AP Calculus BC - College Board

1 AP Calculus BCScoring Guidelines 2017 The College Board . College Board , Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board . Visit the College Board on the Web: Central is the official online home for the AP Program: Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 1 1 : units in parts (a), (c), and (d) (a) ( )() ( ) () ( ) ()()10020052210V552 cubic foeetlumeAhdhAAA + + = + + == { 1 : left Riemann sum2 : 1 : approximation (b) The approximation in part (a) is an overestimate because a left Riemann sum is used and A is decreasing. 1 : overestimate with reason (c) ( ) The volume is cubic feet. { 1 : integral2 : 1 : answer (d) Using the model, ( )( ) ( )( )5555 = = = = When 5,h= the volume of water is changing at a rate of cubic feet per minute.}}

2 2 : 3 : 1 : answerdVdt AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 2 (a) ()() = The area of R is { 1 : integral2 : 1 : answer (b) ( )()( )()()( )()( )()()222220012kgfdgfd = OR ( )()( )()()( )()( )()()222220kkgfdgfd = 1 : integral expression2 : for one region 1 : equation (c) ( )( ) ( )wgf = ( ) == The average value of ( )w on the interval 0,2 is ( ) 1 : 3 : 1 : integral 1 : average valuew (d) ( )Aww = for 02 = ( )Aww = at (or ). = () < ( )w is decreasing at = ( ) 1 : solves 2 : 1 : answer with reasonAww = AP Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 3 (a) ( )( )( )( )622662774 3fff x dxf x dx = += = = ( )( )( )5252723102fff x dx = += += ( )( ) 1 : uses initial condition3 : 1 : 6 1 : 5ff (b) ( )0fx > on the intervals [)6, 2 and ( )2, 5.]}

3 Therefore, f is increasing on the intervals []6, 2 and [ ]2, 5 . 2 : answer with justification (c) The absolute minimum will occur at a critical point where ( )0fx = or at an endpoint. ( )202,xfxx= = = x ( )fx 6 3 2 7 2 72 5 102 The absolute minimum value is ( )27 = { 1 : considers 2 : 1 : answer with justifica on2tix= (d) ( )( )2015262f == ( )( )33lim23xfx fx = and ( )( )33lim13xfx fx+ = ( )3f does not exist because ( )( )( )( ) ffx fxx + ( )( ) 1 : 52 : 1 : 3 does not exist,with explanationff AP Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 4 (a) ( )()109127164H = = ( )091H= An equation for the tangent line is 91 16 .yt= The internal temperature of the potato at time 3t= minutes is approximately 91 16 343 = degrees Celsius.}

4 1 : slope3 : 1 : tangent line 1 : approximation (b) ()()()()221111272744416dHdHHHdtdt= = = 27H> for 0t> ()22127016dHHdt= > for 0t> Therefore, the graph of H is concave up for > Thus, the answer in part (a) is an underestimate. 1 : underestimate with reason (c) ()()( )()()()( )()23231313133271273273 9127012327121227 for 1030dGdtGdGdtGGtCCCGttGtt= = = + =+ = = = + < The internal temperature of the potato at time 3t= minutes is ()312327543 += degrees Celsius. ( )( ) 1 : separation of variables 1 : antiderivatives 1 : constant of integration and5 : uses initial condition 1 : equation involving and 1 : and 3 GtGtG Note: max 25 [1-1-0-0-0] if no constant of integration Note: 05 if no separation of variables AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 5 (a) ( )()()223472 75xfxxx = + ( )( )( )()235182151534f =+ = ( )2 : 3f (b) ()()()223477042 75xxxxxf == =+ The only critical point in the interval << has x-coordinate f changes sign from positive to negative at Therefore, f has a relative maximum at 1 : -coordinate2 : 1 : relative maximum with justificationx (c) ( )()()()()()( )()( )255555321limlim2512 7525lim ln 25ln1255lnlnlim ln158limln 2lnln5144bbbbbbbbbfxdxdxdxxxxxxxxxbb == + = = == = 1 : antiderivative3 : 1 : limit expression 1 : answer (d) f is continuous, positive, and decreasing on [)5.]

5 The series converges by the integral test since 2532 75dxxx + converges. OR 2302 75nn> + and 210n> for Since 22332 75lim12nnnn += and the series 251nn= converges, the series 2532 75nnn= + converges by the limit comparison test. 2 : answer with conditions AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 6 (a) ( )00f= ( )01f = ( )( )0111f = = ( )( )021 2f = = ()()( )40326f= = The first four nonzero terms are !2!3!234xxxxxxxx ++= + ++ The general term is 1( 1).nnxn+ ( )( )( )( )4 1 : 0 ,0 , and 03 : 1 : verify terms 1 : general termfff (b) For 1,x= the Maclaurin series becomes () += The series does not converge absolutely because the harmonic series diverges. The series alternates with terms that decrease in magnitude to 0, and therefore the series converges conditionally.

6 2 : converges conditionally with reason (c) ( )( )( )()( )()100123423 4 5234510114151232 32 432411260121xnnxtxnntnnttftdtdtntttttttx xxtnnxxnn+=++=++ + = +++ = +++ + = + +++++ 1 : two terms3 : 1 : remaining terms 1 : general term (d) The terms alternate in sign and decrease in magnitude to 0. By the alternating series error bound, the error ()()41122Pg is bounded by the magnitude of the first unused term, ( ) Thus, ()()( )5412111 20500Pg =< 1 : error bound