2006 AP6 CALCULUS AB FREE-RESPONSE QUESTIONS
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1 2006 AP6 CALCULUS AB FREE-RESPONSE QUESTIONS dy 1+ y 5. Consider th e differential equation =, where x 0. dx x (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the axes provided in the pink exam booklet.) (b) Find the particular solution y = f (x) to the differential equation with the initial condition f( l) = 1 and state its domain. - _
2 AO AY CALCULUS AB FREE-RESPONSE QUESTIONS 6. Consider the differential equation (4) = -- 2x. dx (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 pink test booklet.) 2. (b) Let y = f (x) be the particular solution to the differential equation with the initial condition f(i) = Write an equation for the line tangent to the graph of f at (1, 1) and use it to approximate f(1.1). (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f(1) = 1. WRITE ALL WORK IN THE TEST BOOKLET. + END OF EXAM 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents).
3 2004 AP CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 5. Consider the differential equation dy = x 4 (y 2). cbc (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.) Y (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are negative. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f(0) = 0. 4 ' i 0 - Copyright 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboanicom (for AP professionals) and (for AP students and parents). C-4 5 GO ON TO THE NEXT PAGE. r '
4 2006 AP CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 5. Consider the differential equation -- ci k= (y-1)2 cos(irx). c (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.) (b) There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c. (c) Find the particular solution y = f (x) to the differential equation with the initial condition f(1) = The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). \ 6 GO ON TO THE NEXT PAGE.
5 2006 AP CALCULUS AB FREE-RESPONSE QUESTIONS Consider the differential equation d -i= Y-, where x 0. dx x (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the axes provided in the pink exam booklet.) 2_ 0/!/ 2 (b) Find the particular solution y = f (x) to the differential equation with the initial conditio i f (1.) = 1 and state its domain fl r_1,/ d,x + I j e 14 a e orw,c4a/1&_, OR 0 - x 2--
6 2005 AY CALCULUS AB FREE-RESPONSE QUESTIONS dt, 2x 6. Consider the differential equation dx Y (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 pink test booklet.) (b) Let y = f (x) be the particular solution to the differential equation with the initial condition f(1) = 1. Write an equation for the line tangent to the graph of f at (1, 1) and use it to approximate f (1.1). (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (1) = 1. RITE ALL WORK IN THE TEST BOOKLET.. E ND OF EXAM Copyright 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 7
7 \ 4") 2004 AP CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) Consider the differential equation 4-1, = x4 (y 2). (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.) Ct.') X -I 0 0 0,0 ( (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are negative. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f(0) = 0. ).54 -e Copyright 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 5 GO ON TO THE NEXT PAGE.
8 "s\ 2006 AY CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) onsider the differential equation dy (y 1)2 cos(rx). (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.) >< 0)=- (4)(c - (b) There is a horizontal line with equation y = c that satisfies this differential equation. (c) Find the particular solution y = f (x) to the differential equation with the initial condition f(1) = 0. _ Ctre _ I. ( ost (o) cos o ) CoS 0 = (L- ) CroS0 ): d the value of c ) 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 6 GO ON TO THE NEXT PAGE.
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AP Calculus AB AP Calculus BC Free-Response Questions and Solutions 969 978 Copyright 4 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central,
Formulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x
1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6