9 Differential Equations. 9.1 Modeling with Differential Equations.
Differential Equations and Slopefields - crunchy math - Home
Transcript of Differential Equations and Slopefields - crunchy math - Home
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
Copyright e 200' by College Board. All rights reserved.Visit apcentral.ccllegeboard.com (for AP professionals) and www.roUegeboard.com/apstudents (for AP students and parents).
7
5
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),
6
6
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
© 2008 The College Board. All rights reserved.Visit the College Board on the Web: www.collegeboard.coffi.
8