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AP Calculus AB BC FRQ Free Response 2014 #2 Solutions Areas Volumes

2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS

© 2014 The College Board.

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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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