ap10_calculus_ab_form_b_q2

ap10_calculus_ab_form_b_q2 - AP CALCULUS AB 2010 SCORING...

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Unformatted text preview: AP CALCULUS AB 2010 SCORING GUIDELINES (Form B) Question 2 1 1 1 , and g ( x ) = 1 - 2 cos x + . x x x (a) Find all values of x in the interval 0.12 x 1 at which the graph of g has a horizontal tangent line. The function g is defined for x > 0 with g (1) = 2, g ( x ) = sin x + ( ) ( ) (b) On what subintervals of ( 0.12, 1) , if any, is the graph of g concave down? Justify your answer. (c) Write an equation for the line tangent to the graph of g at x = 0.3. (d) Does the line tangent to the graph of g at x = 0.3 lie above or below the graph of g for 0.3 < x < 1 ? Why? (a) The graph of g has a horizontal tangent line when g ( x ) = 0. This occurs at x = 0.163 and x = 0.359. 1 : sets g ( x ) = 0 2: 1 : answer (b) g ( x ) = 0 at x = 0.129458 and x = 0.222734 The graph of g is concave down on ( 0.1295, 0.2227 ) because g ( x ) < 0 on this interval. 2: { 1 : answer 1 : justification (c) g ( 0.3) = -0.472161 g ( 0.3) = 2 + 1 0.3 g ( x ) dx = 1.546007 An equation for the line tangent to the graph of g is y = 1.546 - 0.472 ( x - 0.3) . 1 : g ( 0.3) 1 : integral expression 4: 1 : g ( 0.3) 1 : equation (d) g ( x ) > 0 for 0.3 < x < 1 Therefore the line tangent to the graph of g at x = 0.3 lies below the graph of g for 0.3 < x < 1. 1 : answer with reason 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. AP CALCULUS AB 2010 SCORING COMMENTARY (Form B) Question 2 Sample: 2A Score: 9 The student earned all 9 points. Sample: 2B Score: 6 The student earned 6 points: 2 points in part (a), 2 points in part (b), 1 point in part (c), and 1 point in part (d). In parts (a) and (b), the student's work is correct. In part (c) the student earned the slope point for g ( 0.3) . In part (d) the student's work is correct. Sample: 2C Score: 3 The student earned 3 points: 2 points in part (a), no points in part (b), 1 point in part (c), and no points in part (d). In part (a) the student's work is correct. In part (b) the student's concavity statement is incorrect. In part (c) the student earned the slope point for g ( 0.3) . In part (d) the student does not include a statement about the tangent line. 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. ...
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