AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
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1 SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the fuctio give by g ftdt. (a) Fid g (, ) g ( ), ad g ( ). (b) Fid the average rate of chage of g o the iterval. (c) For how may values c, where < c <, is g () c equal to the average rate foud i part (b)? Eplai your reasoig. (d) Fid the -coordiate of each poit of iflectio of the graph of g o the iterval < < 7. Justify your aswer. (a) g() f( t) dt ( 4 + ) g () f() 4 g() f() 4 : : g() : g() : g() (b) g() g() () ftdt 7 ()(4) + (4 + ) : : g() g() f( t) dt : aswer (c) There are two values of c. We eed 7 g( c) f( c) The graph of f itersects the lie places betwee ad. 7 y at two : aswer of : : reaso Note: / if aswer is by MVT (d) ad 5 because g f chages from icreasig to decreasig at, ad from decreasig to icreasig at 5. : ad 5 oly : : justificatio (igore discussio at 4) Copyright by College Etrace Eamiatio Board. All rights reserved. Available at apcetral.collegeboard.com. 6
2 SCORING GUIDELINES (Form B) Questio 6 Let g be the piecewise-liear fuctio defied o [, 4 ] whose graph is give above, ad let f g 4 cos. (a) Fid f d. Show the computatios that lead to your aswer. (b) Fid all -values i the ope iterval (, 4 ) for which f has a critical poit. (c) Let h gt dt. Fid h. 4 4 (a) ( cos ) 4 f d g d 6 si 6 : { : atiderivative : aswer (b) f g si + si for < < + + si for < < 4 f does ot eist at. For < <, f. For < < 4, f whe. ( ) d : cos d 4 : : g : : f has critical poits at ad. (c) h g h ( ) g( ) : h : : aswer The College Board. Visit the College Board o the Web:
3 9 SCORING GUIDELINES (Form B) Questio A cotiuous fuctio f is defied o the closed iterval 4 6. The graph of f cosists of a lie segmet ad a curve that is taget to the -ais at, as show i the figure above. O the iterval < < 6, the fuctio f is twice differetiable, with f >. (a) Is f differetiable at? Use the defiitio of the derivative with oe-sided limits to justify your aswer. (b) For how may values of a, 4 a 6, equal to? Give a reaso for your aswer. (c) Is there a value of a, 4 a 6, < is the average rate of chage of f o the iterval [ a,6] < for which the Mea Value Theorem, applied to the iterval [ a ] guaratees a value c, a < c < 6, at which f ( c)? Justify your aswer. (d) The fuctio g is defied by g f( t) dt for 4 6. O what itervals cotaied i [ 4, 6] is the graph of g cocave up? Eplai your reasoig. f( h) f (a) lim h h f( h) f lim < + h h Sice the oe-sided limits do ot agree, f is ot differetiable at. f( 6) f( a) (b) whe f( a) f( 6. ) There are 6 a two values of a for which this is true. (c) Yes, a. The fuctio f is differetiable o the iterval < < 6 ad cotiuous o 6. f( 6) f Also,. 6 6 By the Mea Value Theorem, there is a value c, < c < 6, such that f ( c). (d) g f, g f g > whe f > This is true for 4 < < ad < < 6.,6, : sets up differece quotiet at : { : aswer with justificatio : epressio for average rate of chage : { : aswer with reaso : aswers yes ad idetifies a : { : justificatio : g f : : cosiders g > : aswer 9 The College Board. All rights reserved. Visit the College Board o the Web:
4 8 SCORING GUIDELINES (Form B) Questio 4 The fuctios f ad g are give by f 4 + t dt ad g f( si ). (a) Fid f ad g. (b) Write a equatio for the lie taget to the graph of y g at. (c) Write, but do ot evaluate, a itegral epressio that represets the maimum value of g o the iterval. Justify your aswer. (a) f 4 + g f ( si ) cos 4 : : : f g 4 + ( si ) cos (b) g( ), g ( ) 6 Taget lie: y 6( ) : g or g : : taget lie equatio (c) For < <, g oly at. g g( ) + > g 4 t dt, is The maimum value of g o [ ] 4 + t dt. : : sets g : justifies maimum at : itegral epressio for g 8 The College Board. All rights reserved. Visit the College Board o the Web:
5 5 SCORING GUIDELINES (Form B) Questio 4 The graph of the fuctio f above cosists of three lie segmets. (a) Let g be the fuctio give by g f( t) dt. 4 For each of g(, ) g (, ) ad g (, ) fid the value or state that it does ot eist. (b) For the fuctio g defied i part (a), fid the -coordiate of each poit of iflectio of the graph of g o the ope iterval 4 < <. Eplai your reasoig. (c) Let h be the fuctio give by h f( tdt ). Fid all values of i the closed iterval 4 for which h. (d) For the fuctio h defied i part (c), fid all itervals o which h is decreasig. Eplai your reasoig. 5 (a) g( ) f( t) dt ( 5) 4 g ( ) f( ) g ( ) does ot eist because f is ot differetiable at. : : g( ) : g ( ) : g ( ) (b) g f chages from icreasig to decreasig at. : (oly) : : reaso (c),, : correct values each missig or etra value (d) h is decreasig o [, ] h f < whe f > : iterval : : reaso Copyright 5 by College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for AP studets ad parets). 5
6 Let be the lie taget to the graph of y where >, as show above. (a) Fid d i terms of. AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Questio 6 at the poit (, ), (b) Let T be the triagular regio bouded by, the -ais, ad the lie. Show that the area of T is. (c) Let S be the regio bouded by the graph of y, the lie, ad the -ais. Epress the area of S i terms of ad determie the value of that maimizes the area of S. (a) + d : + + : atiderivative of : aswer (b) Let b be the legth of the base of triagle T. b is the slope of lie, which is Area( T) b() : : slope of lie is : base of T is : shows area is (c) Area( S) d Area( T) + d Area( S ) + d ( + ) 4 : : area of S i terms of : derivative : sets derivative equal to : solves for ( + ) ( + ) + Copyright 4 by College Etrace Eamiatio Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for AP studets ad parets). 7
7 AP CALCULUS BC SCORING GUIDELINES (Form B) Questio 4 6DACHFDB@EBBAHAJE>ABK?JEBJDA?IA@ EJAHLГ! # EIIDMEJDABECKHA>LA6DACHFDB BDIDHEJJCAJEAJN$AJ CN # BJ@J BH Г! > N > # $ N.E@C$ C $ > MDJEJAHLIEIC@A?HAIECKIJEBOOKHIMAH? MDJEJAHLIEIJDACHFDBC??LA@MKIJEBOOKHIMAH BJ@J KIECIENIK>EJAHLIBACJD J! Г! $ C$ # # C$ $! C C $ $ B$! C $ C$ B$ > # IE?A Г! C N BN # KIJEBE?JE? 6DACHFDBCEI??LA@M$# IE?A EJAHL C B EI@A?HAIECJDEIEJAHL Г! Г JHFAE@AJD@ Copyright by College Etrace Eamiatio Board. All rights reserved. Advaced Placemet Program ad AP are registered trademarks of the College Etrace Eamiatio Board. 5
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