AP Calculus AB 2001 Scoring Guidelines

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1 P Calculus Scing Guidelines The materials included in these files are intended f non-commercial use by P teachers f course and eam preparation; permission f any other use must be sought from the dvanced Placement Program. Teachers may reproduce them, in whole in part, in limited quantities, f face-to-face teaching purposes but may not mass distribute the materials, electronically otherwise. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party copyrights contained herein. These materials were produced by Educational Testing Service (ETS), which develops and administers the eaminations of the dvanced Placement Program f the College oard. The College oard and Educational Testing Service (ETS) are dedicated to the principle of equal opptunity, and their programs, services, and employment policies are guided by that principle. The College oard is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opptunity. Founded in 9, the association is composed of me than,9 schools, colleges, universities, and other educational ganizations. Each year, the College oard serves over three million students and their parents,, high schools, and,5 colleges, through maj programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. mong its best-known programs are the ST, the PST/NMSQT, the dvanced Placement Program (P ), and Pacesetter. The College oard is committed to the principles of equity and ecellence, and that commitment is embodied in all of its programs, services, activities, and concerns. Copyright by College Entrance Eamination oard. ll rights reserved. College oard, dvanced Placement Program, P, and the acn logo are registered trademarks of the College Entrance Eamination oard.

2 SCORING GUIDELINES Question Let R and S be the regions in the first quadrant shown in the figure above. The region R is bounded by the -ais and the graphs of y Г and y tan. The region S is bounded by the y-ais and the graphs of y Г and y tan. (a) Find the area of R. (b) Find the area of S. (c) Find the volume of the solid generated when S is revolved about the -ais. Point of intersection Г tan at (, ) (.955,.575) (a) rea R = rea R = Г =.79 tan / Г ( Гy) Г tan y =.79 : : limits : integrand : answer =.79 rea R = Г Г Г Г tan (b) rea S = rea S = Г Г tan =.. Г / tan y ( Гy) =.. : : limits : integrand : answer rea S / / Г = ( Гy) Г ( Гy) Г tan y =.. (c) Volume = Г Г tan = : limits and constant : : integrand : answer Copyright by College Entrance Eamination oard. ll rights reserved.

3 SCORING GUIDELINES Question The temperature, in degrees Celsius ( C), of the water in a pond is a t differentiable function W of time t. The table above shows the water (days) temperature as recded every days over a 5-day period. (a) Use data from the table to find an approimation f W = (). Show the 9 computations that lead to your answer. Indicate units of measure. (b) pproimate the average temperature, in degrees Celsius, of the water 5 over the time interval > t > 5 days by using a trapezoidal approimation with subintervals of length t days. ( t /) (c) student proposes the function P, given by Pt ( ) te Г, as a model f the Wt () ( C) temperature of the water in the pond at time t, where t is measured in days and Pt () is measured in degrees Celsius. Find P= (). Using appropriate units, eplain the meaning of your answer in terms of water temperature. (d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of Pt () over the time interval > t > 5 days. 8 (a) Difference quotient; e.g. W(5) ГW() W = () N Г 5 Г C/day W() ГW(9) W = () N Г Г 9 C/day : : difference quotient : answer (with units) W(5) ГW(9) W = () N Г 5 Г 9 C/day (b) () (8) () () 7.5 verage temperature N (7.5) 5. C 5 : : trapezoidal method : answer (c) P= () e Г te Гt/ Гt/ t e Г Г Г.59 C/day : : P = () (with without units) : interpretation This means that the temperature is decreasing at the rate of.59 C/day when t = days. (d) 5 5 Гt / te dt C : : integrand : limits and average value constant : answer Copyright by College Entrance Eamination oard. ll rights reserved.

4 SCORING GUIDELINES Question car is traveling on a straight road with velocity 55 ft/sec at time t =. F > t > 8 seconds, the car s acceleration at (), in ft/sec, is the piecewise linear function defined by the graph above. (a) Is the velocity of the car increasing at t = seconds? Why why not? (b) t what time in the interval > t > 8, other than t =, is the velocity of the car 55 ft/sec? Why? (c) On the time interval > t > 8, what is the car s absolute maimum velocity, in ft/sec, and at what time does it occur? Justify your answer. (d) t what times in the interval > t > 8, if any, is the car s velocity equal to zero? Justify your answer. (a) Since v= () a() and a() 5, the velocity is increasing at t =. : answer and reason (b) t time t = because v() Гv() a( t) dt. : t : : reason (c) The absolute maimum velocity is 5 ft/sec at t =. The absolute maimum must occur at t = at an endpoint. v() 55 a( t) dt 8 55 (5) ()(5) 5 v() at () dt so v(8) v() : : t : absolute maimum velocity : identifies t and t 8 as candidates indicates that v increases, decreases, then increases : eliminates t 8 (d) The car s velocity is never equal to. The absolute minimum occurs at t = where v() 5 a( t) dt 5 Г5. : answer : : reason Copyright by College Entrance Eamination oard. ll rights reserved.

5 SCORING GUIDELINES Question Let h be a function defined f all L such that h() Г and the derivative of h is given Г by h= ( ) f all L. (a) Find all values of f which the graph of h has a hizontal tangent, and determine whether h has a local maimum, a local minimum, neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation f the line tangent to the graph of h at =. (d) Does the line tangent to the graph of h at = lie above below the graph of h f? Why? (a) h= ( ) at h= ( ) + und + Г : : : analysis : conclusions Г > not dealing with discontinuity at Local minima at Г and at (b) h== ( ) f all L. Therefe, the graph of h is concave up f all L. : : h== ( ) : h== ( ) : answer (c) Г 7 h= () 7 y ( Г ) : tangent line equation (d) The tangent line is below the graph because the graph of h is concave up f. : answer with reason Copyright by College Entrance Eamination oard. ll rights reserved. 5

6 cubic polynomial function f is defined by P CLCULUS SCORING GUIDELINES Question 5 f ( ) a b k where a, b, and k are constants. The function f has a local minimum at Г, and the graph of f has a point of inflection at Г. (a) Find the values of a and b. (b) If f ( ) =, what is the value of k? (a) f= () a b f== () a f= ( Г) Гa b f== ( Г) Г8 a a b Г a 5 : : f= ( ) : f== ( ) : f = ( Г) : f == ( Г) : a, b (b) k 8 8 k 7 k 7 k k 5 : antidifferentiation < Г > each err : : epression in k : k Copyright by College Entrance Eamination oard. ll rights reserved.

7 SCORING GUIDELINES Question The function f is differentiable f all real numbers. The point, is on the graph of y f( ), and the slope at each point y, on the graph is given by y Г. (a) Find and evaluate it at the point,. (b) Find y f( ) by solving the differential equation y Г with the initial condition f (). (a) y ( Г ) Г y y ( Г ) Г y = Г Г, 8 : : < Г > product rule : value at, chain rule err (b) y ( Г ) Г Г C y Г 8 Г 9 C 9 C C Г : : separates variables : antiderivative of term : antiderivative of term : constant of integration : uses initial condition f () : solves f y y Г Note: ma / [-----] if no constant of integration Note: / if no separation of variables Copyright by College Entrance Eamination oard. ll rights reserved. 7

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