AP Calculus AB. Scoring Guidelines
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1 17 AP Calculus AB Scoring Guidelines 17 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home for the AP Program: apcentral.collegeboard.org
2 AP CALCULUS AB/CALCULUS BC 17 SCORING GUIDELINES uestion 1 1 : units in parts (a), (c), and (d) 1 : { (a) Volume A( h) dh ( ) A( ) + ( 5 ) A( ) + ( 1 5) A( 5) cubic feet 1 : left Riemann sum 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 1 (c) f( h) dh The volume is 11.5 cubic feet. : { 1 : integral 1 : answer (d) Using the model, V ( h) f dx. dv dv dh dt dh dt h 5 h 5 dh f( h) dt h 5 f ( 5) h : dv : dt 1 : answer When h 5, the volume of water is changing at a rate of cubic feet per minute. 17 The College Board.
3 AP CALCULUS AB 17 SCORING GUIDELINES uestion (a) f ( t) dt pounds of bananas are removed from the display table during the first hours the store is open. : { 1 : integral 1 : answer (b) f ( 7) 8.1 (or ) After the store has been open 7 hours, the rate at which bananas are being removed from the display table is decreasing by 8.1 (or 8.119) pounds per hour per hour. : { 1 : value 1 : meaning (c) g( 5) f( 5).61 < Because g( 5) f( 5) <, the number of pounds of bananas on the display table is decreasing at time t 5. 1 : considers f( 5 ) and g( 5) : 1 : answer with reason 8 8 (d) 5 + g( t) dt f ( t) dt pounds of bananas are on the display table at time t 8. : { : integrals 1 : answer 17 The College Board.
4 AP CALCULUS AB/CALCULUS BC 17 SCORING GUIDELINES uestion 6 (a) f ( 6) f ( ) + f dx 7 f dx f ( 5 ) f ( ) + f dx 7 π + 1 π 1 : uses initial condition : 1 : f ( 6) 1 : f ( 5) (b) f > on the intervals [ 6, ) and (, 5 ). Therefore, f is increasing on the intervals [ 6, ] and [, 5 ]. : answer with justification (c) The absolute minimum will occur at a critical point where f or at an endpoint. : { 1 : considers x 1 : answer with justification f x, x x f( x ) π 5 1 π The absolute minimum value is f ( ) 7 π. 1 (d) f ( 5) 6 ( ) f f ( ) lim and x x f f ( ) lim 1 + x x 1 : f ( 5) : 1 : f ( ) does not exist, with explanation f ( ) does not exist because f f ( ) f f ( ) lim lim. x + x x x 17 The College Board.
5 AP CALCULUS AB/CALCULUS BC 17 SCORING GUIDELINES uestion 4 1 (a) H ( ) ( 91 7) 16 4 H ( ) 91 An equation for the tangent line is y t. 1 : slope : 1 : tangent line 1 : approximation The internal temperature of the potato at time t minutes is approximately degrees Celsius. d H dt 1 dh dt (b) ( )( )( H 7) ( H 7) 1 : underestimate with reason H > 7 for d H 1 t > ( H 7) > for t > dt 16 Therefore, the graph of H is concave up for t >. Thus, the answer in part (a) is an underestimate. (c) dg dt ( ) G 7 dg ( 1) dt ( ) G 7 1 ( G 7) t + C 1 ( 91 7) + C C 1 1 ( G 7) 1 t 1 t Gt ( ) 7 + ( ) for t < 1 The internal temperature of the potato at time t minutes is 1 ( ) degrees Celsius. 5 : 1 : separation of variables 1 : antiderivatives 1 : constant of integration and uses initial condition 1 : equation involving G and t 1 : Gt ( ) and G( ) Note: max 5 [1-1---] if no constant of integration Note: 5 if no separation of variables 17 The College Board.
6 AP CALCULUS AB 17 SCORING GUIDELINES uestion 5 t ( t 1) (a) x P ( t) t t + 1 t t + 1 t t + 1 > for all t. 1 : x P( t) : 1 : interval x P ( t) t 1 xp( t) t < for < 1. Therefore, the particle is moving to the left for t < 1. (b) v ( t) ( t 5)( t ) v( t) t, t 5 : 1 : intervals 1 : analysis using vp( t) and v( t) Note: 1 if only one interval with analysis Note: if no analysis Both particles move in the same direction for 1 < t < and 5 < t 8 since vp( t) xp ( t) and v ( t ) have the same sign on these intervals. (c) a ( t) v ( t) t 8 a ( ) 8 4 : 1 : a ( ) 1 : speed decreasing with reason a ( ) < and v ( ) > At time t, the speed of the particle is decreasing because velocity and acceleration have opposite signs. (d) Particle first changes direction at time t. ( ) ( ) + ( ) 5 + ( ) x x v t dt t t dt t t 4t + 15t 5 + ( ) t 1 : antiderivative : 1 : uses initial condition 1 : answer 17 The College Board.
7 AP CALCULUS AB 17 SCORING GUIDELINES uestion 6 (a) f sin + cos xe sin x : f ( π ) sin π f ( π) sin ( π) + cos π e 1 (b) k h ( f ) f k ( π) h ( f( π) ) f ( π) h ( ) ( 1) 1 1 ( )( 1) : 1 : k 1 : k ( π ) (c) m g ( x) h + g( x) h m ( ) g ( 4) h( ) + g( 4) h ( ) 1 ( 1)( ) + 5( ) : : m 1 : m ( ) (d) g is differentiable. g is continuous on the interval [ 5, ]. g( ) g( 5) 1 4 ( 5) Therefore, by the Mean Value Theorem, there is at least one value c, 5 < c <, such that g ( c) 4. g( ) g( 5) 1 : ( 5) : 1 : justification, using Mean Value Theorem 17 The College Board.
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