**Unformatted text preview: **AP® CALCULUS BC
2011 SCORING GUIDELINES (Form B)
Question 4
The graph of the differentiable function y = f ( x ) with domain
0 ≤ x ≤ 10 is shown in the figure above. The area of the region enclosed
between the graph of f and the x-axis for 0 ≤ x ≤ 5 is 10, and the area of
the region enclosed between the graph of f and the x-axis for 5 ≤ x ≤ 10
is 27. The arc length for the portion of the graph of f between x = 0 and
x = 5 is 11, and the arc length for the portion of the graph
of f between x = 5 and x = 10 is 18. The function f has exactly two
critical points that are located at x = 3 and x = 8. (a) Find the average value of f on the interval 0 ≤ x ≤ 5.
(b) Evaluate 10 ∫0 ( 3 f ( x ) + 2 ) dx. Show the computations that lead to your answer.
(c) Let g ( x ) = x ∫5 f ( t ) dt. On what intervals, if any, is the graph of g both concave up and decreasing? Explain your reasoning.
(d) The function h is defined by h( x ) = 2 f ( 2x ). The derivative of h is h′( x ) = f ′( 2x ). Find the arc length of the graph of y = h( x ) from x = 0 to x = 20. (a) Average value = (b) 10 ∫0 1 5
−10
f ( x ) dx =
= −2
5 ∫0
5 ( 3 f ( x ) + 2 ) dx = 3 (∫ 5 0 f ( x ) dx + 10 ∫5 1 : answer ) f ( x ) dx + 20 2 : answer = 3 ( −10 + 27 ) + 20 = 71
(c) g ′( x ) = f ( x )
g ′( x ) < 0 on 0 < x < 5
g ′( x ) is increasing on 3 < x < 8.
The graph of g is concave up and decreasing on 3 < x < 5. (d) Arc length =
Let u =
20 ⌠
⎮
⌡0 20 ∫0 1 + ( h′( x ) )2 dx = ⌠
⎮ 20 ⌡0 ( ( 2x )) 1+ f′ 2 dx x
1
. Then du = dx and
2
2 ( ( )) 1+ f′ x
2 2 dx = 2∫ 10 0 ⎧ 1 : g ′( x ) = f ( x )
⎪
3 : ⎨ 1 : analysis
⎪⎩ 1 : answer and reason ⎧ 1 : integral
⎪
3 : ⎨ 1 : substitution
⎪⎩ 1 : answer 1 + ( f ′( u ) )2 du = 2 (11 + 18 ) = 58 © 2011 The College Board.
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Visit the College Board on the Web: . © 2011 The College Board.
Visit the College Board on the Web: . © 2011 The College Board.
Visit the College Board on the Web: . © 2011 The College Board.
Visit the College Board on the Web: . © 2011 The College Board.
Visit the College Board on the Web: . AP® CALCULUS BC
2011 SCORING COMMENTARY (Form B)
Question 4
Sample: 4A
Score: 9
The student earned all 9 points.
Sample: 4B
Score: 6
The student earned 6 points: no point in part (a), 1 point in part (b), 3 points in part (c), and 2 points in part (d). In
parts (a) and (b) the student does not work correctly with the region below the x-axis. The student’s work earned
1 of the 2 points in part (b). In part (c) the student’s work is correct. In part (d) the student may be attempting a
substitution because there are new correct limits of integration, but the factor of 2 is missing, so only 2 of the 3
possible points were earned.
Sample: 4C
Score: 3
The student earned 3 points: no point in part (a), 2 points in part (b), no points in part (c), and 1 point in part (d).
In part (a) the student does not work correctly with the region below the x-axis. In part (b) the student’s work is
correct. In part (c) the student’s work is incorrect. In part (d) the student has the correct answer, which is achieved
by doubling the arc length of the graph of y = f ( x ) from x = 0 to x = 10 but gives no indication as to why
that method works. The student earned 1 of the 3 possible points. © 2011 The College Board.
Visit the College Board on the Web: . AP® CALCULUS BC
2011 SCORING COMMENTARY (Form B)
Question 4
Sample: 4A
Score: 9
The student earned all 9 points.
Sample: 4B
Score: 6
The student earned 6 points: no point in part (a), 1 point in part (b), 3 points in part (c), and 2 points in part (d). In
parts (a) and (b) the student does not work correctly with the region below the x-axis. The student’s work earned
1 of the 2 points in part (b). In part (c) the student’s work is correct. In part (d) the student may be attempting a
substitution because there are new correct limits of integration, but the factor of 2 is missing, so only 2 of the 3
possible points were earned.
Sample: 4C
Score: 3
The student earned 3 points: no point in part (a), 2 points in part (b), no points in part (c), and 1 point in part (d).
In part (a) the student does not work correctly with the region below the x-axis. In part (b) the student’s work is
correct. In part (c) the student’s work is incorrect. In part (d) the student has the correct answer, which is achieved
by doubling the arc length of the graph of y = f ( x ) from x = 0 to x = 10 but gives no indication as to why
that method works. The student earned 1 of the 3 possible points. © 2011 The College Board.
Visit the College Board on the Web: . ...

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