AP Calculus BC 2005 Scoring Guidelines

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1 AP Calculus BC 5 Scorig Guidelies The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad opportuity. Fouded i 9, the associatio is composed of more tha 4,7 schools, colleges, uiversities, ad other educatioal orgaizatios. Each year, the College Board serves over three ad a half millio studets ad their parets,, high schools, ad,5 colleges through major programs ad services i college admissios, guidace, assessmet, fiacial aid, erollmet, ad teachig ad learig. Amog its best-kow programs are the SAT, the PSAT/NMSQT, ad the Advaced Placemet Program (AP ). The College Board is committed to the priciples of excellece ad equity, ad that commitmet is embodied i all of its programs, services, activities, ad cocers. Copyright 5 by College Board. All rights reserved. College Board, AP Cetral, APCD, Advaced Placemet Program, AP, AP Vertical Teams, Pre-AP, SAT, ad the acor logo are registered trademarks of the College Etrace Examiatio Board. Admitted Class Evaluatio Service, CollegeEd, Coect to college success, MyRoad, SAT Professioal Developmet, SAT Readiess Program, ad Settig the Corerstoes are trademarks owed by the College Etrace Examiatio Board. PSAT/NMSQT is a registered trademark of the College Etrace Examiatio Board ad Natioal Merit Scholarship Corporatio. Other products ad services may be trademarks of their respective owers. Permissio to use copyrighted College Board materials may be requested olie at: Visit the College Board o the Web: AP Cetral is the official olie home for the AP Program ad Pre-AP: apcetral.collegeboard.com.

2 5 SCORING GUIDELINES Questio x Let f ad g be the fuctios give by f ( x) = + si ( x) ad gx ( ) = 4. Let 4 R be the shaded regio i the first quadrat eclosed by the y-axis ad the graphs of f ad g, ad let S be the shaded regio i the first quadrat eclosed by the graphs of f ad g, as show i the figure above. (a) Fid the area of R. (b) Fid the area of S. (c) Fid the volume of the solid geerated whe S is revolved about the horizotal lie y =. x f ( x) = g( x) whe + si ( x) = 4. 4 f ad g itersect whe x =.7 ad whe x =. Let a =.7. a (a) ( g ( x ) f ( x )) =.64 or.65 : : limits : itegrad : aswer (b) ( f ( x ) g ( x )) =.4 a : : limits : itegrad : aswer ( ) (c) ( f( x) + ) ( g( x) + ) = 4.55 or a : { : itegrad : limits, costat, ad aswer Copyright 5 by College Board. All rights reserved.

3 5 SCORING GUIDELINES Questio The curve above is draw i the xy-plae ad is described by the equatio i polar coordiates r = θ + si ( θ) for θ, where r is measured i meters ad θ is measured i radias. The derivative of r with respect to θ is dr give by cos( θ ). dθ = + (a) Fid the area bouded by the curve ad the x-axis. (b) Fid the agle θ that correspods to the poit o the curve with x-coordiate. (c) For < θ < dr, is egative. What does this fact say about r? What does this fact say about the curve? dθ (d) Fid the value of θ i the iterval θ that correspods to the poit o the curve i the first quadrat with greatest distace from the origi. Justify your aswer. (a) Area = r dθ = ( θ + si( θ) ) dθ = 4. : : limits ad costat : itegrad : aswer (b) = r cos( θ) = ( θ + si( θ) ) cos( θ) θ =.76 : { : equatio : aswer dr (c) Sice dθ < for < θ <, r is decreasig o this iterval. This meas the curve is gettig closer to the origi. : iformatio about r : { : iformatio about the curve (d) The oly value i, where dr dθ = is θ =. θ r.9.57 : : θ = or.47 : aswer with justificatio The greatest distace occurs whe θ =. Copyright 5 by College Board. All rights reserved.

4 5 SCORING GUIDELINES Questio Distace x (cm) Temperature T( x ) ( C) A metal wire of legth cetimeters (cm) is heated at oe ed. The table above gives selected values of the temperature T( x ), i degrees Celsius ( C, ) of the wire x cm from the heated ed. The fuctio T is decreasig ad twice differetiable. (a) Estimate T ( 7. ) Show the work that leads to your aswer. Idicate uits of measure. (b) Write a itegral expressio i terms of T( x ) for the average temperature of the wire. Estimate the average temperature of the wire usig a trapezoidal sum with the four subitervals idicated by the data i the table. Idicate uits of measure. (c) Fid T ( x), ad idicate uits of measure. Explai the meaig of ( ) T x i terms of the temperature of the wire. (d) Are the data i the table cosistet with the assertio that T ( x) > for every x i the iterval < x <? Explai your aswer. (a) T( ) T( 6) = = Ccm 6 : aswer (b) ( ) T x Trapezoidal approximatio for T( x) : A = Average temperature C A = (c) T ( x) = T( ) T( ) = 55 = 45 C The temperature drops 45 C from the heated ed of the wire to the other ed of the wire., 5 is 7 9 = , 6 is 6 7 =. 6 5 T c = 5.75 for some c i the iterval (, 5 ) T c = for some c i the iterval ( 5, 6 ). It follows that c, c. Therefore T (d) Average rate of chage of temperature o [ ] Average rate of chage of temperature o [ ] No. By the MVT, ( ) ad ( ) T must decrease somewhere i the iterval ( ) is ot positive for every x i [, ]. : : T( x) : trapezoidal sum : aswer : { : value : meaig : two slopes of secat lies : { : aswer with explaatio Uits of Ccmi (a), ad C i (b) ad (c) : uits i (a), (b), ad (c) Copyright 5 by College Board. All rights reserved. 4

5 5 SCORING GUIDELINES Questio 4 dy Cosider the differetial equatio x y. = (a) O the axes provided, sketch a slope field for the give differetial equatio at the twelve poits idicated, ad sketch the solutio curve that passes through the poit (, ). (Note: Use the axes provided i the pik test booklet.), has a local miimum at ( ) (b) The solutio curve that passes through the poit ( ) x = l. What is the y-coordiate of this local miimum? (c) Let y = f ( x) be the particular solutio to the give differetial equatio with the iitial coditio f ( ) =. Use Euler s method, startig at x = with two steps of equal size, to approximate f (.4 ). Show the work that leads to your aswer. d y (d) Fid i terms of x ad y. Determie whether the approximatio foud i part (c) is less tha or greater tha f (.4 ). Explai your reasoig. (a) : : zero slopes : ozero slopes : curve through (, ) dy (b) = whe x y = The y-coordiate is ( ) l. dy : sets = : : aswer (c) f (.) f ( ) + f ( )(.) = + ( )(.) =. f(.4) f(.) + f (.)(.). + (.6)(.) =.5 : : Euler's method with two steps : Euler approximatio to f (.4) (d) d y dy = = x + y d y is positive i quadrat II because x < ad y >..5 < f (.4) sice all solutio curves i quadrat II are cocave up. : d y : : aswer with reaso Copyright 5 by College Board. All rights reserved. 5

6 5 SCORING GUIDELINES Questio 5 A car is travelig o a straight road. For t 4 secods, the car s velocity vt (), i meters per secod, is modeled by the piecewise-liear fuctio defied by the graph above. 4 4 (a) Fid vt () dt. Usig correct uits, explai the meaig of () vt dt. (b) For each of v ( 4) ad v ( ), fid the value or explai why it does ot exist. Idicate uits of measure. (c) Let at () be the car s acceleratio at time t, i meters per secod per secod. For < t < 4, write a piecewise-defied fuctio for at (). (d) Fid the average rate of chage of v over the iterval t. Does the Mea Value Theorem guaratee a value of c, for < c <, such that v ( c) is equal to this average rate of chage? Why or why ot? 4 (a) vt () dt= ( 4)( ) + ( )( ) + ( )( ) = 6 The car travels 6 meters i these 4 secods. : { : value : meaig with uits (b) v ( 4) does ot exist because vt () v( 4) vt () v( 4) lim = 5 = lim. t 4 t 4 + t 4 t 4 5 v ( ) = = m sec 6 4 : v ( 4 ) does ot exist, with explaatio : : v ( ) : uits (c) 5 if < t < 4 at () = if 4 < t < 6 5 if 6 < t < 4 at () does ot exist at t = 4 ad t = 6. : 5 : fids the values 5,, : idetifies costats with correct itervals (d) The average rate of chage of v o [, ] is v( ) v( ) 5 = m sec. 6 No, the Mea Value Theorem does ot apply to v o [, ] because v is ot differetiable at t = 6. : : average rate of chage of v o [, ] : aswer with explaatio Copyright 5 by College Board. All rights reserved. 6

7 5 SCORING GUIDELINES Questio 6 Let f be a fuctio with derivatives of all orders ad for which f ( ) = 7. Whe is odd, the th derivative ( of f at x = is. Whe is eve ad, the th derivative of f at x = is give by ) (! ) f ( ) =. (a) Write the sixth-degree Taylor polyomial for f about x =. (b) I the Taylor series for f about x =, what is the coefficiet of ( x ) for? (c) Fid the iterval of covergece of the Taylor series for f about x =. Show the work that leads to your aswer. (a) P 6( x!! 5! ) = 7 + ( ) ( ) ( )! 4 4! 6 x + x + 6! x 4 6 : polyomial about x = : P6( x) : each icorrect term max for all extra terms, +, misuse of equality (b) ( )! = ( )! : coefficiet ( ) (c) The Taylor series for f about x = is f ( x ) 7 ( ) = +. x = ( ) ( ) ( ) ( + x ) + + L = lim ( x ) ( x ) = lim ( x ) = ( + ) 9 L < whe x <. Thus, the series coverges whe < x < 5. Whe x = 5, the series is 7 + = 7 +, = = which diverges, because, the harmoic series, diverges. = Whe x =, the series is ( ) 7 + = 7 +, = = which diverges, because, the harmoic series, diverges. = The iterval of covergece is (, 5 ). 5 : : sets up ratio : computes limit of ratio : idetifies iterior of iterval of covergece : cosiders both edpoits : aalysis/coclusio for both edpoits Copyright 5 by College Board. All rights reserved. 7

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success