AP Calculus AB 2006 Scoring Guidelines Form B

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3 6 SCORING GUIDELINES (Form B) Questio Let f be the fuctio defied for x with f ( ) = 5 ad f, the ( x ) first derivative of f, give by f = e si ( x ). The graph of y = f is show above. (a) Use the graph of f to determie whether the graph of f is cocave up, cocave dow, or either o the iterval.7 < x <.9. Explai your reasoig. (b) O the iterval x, fid the value of x at which f has a absolute maximum. Justify your aswer. (c) Write a equatio for the lie taget to the graph of f at x =. (a) O the iterval.7 < x <.9, f is decreasig ad thus f is cocave dow o this iterval. : { : aswer : reaso (b) f = whe x =,,,, K O [, ] f chages from positive to egative oly at. The absolute maximum must occur at x = or at a edpoit. f ( ) = 5 ( ) ( ) ( ) f = f + f x dx = f( ) = f( ) + f dx = : idetifies ad as cadidates - or - idicates that the graph of f : icreases, decreases, the icreases : justifies f( ) > f( ) : aswer This shows that f has a absolute maximum at x =. (c) f( ) = f( ) + f dx = 5.6 f ( ) = e.5 si( ) =.59 y 5.6 = (.59)( x ) : f ( ) expressio : itegral : : icludig f ( ) term : f ( ) : equatio 6 The College Board. All rights reserved.

5 6 SCORING GUIDELINES (Form B) Questio The rate, i calories per miute, at which a perso usig a exercise machie burs calories is modeled by the fuctio f. I the figure above, f () t = t + t + for t ad f is piecewise liear for t. (a) Fid f ( ). Idicate uits of measure. (b) For the time iterval t, at what time t is f icreasig at its greatest rate? Show the reasoig that supports your aswer. (c) Fid the total umber of calories bured over the time iterval 6 t 8 miutes. (d) The settig o the machie is ow chaged so that the perso burs f () t + c calories per miute. For this settig, fid c so that a average of 5 calories per miute is bured durig the time iterval 6 t 8. 5 (a) f ( ) = = calories/mi/mi : f ( ) ad uits (b) f is icreasig o [, ] ad o [, 6 ]. 5 9 O (, 6 ), f () t = = sice f has 6 costat slope o this iterval. O (, ), f () t = t + t ad f () t = t + = whe t =. This is where f has a maximum o [, ] sice f > o (, ) ad f < o (, ). O [, ], f is icreasig at its greatest rate whe t = because f ( ) = >. 8 (c) f() t dt = 69 ( ) + ( )( 9+ 5) + 5 ( ) 6 = calories 8 (d) We wat ( () ) 5. f t + c dt = 6 This meas + c = 5(). So, c =. OR Curretly, the average is = calories/mi. Addig c to f () t will shift the average by c. So c = to get a average of 5 calories/mi. : f o (, ) : shows f has a max at t = o (, ) : : shows for < t < 6, f () t < f ( ) : aswer : { : method : aswer : { : setup : value of c 6 The College Board. All rights reserved. 5

6 6 SCORING GUIDELINES (Form B) Questio 5 dy Cosider the differetial equatio = ( y ) cos ( x). dx (a) O the axes provided, sketch a slope field for the give differetial equatio at the ie poits idicated. (Note: Use the axes provided i the exam booklet.) (b) There is a horizotal lie with equatio y = c that satisfies this differetial equatio. Fid the value of c. (c) Fid the particular solutio y = f to the differetial equatio with the iitial coditio f () =. (a) : zero slopes : { : all other slopes (b) The lie y = satisfies the differetial equatio, so c =. : c = (c) dy = cos( x) dx ( y ) ( y ) = si( x) + C = si ( x ) + C y = si( ) + C = C = si ( x) + y y = si + y = for si + < x < 6 : : separates variables : atiderivatives : costat of itegratio : uses iitial coditio : aswer Note: max 6 [----] if o costat of itegratio Note: 6 if o separatio of variables 6 The College Board. All rights reserved. 6

7 6 SCORING GUIDELINES (Form B) Questio 6 t (sec) vt () ( ft sec ) at () ( ft sec ) A car travels o a straight track. Durig the time iterval t 6 secods, the car s velocity v, measured i feet per secod, ad acceleratio a, measured i feet per secod per secod, are cotiuous fuctios. The table above shows selected values of these fuctios. 6 (a) Usig appropriate uits, explai the meaig of vt () dti terms of the car s motio. Approximate 6 vt () dtusig a trapezoidal approximatio with the three subitervals determied by the table. (b) Usig appropriate uits, explai the meaig of at () dti terms of the car s motio. Fid the exact value of at () dt. (c) For < t < 6, must there be a time t whe vt () = 5? Justify your aswer. (d) For < t < 6, must there be a time t whe at () =? Justify your aswer. 6 (a) vt () dtis the distace i feet that the car travels from t = sec to t = 6 sec. Trapezoidal approximatio for 6 vt () dt: A = ( + ) 5 + ( )( 5) + ( )( ) = 85 ft (b) at () dtis the car s chage i velocity i ft/sec from t = sec to t = sec. a() t dt = v () t dt = v( ) v( ) = ( ) = 6 ft/sec (c) Yes. Sice v( 5) = < 5 < = v( 5 ), the IVT guaratees a t i ( 5, 5 ) so that vt () = 5. (d) Yes. Sice v( ) = v( 5 ), the MVT guaratees a t i (, 5 ) so that at () = v () t =. Uits of ft i (a) ad ft/sec i (b) : { : explaatio : value : { : explaatio : value : v( 5) < 5 < v( 5) : : Yes; refers to IVT or hypotheses : v( ) = v( 5) : : Yes; refers to MVT or hypotheses : uits i (a) ad (b) 6 The College Board. All rights reserved. 7

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

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a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x 0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio

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AP Calculus AB 2009 Free-Response Questions AP Calculus AB 2009 Free-Response Questions 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 Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two