# Interactive video lesson plan for: 2014 ap Calculus frq Free Response question #6 Differential Equations slope field

#### Activity overview:

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2014 ap Calculus frq Free Response question #6 Differential Equations slope field

2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
Visit the College Board on the Web: www.collegeboard.org.
-7-
6. Consider the differential equation
the derivative of y with respect to x, equals open parenthesis 3 minus close times cosine x the derivative of y with respect to x, equals open parenthesis 3 minus close times cosine x the derivative of y with respect to x, equals open parenthesis 3 minus close times cosine xthe derivative of y with respect to x, equals open parenthesis 3 minus close times cosine xthe derivative of y with respect to x, equals open parenthesis 3 minus close times cosine x the derivative of y with respect to x, equals open parenthesis 3 minus close times cosine xthe derivative of y with respect to x, equals open parenthesis 3 minus close times cosine xthe derivative of y with respect to x, equals open parenthesis 3 minus close times cosine x the derivative of y with respect to x, equals open parenthesis 3 minus close times cosine x
. Let
y equals f of x y equals f of xy equals f of x y equals f of x y equals f of x
be the particular solution to the differential
equation with the initial condition
f of zero equals 1 f of zero equals 1 f of zero equals 1f of zero equals 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
open parenthesis zero comma one close open parenthesis zero comma one close open parenthesis
zero comma one close .
The figure shows a slope field. horizontal axis is labeled x and the vertical y.
x-has three tickmarks to left of origin 3 right; leftmost tickmark labelled negative rightmost 3.
y-above below; bottommost topmost segments at points with y equals are horizontal.
less than one point five have slopes.
between positive greater At zero, slopes increase as increases.
decrease (b) Write an equation for the line tangent to the solution curve in part (a) at the point
open parenthesis zero comma one close open parenthesis zero comma one close open parenthesis
zero comma one close . Use the equation to
approximate
f of zero point two f of zero point two f of zero point two
.
(c) Find
y equals f of x y equals f of xy equals f of x y equals f of x
, the particular solution to the differential equation with the initial condition
f of zero equals 1 f of zero equals 1 f of zero equals 1 f of zero equals 1
.
STOP
END OF EXAM

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