AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 3. Scoring Guideline.

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1 208 AP Calculus BC Sample Student Responses and Scoring Commentary nside: Free Response Question RR Scoring Guideline RR Student Samples RR Scoring Commentary 208 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 208 SCORNG GUDELNES Question 5 { : integral (a) f ( 5) = f ( ) + g( x) dx = f ( ) g( x) dx 5 ( ) ( ) 9 25 = 9 + = = : : answer 6 6 (b) g( x) dx = g( x) dx + g( x) dx 6 2 = 2 dx + 2( x 4) dx x= 6 ( x ) ( ) = = 4 + = 0 x= : split at x = : : antiderivative of 2( x 4) 2 : answer (c) The graph of f is increasing and concave up on 0 < x < and 4 < x < 6 because f ( x) = g( x) > 0 and f ( x) = g( x) is increasing on those intervals. 2 : { : intervals : reason (d) The graph of f has a point of inflection at x = 4 because f ( x) = g( x) changes from decreasing to increasing at x = 4. 2 : { : answer : reason 208 The College Board.

$NO CALCULATOR ALLOWED y 8 7 6. 5 A l J \ J - \ -s-4 - -2 Ao 2 4 5 6 t _2,._ - Graph of g X. The graph of the continuous function g, the derivative of the function f, is shown above.$

3 NO CALCULATOR ALLOWED y A l J \ J - \ -s Ao t _2,._ - Graph of g X. The graph of the continuous function g, the derivative of the function f, is shown above. The function g is piecewise linear for -5 f x <, and g(x) = 2(x - 4) 2 f or f x::; 6. (a) f f(l) =, whatis thevalue of f(-5)?. J\ ')::.9l ) ¼ ( 'i ) J): ({)Jt -t 8(-5) :-S:,( l)j-b t - (--S)= - ( (Ot2)-t(\)( )-( )( ))+?> (b) Evaluate Ji 6 g(x) dx. T r r: (x \ch jl'i \ (x )Jx t t C-x )J-x :: {Z)('Z} + l 4lJx 5t-J J <-/ ( i:.'i) )\; S; l).lj).; Lf + ( ½ ( i) - (-i)) -f (- -C- -9) f fc.. s-) +l J s ll +½ p c J, 'J Lf '.,.. l x- + _ 5(xlih-= \(') '¾.:G Unauthorized copying or rauae of any pert of this page llegal. 4- Continue question on page The College Board.

4 NO CALCULATOR.A;LOWED ( c) For - 5 < x < 6, on what open intervals, if any, is the graph off both increasing and concave up? Give a reason for your answer.?,ft On +k vrtuvt) {O) \) v( LL -) s bo-th lvl ttdn5 tmj co-n vp sin ve. f l ).:: C<) 'mj is -poi>ltve on ik-½m,intt,,vtlj f \$' n {n.5 on tu, t tmj ;Oj is fo /05 fn itj fnt-mt/ ) i :, -r'\"x)-,o <m 4htd- ; t w-t,s CA>'\ Of <n ih<tt,rjm-q/ (d) Find the x-coordinate of each point of inflection of the graph off Give a reason for your answer. t k Q y,oinl at ;,.,fk.c,.\-;w\,j )(a l / Sin<:<, f'c-x-= sc-x)..,j s,nu swrkb()s TTJM ir.j -h ' \.i; cj -= Y ) ibef {'.\4) o dt t,oin{ cttd~-wwlj c s 'h 'tf.om0-lu@qh=y --ikj x Y,s Qn rdleoh'e-n p:,{j. Unauthorized copying or reu&e of any part of this page is llegal. -5- GO ON TO THE NEXT PAGE. 208 The College Board.

$\ Ao 2 4 s 6 ''2 / X Graph of g.$ The graph of the continuous f u nction g, the derivative of tht: function f, is

The function g is piecewise linear for -5 x <, and g(x) = 2(x - 4) 2 for x $ 6.

5 j NO CALCULATOR ALLOWED y f i _, 2 -,/. \ Ao 2 4 s 6 ''2 / X Graph of g. The graph of the continuous f u nction g, the derivative of tht: function f, is shown above. The function g is piecewise linear for -5 x <, and g(x) = 2(x - 4) 2 for x $ 6. - '" - l- (a) f f(l) =, what is the value of J.(-5)? C,.L -:.-;...,_ 4,,?,, (b) Evaluate J6 g(x) dx. Unauthorized copying or reuse of any part of this page ls llegal. -4- Continue question on page The College Board.

$f' i& f f f"b -t,s :t.nct ro w\.c,,n f '(x) :::- e<x) ($ f's;+itc.$ $\\t, ttafh t is 'Vt.$ $of inflection of the graph off Give a reason for your answer. i o.. f &\fl" t> \!$

6 - NO CALCULATiOR ALLOWED (c) For -5 < x < 6, on what open intervals, if any, is the graph off both increasing and concave up? Give a reason for your answer. f' i& f f f"b -t,s :t.nct ro w\.c,,n f '(x) :::- e<x) ($ f's;+itc. 't \ (o A\'<t v\f Wht,,'\ "t'()'j(.)- G<.) C;) \r\o\cc,- l" j '? \\t, ttafh t is 'Vt."' '"d '"'CKCJle, o V\ (o >,; \) ( '\, ) (d) Find the x-coordinate of each point of inflection of the graph off Give a reason for your answer. i o.. f &\fl" t> \!'\\fl h,rr\ "t 'W::. '{)Gt) CA \mu-.m o ir w,'n ""'( }oc.o.l / Unauthorized copying or reu of any part of this page ls llegal. -5- GO ON TO THE NEXT PAGE. 208 The College Board.

$NO CALCULATOR ALLOWED y 8 7 ' 5 A ' l -,j r \ --d$ . The graph of the continuous function g, the

The function g is piecewise linear for -::. _s;_.

$(a) f f(l) =, whatistbevalueof f(-5)? \ ( '><.$ $:J (')(_) X <; ( _ s- 2 l - s - -r\ {b) Evaluate$ '2) lx oy- :, 2 -y"') r L, SY + - - -:: C J

7 NO CALCULATOR ALLOWED y 8 7 ' 5 A ' l -,j r \ --d _" :.fi / - 2 l - Graph Of g.. The graph of the continuous function g, the derivative of the function f, is shown above. The function g is piecewise linear for -::. _s;_.x <, and g(x) = 2.(x - 4) 2 for s; x s; 6. (a) f f(l) =, whatistbevalueof f(-5)? \ ( '><. \ :::-?( "{. - y \ r\( \ -:. :J (')(_) X <; ( _ s- 2 l - s - -r\ {b) Evaluate Ji6 g(x) dx. J i! '2) lx oy- :, 2 -y"') r L, SY :: C J Unauthorized copying or reuse of any part.ot this pa_ge ls llegal -4- Continue question on page The College Board.

$C t'y'dm O 'x.. L. \ -\-'r-' e ro.$ $p'n ' \'"'\0+ '-on c a \J e ',5 C) t; Ob0" -e and L x l_ lt> t.$ $o,,- 'v-e ) ''C"'.c\ "=> on \ c v o, "'t'.$ $graph off Give a reason for your answer. tnere t C\?Dlf\'t \/.$

8 NO CALCULATOR ALLOWED. ' t (c) For -5 < x < 6, on what open intervals, if any, is the graph off both increasing and concave up? Give a reason for your answer. C t'y'dm O 'x.. L. \ -\-'r-' e ro. p'n ' \'"'\0+ '-on c a \J e ',5 C) t; Ob0" -e and L x l_ lt> t..\_s bo+h \nc. v? be,c ' u.se + h e,, - a x.. \ s l? o,,- 'v-e ) ''C"'.c\ "=> on \ c v o, "'t'. 6\ pe (d) Find the x-coordinate of each point of inflection of the graph off Give a reason for your answer. tnere t C\?Dlf\'t \/. -: Y be c o.u<oe c..,;-e s r C 'f lr" of \\ l-e"c.+\on i-h s\o of f \ c\,ecr e o. "'"' 5 t-o l-j Unauthorized.copying or reuse of Anv rurt nt this naae is Jlleaal. -5- GO_ ON TO THE NEXT PAGE. 208 The College Board.

9 AP CALCULUS AB/CALCULUS BC 208 SCORNG COMMENTARY Question Overview n this problem the graph of the continuous function g is provided; g is piecewise linear for 5 x, and 2 g x 2( x 4) for x 6. t is also given that g is the derivative of the function f. n part (a) students were given that f and asked for the value of f 5. A correct response should demonstrate knowledge that f is 5 an antiderivative of g, so that f 5 f g x dx. The integral g x dx should then be evaluated using properties of definite integrals and computation of areas of the regions between the graph of g and the x-axis using geometry. n part (b) students were asked to evaluate g x dx. A correct response should use the property of integrals to split the interval of integration into the sum of integrals across adjacent intervals, and, 6. One of the resulting integrals can be computed using geometry and the other using an antiderivative of 2 g x 2( x 4) on the interval x 6. n part (c) students were asked for the open intervals on 5 x 6 where the graph of f is both increasing and concave up and to give a reason for their answer. A correct response should demonstrate the connection between properties of the derivative of f and the properties of monotonicity and concavity for the graph of f. The graph of f is strictly increasing where g f is positive, and the graph of g is concave up where the graph of g f is increasing. n part (d) students were asked for the x-coordinate of each point of inflection of the graph of f and to give a reason for their answer. A correct response should convey that a point of inflection of the graph of f occurs at a point where the derivative of f changes from increasing to decreasing, or from decreasing to increasing. This can be obtained from the supplied graph of g f, which changes from decreasing to increasing at x 4. For part (a) see LO.2C/EK.2C, LO.2C/EK.2C2. For part (b) see LO.2C/EK.2C, LO.2C/EK.2C2, LO.B(b)/EK.B2, LO.B(b)/EK.B5. For parts (c) and (d), see LO 2.2A/EK 2.2A. This problem incorporates the following Mathematical Practices for AP Calculus (MPACs): reasoning with definitions and theorems, connecting concepts, implementing algebraic/computational processes, connecting multiple representations, building notational fluency, and communicating. Sample: A Score: 9 The response earned all 9 points: 2 points in part (a), points in part (b), 2 points in part (c), and 2 points in part (d). n part (a) the response earned the first point with the expression g t dt in line on the left. The 5 second point would have been earned by the numerical expression in line 4 with no simplification. n this case, correct simplification to earned the second point. n part (b) the response earned the first point with the 2 sum of the two integrals in line on the left. The second point was earned with the antiderivative expression 2 ( 4) x in line. The second point would have been earned by the numerical expression in line 4 with no simplification. n this case, correct simplification to 0 earned the third point. n part (c) the union of intervals 0, 4, 6 earned the first point. The second point was earned with the reason f x g x, g is positive, and g is increasing on that interval. n part (d) the first point was earned by identifying the x-coordinate of a point of inflection at x 4. The second point was earned with the reason f x g x and g switches from decreasing to increasing at x The College Board.

10 AP CALCULUS AB/CALCULUS BC 208 SCORNG COMMENTARY Question (continued) Sample: B Score: 6 The response earned 6 points: 2 points in part (a), no points in part (b), 2 points in part (c), and 2 points in part (d). n part (a) the response earned the first point with the expression g x dx in line. The second point would 5 have been earned by the numerical expression in line with no simplification. n this case, correct simplification to 25 2 earned the second point. n part (b) the integral expression 6 () g( x) dx in line did not earn the first 6 2 point because g x dx 4, not 9. This response used substitution of variables to write 2 x 4 dx in an equivalent form. The antiderivative in line 4 is incorrect, and the response did not earn the second point. As a result of this error, the response is not eligible for the answer point. n part (c) the union of intervals 0, 4, 6 earned the first point. The second point was earned with the reason f x g x is positive in line and f x g x is increasing in line 2. n part (d) the first point was earned by identifying the x-coordinate of a point of inflection at x 4. The second point was earned with the reason f x g x has a maximum or minimum (local). Sample: C Score: The response earned points: no points in part (a), point in part (b), point in part (c), and point in part (d). n part (a) an integral expression is not presented nor is its numerical equivalent, so the first point was not earned. The value given for f 5 is incorrect, so the second point was not earned. n part (b) the response did not earn 6 the first point because g x dx is not written as the sum of two integrals or the equivalent. The second point was earned with the antiderivative expression in line 2. Because the second point was earned, the response is eligible for the third point. The answer is incorrect, however, so the third point was not earned. n part (c) the first point was earned with the intervals 0 x and 4 x 6. Although f is above the x-axis is a valid reason for why f is increasing on those intervals, f has an increasing slope is not a valid reason to explain why the graph of f is concave up on those intervals. The second point was not earned. n part (d) the first point was earned by identifying the x-coordinate of a point of inflection at x 4. The slope of f changes from decreasing to increasing is not a valid reason to explain why the graph of f has a point of inflection at x 4, so the second point was not earned. 208 The College Board.

AP Calculus AB. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 1. Scoring Guideline.

AP Calculus AB. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 1. Scoring Guideline. 217 AP Calculus AB Sample Student Responses and Scoring Commentary Inside: RR Free Response Question 1 RR Scoring Guideline RR Student Samples RR Scoring Commentary 217 The College Board. College Board,