Differential Equations and Slope Fields

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1 Differential Equations and Slope Fields. Population grows according to the equation : = k where k is a constant and t is measured in ears. f the population doubles ever 10 ears, then the value of k is a b c d. 3.3 e d. f dx = k and k is a nonzero constant, then could be a. e kljl b. ekj c. e" +3 d. kt+5 e. k f:t = sjnxcos/x and if = 0 when x = ~, what is the value of when x = O? a. - b c. 0 d. 3 e The rate of change of the volume, V; of water in a tank with respect to time, t. is directl proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship? a. V(t) = k,ji b. V(t) =k,.[v dv c. -=k,jit d. e. dt dv k di=.jv dv =k,.[v dt 5. A curve has slope x + 3 at each point (z, ) on the curve. Which of the following is an equation for this curve if it passes through the point (1,)? a. =5x-3 b. =x + 1 c. Y =x +3x d. =X +3x- e. =x +3x-3

2 6. -.,..// '-...--"'/ '''''-// ""-....,"\.'''- - \ \. \ \ \ \, -, \ \ \ \ \ \ \ \ \ \ \ \ \ ~ / / / / / / / / / / / -/ / /,-~,/ /, ,.,/ " ". \\., ,, Shown above is a slope field for which of the following differential equations? d a, dx = 1+x d b. dx =X d c. dx =x+ d x d. dx = d e. dx = n 7. \ \,... 3 \\"--,\ "....-, \ \,- " \- \ -31 \ \ \ -,, \ \,-,\ \,-.- \ \,,-- \ \ '''--:3 --'/11 --",/11 -,-/1" -""", -,,, 13 -'" X 1 -"'1/" ---//1' --"'/11 --'-/11 Shown above is a slope field for which of the following differential equations?

3 00& AP CALCULUS AB FREE-RESPONSE QUESTONS 6. Consider the differential equation : = - ~. X:".3.'S- (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use tbe axes provided n the pink test booklet.) s x (b) Let = f(x) be the particular solution to the differential equation with the initial condition f(l) = -l. Write an equation for the line tangent to the graph of fat (1.-1) and use it to approximate fell). (c) Find the particular solution = (x) fell = -1. to the given differential equation with the initial condition 006 Ap CALCULUS AD FREE-RESPONSE QUESTONS 5. Consider the differential equation : = 1 : Y, where x '* o. (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.) r (b) Find the particular solution = j(x) to the differential equation with the initial condition j(-l) = 1 and state its domain.

4 008 AP~ CALCULUS AD FREE-RESPONSE QUESTONS ><.: Consider the differential equation : = x~ 1, where x :;:.O. (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.) --~--~----~--~~~.x o -1 (b) Find the particular solution = f(x) to the differential equation with the initial condition f() = O. (c) For the particular solution = f(x) described in part (b), find lim f(x). x---)oo

6 Ape CALCULUS AS 005 SCORNG GUDELNES Consider the differential equation : = - ~. Question 6 (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the axes provided D the pidk test booldet.) (b) Let = f(x) be the particular solution to the differentiaj equation with the initial condition f(l) = -. Write an equation for the line tangent to the graph off at (1,-1) and use it to approximate f(1.1). (c) Find the particular solution = f(x) condition f(l) = -. to the given differential equation with the initial o 1 -_... -~--_x, i - - (a) 0 / "'- { o : zero slopes : nonzero slopes \ -1 0 \ - JC "- (b) The line tangent to f at (1, -1) is + = (x -1). / Thus. f(1.1) is approximatel { : equation of the tangent line. :approximation for f(1.1) (c) fie = _ x dx fl:. -x dx L. = _x +C ~=-l+c; C=% i = _x +3 Since the particular solution goes through (1. -). must be negative. Thus the particular solution is = -~ 3 - x. 1 : separates variables :antiderivatives 5 : : constant ofintegration 1 : uses initial condition 1 : solves for Note: Max /5 [ ] ifno constant of integration Note: 0/5 if no separation of variables Copright e 00' b College Board. All rights reserved. Visit apcentral.ccllegeboard.com (for AP professionals) and (for AP students and parents). 7 5

7 Ape CALCULUS AS 008 SCORNG GUDEUNES Question 6 Consider the differential equation ~ = 1:.Y. where x _ O. (a) On the axes provided. sketch a slope field for the given differential equation at the eight points indicated. (Note: Use tbe axes provided Dthe pldk eum booklet.) x - " (b) Find the particular solution = [(x) state its domain. to the differential equation with the initial condition [(-) = 1and (a) \ --~--~ ~~~-x - : sign of slope at each point and relative steepness ofslope lines in rows and columns (b) d = x en nlt + l = nlxl + K 11 + l = e1n/zl+ K 1+ Y = Clxl =C 1+ Y = 1xl = 1xl- 1 and x < 0 or = -.%- 1 and x < 0 7: 1 : separates variables : antiderivatives 6 : 1: constant of integration 1 : uses initial condition 1 : solves for ) Note: max 3/6 [-..().().()]if no constant of integration Note: 0/6 ifno separation of variables :domain o 006 The College Board. All eights reserved. Visit apcental.colleqeboard.com (for AP professionals) and (for AP students and parents), 6 6

8 Ap CALCULUS AB 008 SCORNG GUDELNES Question 5 Consider the differential equation : = ~ 1, where x '# O. x (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) Find the particular solution = f(x) the initial condition f() = O. (c) For the particular solution = f(x) lim f(x). x--)oo to the differential equation with described in part (b), find ~--~r ~--x -1 0 (a) / / : zero slopes '. { : all other slopes --~~ ~--~--~x -1 1 (b) _1_ d = _1 dx - x nl -11 = _ + C x -..!.+c l-ll=e x - = ece--; - = ke --;, -1 = ke-" 1 k = -e where k = ±e c 1 : separates variables : antidifferentiates 6 : 1: includes constant of integration :uses initial condition 1 : solves for Note: max 3/6 [ ] ifno constant of integration Note: 0/6 if no separation of variables f(x) = 1-)~-±),x > 0 :limit 008 The College Board. All rights reserved. Visit the College Board on the Web: 8

2006 AP6 CALCULUS AB FREE-RESPONSE QUESTIONS

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