AP® CALCULUS BC
2007 SCORING GUIDELINES (Form B)
Question 5
Consider the differential equation
(a) Find
dy
= 3 x + 2 y + 1.
dx
d2y
in terms of x and y.
dx 2
(b) Find the values of the constants m, b, and r for which y = mx + b + e rx is a solution to the differential
equation.
(c) Let y = f ( x ) be a particular solution to the differential equation with the initial condition f ( 0) = −2. Use
1
Euler’s method, starting at x = 0 with a step size of , to approximate f (1) . Show the work that leads to
2
your answer.
(d) Let y = g ( x ) be another solution to the differential equation with the initial condition g ( 0) = k , where k is
a constant. Euler’s method, starting at x = 0 with a step size of 1, gives the approximation g (1) ≈ 0. Find the
value of k.
(a)
d2y
dy
= 3+2
= 3 + 2 ( 3x + 2 y + 1) = 6 x + 4 y + 5
2
dx
dx
(b) If y = mx + b + e rx is a solution, then
(
)
m + re rx = 3 x + 2 mx + b + e rx + 1.
If r ≠ 0: m = 2b + 1, r = 2, 0 = 3 + 2m,
3
5
so m = − , r = 2, and b = − .
2
4
⎧⎪ 1 : 3 + 2 dy
2: ⎨
dx
⎪⎩ 1 : answer
⎧ 1 : dy = m + re rx
⎪⎪
dx
3: ⎨
1 : value for r
⎪
⎪⎩ 1 : values for m and b
OR
If r = 0: m = 2b + 3, r = 0, 0 = 3 + 2m,
3
9
so m = − , r = 0, b = − .
2
4
(c)
(d)
( 12 ) ≈ f ( 0) + f ′( 0) ⋅ 12 = −2 + ( −3) ⋅ 12 = − 72
1
1
7
9
f ′( ) ≈ 3 ( ) + 2 ( − ) + 1 = −
2
2
2
2
1
1 1
7
9 1
23
f (1) ≈ f ( ) + f ′( ) ⋅ = − + ( − ) ⋅ = −
2
2 2
2
2 2
4
f
⎧ 1 : Euler's method with 2 steps
2: ⎨
⎩ 1 : Euler's approximation for f (1)
g ′( 0 ) = 3 ⋅ 0 + 2 ⋅ k + 1 = 2k + 1
g (1) ≈ g ( 0 ) + g ′( 0 ) ⋅ 1 = k + ( 2k + 1) = 3k + 1 = 0
1
k =−
3
⎧ 1 : g ( 0 ) + g ′( 0 ) ⋅ 1
2: ⎨
⎩ 1 : value of k
© 2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
©2007 The College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
AP® CALCULUS BC
2007 SCORING COMMENTARY (Form B)
Question 5
Sample: 5A
Score: 9
The student earned all 9 points.
Sample: 5B
Score: 6
The student earned 6 points: 2 points in part (a), 1 point in part (b), 1 point in part (c), and 2 points in part (d). The
dy
student presents correct work in parts (a) and (d). In part (b) the student only finds the correct
, and so the first
dx
point was earned. In part (c) the student earned the first point by the use of Euler’s method with two steps to
1
approximate f (1) . The student makes an error in calculating f ′
, so the second point was not earned.
2
()
Sample: 5C
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). The
dy
student presents correct work in part (a). In part (b) the student does not find
. In part (c) the student earned the
dx
first point by the use of Euler’s method with two steps to approximate f (1) . The student makes an error in
1
1
calculating f ′
, so the second point was not earned. In part (d) the student uses instead of 1 for Δ x and
2
2
makes computational errors, so no points were awarded.
()
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Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).