AP Calculus BC 2007 Scoring Guidelines Form B

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1 AP Calculus BC 7 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 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 best-kow programs are the SAT, the PSAT/NMSQT, ad the Advaced Placemet Program (AP ). The College Board is committed to the priciples of ecellece ad equity, ad that commitmet is embodied i all of its programs, services, activities, ad cocers. 7 The College Board. All rights reserved. College Board, Advaced Placemet Program, AP, AP Cetral, SAT, ad the acor logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board ad Natioal Merit Scholarship Corporatio. 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 7 SCING GUIDELINES (Form B) Questio Let R be the regio bouded by the graph of y = e ad the horizotal lie y =, ad let S be the regio bouded by the graph of y = e ad the horizotal lies y = ad y =, as show above. (a) Fid the area of R. (b) Fid the area of S. (c) Write, but do ot evaluate, a itegral epressio that gives the volume of the solid geerated whe R is rotated about the horizotal lie y =. e = whe =.44657,.5594 Let P = ad Q =.5594 R = d =.54 e : P Q (a) Area of : itegrad : limits : aswer (b) e = whe =, Area of S = e d Area of R =.66 Area of R =.546 P ( ) + + ( ) e d Q P e d = =.546 Q : : itegrad : limits : aswer Q = d : { : itegrad P (c) Volume π ( e ) ( ) : costat ad limits 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

3 7 SCING GUIDELINES (Form B) Questio A object movig alog a curve i the y-plae is at positio ( ( t), y( t )) at time t with t d = arcta + t dy ad = l ( t + ) for t. At time t =, the object is at positio (, 4 ). (Note: ta (a) Fid the speed of the object at time t = 4. = arcta ) (b) Fid the total distace traveled by the object over the time iterval t 4. (c) Fid ( 4. ) (d) For t >, there is a poit o the curve where the lie taget to the curve has slope. At what time t is the object at this poit? Fid the acceleratio vector at this poit. (a) Speed = ( 4) + y ( 4) =.9 : speed at t = 4 (b) Distace 4 d dy = + = 6.4 : { : itegral : aswer (c) ( 4) = ( ) + ( t) 4 = =.89 : itegrad : : : uses ( ) = : aswer dy (d) The slope is, so d, t = or ( t + ) = l arcta. + t Sice t >, t = At this time, the acceleratio is ( t), y ( t ) t =.5766 =.5,.955. dy : = d : : t-value : values for ad y 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

4 7 SCING GUIDELINES (Form B) Questio The wid chill is the temperature, i degrees Fahreheit ( F, ) a huma feels based o the air temperature, i degrees Fahreheit, ad the wid velocity v, i miles per hour ( mph ). If the air temperature is F, the the.6 wid chill is give by W( v) = 55.6.v ad is valid for 5 v 6. (a) Fid W ( ). Usig correct uits, eplai the meaig of W ( ) i terms of the wid chill. (b) Fid the average rate of chage of W over the iterval 5 v 6. Fid the value of v at which the istataeous rate of chage of W is equal to the average rate of chage of W over the iterval 5 v 6. (c) Over the time iterval t 4 hours, the air temperature is a costat F. At time t =, the wid velocity is v = mph. If the wid velocity icreases at a costat rate of 5 mph per hour, what is the rate of chage of the wid chill with respect to time at t = hours? Idicate uits of measure. (a).84 W ( ) =..6 =.85 or.86 Whe v = mph, the wid chill is decreasig at.86 F mph. : { : value : eplaatio (b) The average rate of chage of W over the iterval W( 6) W( 5) 5 v 6 is =.5 or W( 6) W( 5) W ( v) = whe v = : average rate of chage : : W ( v) = average rate of chage : value of v dw dw dv = = 5 5 =.89 F hr dv (c) W t= t= W = 55.6.( + 5t).6 dw =.89 F hr t = dv : = 5 : uses v = 5, : or uses vt () = + 5t : aswer Uits of F mph i (a) ad F hr i (c) : uits i (a) ad (c) 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

5 7 SCING GUIDELINES (Form B) Questio 4 Let f be a fuctio defied o the closed iterval 5 5 with f =. The graph of f, the derivative of f, cosists of two semicircles ad two lie segmets, as show above. (a) For 5 < < 5, fid all values at which f has a relative maimum. Justify your aswer. (b) For 5 < < 5, fid all values at which the graph of f has a poit of iflectio. Justify your aswer. (c) Fid all itervals o which the graph of f is cocave up ad also has positive slope. Eplai your reasoig. (d) Fid the absolute miimum value of f ( ) over the closed iterval 5 5. Eplai your reasoig. (a) f = at =,, 4 f chages from positive to egative at ad 4. Thus, f has a relative maimum at = ad at = 4. : { : -values : justificatio (b) f chages from icreasig to decreasig, or vice versa, at = 4,, ad. Thus, the graph of f has poits of iflectio whe = 4,, ad. : { : -values : justificatio (c) The graph of f is cocave up with positive slope where f is icreasig ad positive: 5 < < 4 ad < <. : { : itervals : eplaatio (d) Cadidates for the absolute miimum are where f chages from egative to positive (at = ) ad at the edpoits ( = 5, 5 ). 5 π f( 5) = + f d = + π > f = 5 f( 5) = + f d = + > The absolute miimum value of f o [ 5, 5] is f =. : : idetifies = as a cadidate : cosiders edpoits : value ad eplaatio 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

6 dy Cosider the differetial equatio y d = + +. (a) Fid d y d i terms of ad y. AP CALCULUS BC 7 SCING GUIDELINES (Form B) Questio 5 r (b) Fid the values of the costats m, b, ad r for which y = m + b + e is a solutio to the differetial equatio. (c) Let y = f be a particular solutio to the differetial equatio with the iitial coditio f ( ) =. Use Euler s method, startig at = with a step size of, to approimate f. Show the work that leads to your aswer. (d) Let y = g be aother solutio to the differetial equatio with the iitial coditio g( ) = k, where k is a costat. Euler s method, startig at = with a step size of, gives the approimatio g (). Fid the value of k. d y dy (a) = + = + ( + y + ) = 6 + 4y + 5 d d (b) If r y = m + b + e r is a solutio, the r m + re = + m + b + e +. If r : m = b +, r =, = + m, dy : + : d : aswer dy r : = m + re d : : value for r : values for m ad b so m =, r =, ad 5 b =. 4 If r = : m = b +, r =, = + m, so m =, r =, 9 b = = + = (c) f f( ) f ( ) 7 f + ( ) + = 9 f() f f = + = 4 (d) g ( ) = + k + = k + g g( ) + g ( ) = k + ( k + ) = k + = k = : : Euler's method with steps : Euler's approimatio for f () : g( ) + g ( ) : : value of k 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

7 Let f be the fuctio give by f = 6e for all. AP CALCULUS BC 7 SCING GUIDELINES (Form B) Questio 6 (a) Fid the first four ozero terms ad the geeral term for the Taylor series for f about =. (b) Let g be the fuctio give by g = f( t). Fid the first four ozero terms ad the geeral term for the Taylor series for g about =. (c) The fuctio h satisfies h = kf ( a) for all, where a ad k are costats. The Taylor series for h about = is give by Fid the values of a ad k. h = !!! ( ) (a) f = !!! 6( ) = ! (b) g ( ) = ad g = f, so 4 + ( ) g = ! 4! ( + )! 4 + 6( ) = ( +! ) (c) f = e, so h = ke a h = = e!!! a ke = e a = ad k = a = ad k = f = + +, so h = kf ( a) = k + ak+ h = + + k = ad ak = k = ad a = : : : : two of 6,,, 7 : remaiig terms : geeral term missig factor of 6 : two terms : remaiig terms : geeral term missig factor of 6 : computes kf ( a) : recogizes h = e, or equates series for : values for a ad k h 7 The College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for studets ad parets).

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