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Unformatted text preview: Ž© 1 : solves for y â€² y
1
=
2y âˆ’ x 2
2y = 2y âˆ’ x
x=0
y=Â± 2 y
1
âŽ§1:
=
âŽª
2y âˆ’ x 2
2: âŽ¨
âŽª 1 : answer
âŽ© y
=0
2y âˆ’ x
y=0
The curve has no horizontal tangent since âŽ§1: y = 0
2: âŽ¨
âŽ© 1 : explanation ( 0,
(c) 2 ) , ( 0, âˆ’ 2 ) 02 â‰ 2 + x â‹… 0 for any x. (d) When y = 3, 32 = 2 + 3 x so x = 7
.
3 âŽ§ 1 : solves for x
âŽª
3 : âŽ¨ 1 : chain rule
âŽª 1 : answer
âŽ© dy dy dx
y
dx
=
â‹…
=
â‹…
2 y âˆ’ x dt
dt
dx dt
3
9 dx
dx
At t = 5, 6 =
â‹…
=
â‹…
7 dt 11 dt
6âˆ’
3
dx
22
=
dt t = 5
3 Copyright Â© 2005 by College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6 APÂ® CALCULUS AB
2005 SCORING GUIDELINES (Form B)
Question 6 dy âˆ’ xy 2
. Let
=
dx
2
y = f ( x ) be the particular solution to this differential
equation with the initial condition f ( âˆ’1) = 2. Consider the differential equation (a) On the axes provided, sketch a slope field for the
given differential equation at the twelve points
indicated.
(Note: Use the axes provided in the test booklet.)
(b) Write an equation for the line tangent to the graph of
f at x = âˆ’1.
(c) Find the solution y = f ( x ) to the given differential equation with the initial condition f ( âˆ’1) = 2.
(a) âŽ§ 1 : zero slopes
2: âŽ¨
âŽ© 1 : nonzero slopes âˆ’ ( âˆ’1) 4
=2
2
y âˆ’ 2 = 2 ( x + 1) (b) Slope = (c) 1 : equation 1
x
dy = âˆ’ dx
2
2
y âŽ§ 1 : separates variables
âŽª 2 : antiderivatives
âŽª
6 : âŽ¨ 1 : constant of integration
âŽª 1 : uses initial condition
âŽª
âŽª
âŽ© 1 : solves for y 1
x2
=âˆ’
+C
4
y
1
1
1
âˆ’ = âˆ’ + C; C = âˆ’
2
4
4
1
4
y= 2
=2
1
x
x +1
+
4
4
âˆ’ Note: max 3 6 [12000] if no
constant of integration
Note: 0 6 if no separation of variables Copyright Â© 2005 by College Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 7...
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This note was uploaded on 10/01/2009 for the course OC 9876 taught by Professor Dq during the Spring '09 term at UC Merced.
 Spring '09
 Dq

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