AP CALCULUS AB 2014 SCORING GUIDELINES - College Board
AP? CALCULUS AB 2014 SCORING GUIDELINES
Question 6
Consider the differential equation
dy= dx
(3 - y)cos x. Let y = f ( x) be the particular solution to the differential
equation with the initial condition f (0) = 1. The function f is defined for all real numbers.
(a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the
point (0, 1).
(b) Write an equation for the line tangent to the solution curve in part (a) at the point (0, 1). Use the equation to approximate f (0.2).
(c) Find y = f ( x), the particular solution to the differential equation with the initial condition f (0) = 1.
(a)
1 : solution curve
(b)
dy dx
( x,
y)=(0, 1)
=
2cos 0
=
2
An equation for the tangent line is =y
f (0.2) 2(0.2) + 1 =1.4
2x + 1.
{2 : 1 : tangent line equation 1 : approximation
(c)
dy= dx
(3 - y)cos x
dy 3- y
=
cos
x
dx
- ln 3 - y= sin x + C
- ln 2 =sin 0 + C C =- ln 2
1 : separation of variables
2
:
antiderivatives
6 : 1 : constant of integration
1
:
uses
initial
condition
1 : solves for y
- ln 3 - y= sin x - ln 2
Because y(0) = 1, y < 3, so 3 - y =3 - y
3 - y = 2e-sin x
Note: max 3 6 [1-2-0-0-0] if no constant of integration
y = 3 - 2e-sin x
Note: 0 6 if no separation of variables
Note: this solution is valid for all real numbers.
? 2014 The College Board. Visit the College Board on the Web: .
?2014 The College Board. Visit the College Board on the Web: .
?2014 The College Board. Visit the College Board on the Web: .
?2014 The College Board. Visit the College Board on the Web: .
?2014 The College Board. Visit the College Board on the Web: .
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