Differential Equations and Slope Fields

I. Population y grows according to the equation : = ky where k is a constant and t is measured in years. If the

population doubles every 10 years, then the value of k is

a. 0.069b. 0.200c. 0.301d. 3.322e. 5.000

dy2. If dx = ky and k is a nonzero constant, then y could be

a. 2eklJl

b. 2ekJ

c. e" +3d. kty+5

e. I k 2 1- y +-2 2

3. If:t = sjnxcos/x and if y = 0 when x = ~ , what is the value ofy when x = O?

a. -Ib. I--3c. 0

d. I3

e. 1

4. The rate of change of the volume, V; of water in a tank with respect to time, t. is directly proportional to thesquare root of the volume. Which of the following is a differential equation that describes this relationship?

a. V(t) = k,Jib. V(t) =k,.[V

dV ,Jic. -=k tdt

d. dV kdi=.JV

e. dV =k,.[Vdt

5. A curve has slope 2x + 3 at each point (z, y) on the curve. Which of the following is an equation for thiscurve if it passes through the point (1,2)?

a. y=5x-3b. y =x2 + 1c. Y =x2 +3xd. y =X2 +3x-2

e. y =x2 +3x-3

6.y

-.,..//'-...--"'/'''''-//""- ....•.,"\.'''-

I I I I II I I I I/ / I I I/ / / / I/ / / / /

-2 \ \.\ \ \ \ -,\ \ \ \ ,\ \ \ \ \\ \ \ \ ~2

-/ / / 2,-~,/ /,.•.•.•.. -,.,/" .•.•.•.. -.".\\., .•....•. -

, ,

Shown above is a slope field for which of the following differential equations?

dya, dx = 1+x

dy 2b. dx =X

dyc. dx =x+y

dy xd. dx = y

dye. dx = Iny

7.

y

\ \ , .....__ 3\\"--,\ " ...•.-I , \ \ ,-I I " \-I I I I I \

--'/11--",/11-,-/1"-""I" ,- I I , , II , I I I II I I I I 13 X-I'" 1-"'1/"---//1'--"'/11--'-/11

-31 I I I I \I I I \ \ -, , \ \ ,-,\ \,-.-\ \ ,,--\ \ '''--:3

Shown above is a slope field for which of the following differential equations?

200& AP· CALCULUS AB FREE-RESPONSE QUESTIONS

6. Consider the differential equation : = - ~ . X:" .3.'IS-(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(Note: Use tbe axes provided In the pink test booklet.)

s

• 2 •

• •

0 x-I

I• -1 •

• -2 •

(b) Let y = 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 y = I(x) to the given differential equation with the initial conditionfell = -1.

2006 Ap® CALCULUS AD FREE-RESPONSE QUESTIONS

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.)

y

• •

-2 -I 0 .r2

• -I •

(b) Find the particular solution y = j(x) to the differential equation with the initial condition j(-l) = 1 andstate its domain.

2008 AP~ CALCULUS AD FREE-RESPONSE QUESTIONS

><.:.. 3.705. Consider the differential equation : = yx~ 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.)

y

• 2

• •

• •

•

--~--~----~--~~~.xo-1 2

(b) Find the particular solution y = f(x) to the differential equation with the initial condition f(2) = O.

(c) For the particular solution y = f(x) described in part (b), find lim f(x).x---)oo

Ape CALCULUS AS2005 SCORING GUIDELINES

Question 6

Consider the differential equation : = - ~ .

(a) On the axes provided, sketch a slope field for the given differential equation at thetwelve points indicated.(Note: Use the axes provided ID the piDk test booldet.)

(b) Let y = f(x) be the particular solution to the differentiaJ equation with the initialcondition f(l) = -I. 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 y = f(x) to the given differential equation with the initialcondition f(l) = -I.

o·1

-_...•I-~--_x, i• -I

• -2

(a) y 2 0 { I : zero slopes

2o I: nonzero slopes

/ "'-

I \JC

-1 0

\ -I I

"-/

(b) The line tangent to f at (1, -1) is y + I = 2(x -1).Thus. f(1.1) is approximately -0.8.

(c) fiE = _ 2xdx yy fly:. -2x dx

2L. = _x2 +C2

~=-l+C; C=%i = _2x2 +3Since the particular solution goes through (1. -I).y must be negative.

Thus the particular solution is y = -~ 3 - 2x2 .

2 . { I : equation of the tangent line. I:approximation for f(1.1)

1 : separates variablesI :antiderivatives

5 : I: constant ofintegration1 : uses initial condition1 : solves for y

Note: Max 2/5 [1-1-0-0-0] ifnoconstant of integration

Note: 0/5 if no separation of variables

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Ape CALCULUS AS2008 SCORING GUIDEUNES

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 IDthe plDk eum booklet.)

y

• .1 •

x-2 -I 0 2

• -I • "

(b) Find the particular solution y = [(x) to the differential equation with the initial condition [(-I) = 1andstate its domain.

(a)y

\ I--~--~----+-----~~~-x-2 2

- -I -

1 1(b) 1+ y dy = x en

Inlt + yl = Inlxl + K

11 + yl = e1n/zl+ K

1+ Y = Clxl2=C1+ Y = 21xly = 21xl- 1 and x < 0

ory = -2.%- 1 and x < 0

2 : sign of slope at each point and relativesteepness ofslope lines in rows andcolumns

7:

1 : separates variables2 : antiderivatives

6 : 1: constant of integration1 : uses initial condition1 : solves for )I

Note: max 3/6 [I-2..().().()] if noconstant of integration

Note: 0/6 ifno separation of variables

I:domain

o 2006 The College Board. All eights reserved.Visit apcentIal.colleqeboard.com (for AP professiOnals) and www.collegeboazd.comiapstudents (for AP students and parents),

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Ap® CALCULUS AB2008 SCORING GUIDELINES

Question 5

Consider the differential equation : = y ~ 1, where x '# O.x

y(a) On the axes provided, sketch a slope field for the given differentialequation at the nine points indicated.(Note: Use the axes provided in the exam booklet.)

(b) Find the particular solution y = f(x) to the differential equation withthe initial condition f(2) = O.

(c) For the particular solution y = f(x) described in part (b), findlim f(x).

x--)oo

2

• 2 • •

• •

~--~r---+---~--X-1 0

y(a)

2

/ 2 / ---- ----

--~~---+----~--~--~x-1 1

(b) _1_ dy = _1 dxy -I x2

Inly -11 = _I + Cx

-..!.+cly-ll=e x

IIy-II = eCe--;

Iy -I = ke --;, where k = ±ec

I-1 = ke-"2

1

k = -e2

f(x) = 1- )~-±),x > 0

{I: zero slopes

2'. I: all other slopes

1 : separates variables2 : antidifferentiates

6 : 1: includes constant of integrationI:uses initial condition1 : solves for y

Note: max 3/6 [1-2-0-0-0] ifno constantof integration

Note: 0/6 if no separation of variables

I :limit

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