AP Calculus AB 2006 Scoring Guidelines Form B


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1 AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success ad opportuity. Fouded i 9, the associatio is composed of more tha 5, schools, colleges, uiversities, ad other educatioal orgaizatios. Each year, the College Board serves seve 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 bestkow 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. 6 The College Board. All rights reserved. College Board, AP Cetral, APCD, Advaced Placemet Program, AP, AP Vertical Teams, PreAP, SAT, ad the acor logo are registered trademarks of the College 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 Board. PSAT/NMSQT is a registered trademark of the College Board ad Natioal Merit Scholarship Corporatio. All 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: apcetral.collegeboard.com.
2 6 SCORING GUIDELINES (Form B) Questio x x x Let f be the fuctio give by f = + cos x. Let R be the shaded regio i the secod quadrat bouded by the graph of f, ad let S be the shaded regio bouded by the graph of f ad lie l, the lie taget to the graph of f at x =, as show above. (a) Fid the area of R. (b) Fid the volume of the solid geerated whe R is rotated about the horizotal lie y =. (c) Write, but do ot evaluate, a itegral expressio that ca be used to fid the area of S. For x <, f( x ) = whe x =.7. Let P =.7. (a) Area of R = f dx =.9 P : { : itegral : aswer ( ) (b) Volume = ( f + ) dx = 59.6 P : : limits ad costat : itegrad : aswer (c) The equatio of the taget lie l is y = x. The graph of f ad lie l itersect at A = A Area of = (( ) ( )) S x f x dx : : taget lie : itegrad : limits 6 The College Board. All rights reserved.
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.
4 6 SCORING GUIDELINES (Form B) Questio The figure above is the graph of a fuctio of x, which models the height of a skateboard ramp. The fuctio meets the followig requiremets. (i) At x =, the value of the fuctio is, ad the slope of the graph of the fuctio is. (ii) At x =, the value of the fuctio is, ad the slope of the graph of the fuctio is. (iii) Betwee x = ad x =, the fuctio is icreasig. (a) Let f = ax, where a is a ozero costat. Show that it is ot possible to fid a value for a so that f meets requiremet (ii) above. x (b) Let g = cx, where c is a ozero costat. Fid the value of c so that g meets requiremet (ii) 6 above. Show the work that leads to your aswer. (c) Usig the fuctio g ad your value of c from part (b), show that g does ot meet requiremet (iii) above. x (d) Let hx ( ) =, where k is a ozero costat ad is a positive iteger. Fid the values of k ad so that k h meets requiremet (ii) above. Show that h also meets requiremets (i) ad (iii) above. (a) f ( ) = implies that implies that a = ad f ( ) = a( ) = 6 a =. Thus, f caot satisfy (ii). 8 : a = or a = : 6 8 : shows a does ot work (b) g( ) = 6c = implies that c =. Whe, ( ) c = g ( ) = c( ) = ( )( 6) = 6 x (c) g = x = x( x ) 8 g < for < x <, so g does ot satisfy (iii). (d) h( ) = = implies that = k. k h ( ) = = = = gives = ad k x h = h( ) =. 56 x h = h ( ) = ad h > for < x <. 56 k = = 56. : value of c : g : : explaatio : = k  : : = k : values for k ad : verificatios 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
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