using the given models, at time t =, how much cooler are the biscuits than the tea? H( 5 H( (a H ( = =.666 or.667 degrees Celsius per minute 3 (b ( H t is the average temperature of the tea, in degrees Celsius, over the minutes H t ( ( = 5.95 (c H ( t = H( H( = 3 66 = 3 The temperature of the tea drops 3 degrees Celsius from time t = to time t = minutes. (d B( = + B ( t = 3.875; H( B( = 8.87 The biscuits are 8.87 degrees Celsius cooler than the tea. : answer : meaning of expression : trapezoidal sum : estimate : value of integral : { : meaning of expression : integrand : uses B( = : answer The College Board.

4 SCORING GUIDELINES Question 3 Let R be the region in the first quadrant enclosed by the graphs of f = 8x and g = sin ( π x, as shown in the figure above. (a Write an equation for the line tangent to the graph of f at x =. (b Find the area of R. (c Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y =. 3 (a f ( = f = x, so ( f = 6 An equation for the tangent line is y = + ( x 6. : f : ( : answer (b Area = ( g f dx 3 = ( sin( π x 8x dx = cos ( π x x π = + 8 π : : integrand : antiderivative : answer ( (( 8 3 ( sin( π (c π ( f ( g dx = π x x dx : limits and constant { : integrand The College Board.

5 SCORING GUIDELINES Question The continuous function f is defined on the interval x 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. x Let g = x + f( t. (a Find g( 3. Find g and evaluate g ( 3. (b Determine the x-coordinate of the point at which g has an absolute maximum on the interval x 3. Justify your answer. (c Find all values of x on the interval < x < 3 for which the graph of g has a point of inflection. Give a reason for your answer. (d Find the average rate of change of f on the interval x 3. There is no point c, < c < 3, for which f ( c is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. 3 9π (a g( 3 = ( 3 + f( t = 6 g = + f g ( 3 = + f( 3 = : g( 3 : g : g ( 3 (b g = when f( x =. This occurs at x = g 5 > for < x < and g < for 5 < x < 3. 5 Therefore g has an absolute maximum at x =. 5. : considers g = : identifies interior candidate : answer with justification (c g = f changes sign only at x =. Thus the graph of g has a point of inflection at x =. : answer with reason (d The average rate of change of f on the interval x 3 is f( 3 f( =. 3 ( 7 To apply the Mean Value Theorem, f must be differentiable at each point in the interval < x < 3. However, f is not differentiable at x = 3 and x =. : average rate of change : { : explanation The College Board.

6 SCORING GUIDELINES Question 5 At the beginning of, a landfill contained tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation = ( W 3 for the next years. W is measured in tons, and t is measured in years from 5 the start of. (a Use the line tangent to the graph of W at t = to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of (time t =. (b Find in terms of W. Use to determine whether your answer in part (a is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time t =. (c Find the particular solution W = W( t to the differential equation = ( W 3 with initial 5 condition W ( =. (a = ( W ( 3 = ( 3 = t= 5 5 The tangent line is y = + t. ( ( W + = tons : at : t = : answer (b = = ( W 3 and W 5 65 Therefore > on the interval t. The answer in part (a is an underestimate. : : : answer with reason (c = ( W 3 5 = W 3 5 ln W 3 = t + C 5 ln ( 3 = ( + C ln ( = C 5 W 3 = e t 5 t 5 W( t = 3 + e, t 5 : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for W Note: max 5 [----] if no constant of integration Note: 5 if no separation of variables The College Board.

7 SCORING GUIDELINES Question 6 sin x for x Let f be a function defined by f = x e for x >. (a Show that f is continuous at x =. (b For x, express f as a piecewise-defined function. Find the value of x for which f = 3. (c Find the average value of f on the interval [, ]. (a lim ( sin x = x + x x lim e = f ( = So, lim f = f(. x : analysis Therefore f is continuous at x =. cos x for x (b f = < x e for x > cos x 3 for all values of x <. e x 3 = 3 when x = ln ( >. Therefore f 3 3 = for x = ( ln. : f : value of x (c f dx = f dx + f dx x = ( sin x dx + e dx x = x + cos x + e x x = ( 3 cos( + ( e + = = : x : ( sin x dx and e dx : antiderivatives : answer Average value = ( f x dx 3 = cos( e 8 8 The College Board.

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