1

Name: ____________________ Period: ______ Date: ___________ AP Calc BC

Mr. Mellina/Ms. Lombardi

Chapter 6 – Differential Equations

Topics:6.1 – Slope Fields and Euler’s Method

6.2 – Growth & Decay 6.3 –Logistic Growth

key

2

6.1 – Slope Fields and Euler’s Method Topics

• Use Euler’s Method to approximate solutions of differential equations.

Warm Up! Matcheachslopefieldwithitsdifferentialequation.

SF3 SFG

SF8 SRS

SFI SF7

sF4 SFIO

SEZ SF9

3

Example1:ABReview-FindingGeneralSolutionsFindthegeneralsolutiontotheexactdifferentialequation.a. !"

!# = 5&' − sec, & b. !"!# = sin & − /0# + 8&3

Example2:ABReview-FindingParticularSolutionsSolvetheinitialvalueproblemexplicitly.a. !"

!# = 2/# − cos & , 7 = 3when& = 0 b. !"

!; =<

<=;> + 2; ln 2 , 7 = 3when@ = 0

by 15 4 seok dx

y xs taux cIdf sine e t8x3jd

y cost text 2 4 t C

Jdy 2 ex cosx DX

y Lex sinx t C y 2e sins I

3 Zeo Sino 1 C

C I

dyJ 24n2 dt

y tan t t 2T te

3 tan o t 20 1

L

y tan t 1 Ztt I

4

Example3:ApplyingEuler’sMethodUseEuler’sMethodwithincrementsof∆& = 0.1giventoapproximatethevalueof7when& = 1.3.Comparetheapproximationwiththeexactvalue.a. !"

!# = & − 1, 7 = 2when& = 1.

Euler’s Method Using slope fields, we can graph the particular solution directly, by starting at the given point and piecing together little line segments to build a continuous approximation of the curve. This clever application of local linearity to graph a solution without knowing its equation is called Euler’s Method.

1. Begin at the point (&, 7) specified by the initial condition. This point will beo n the graph, as required.

2. Use the differential equation to find the slope !"!# at the point. 3. Increase & by a small amount, ∆&. Increase 7 by a small

amount,∆7, where ∆7 = F!"!#G ∆&. This defines a new point (& + ∆&, 7 + ∆7) that lies along the linearization. (Figure 7.7)

4. Using this new point, return to step 2. Repeating the process constructs the graph to the right of the initial point.

5. To construct the graph moving to the left from the initial point, repeat the process using negative values for ∆&

Clid 0

X y de dy f O

I 2 0.1 O dy O1 I 2 O I 0.01I 2 2.01 O I 0.02 1.112 If I

DX1 3 2 03 did I

l

yCI 3322.03 dy ol

exeetenalue 1 2,2 01 z 2

Jay c dx y 2 2 5yf z

y 122 Xtc yet33 2 045

dy 022 c 2 It Cc _z s

5

b. !"!# = 2& − 7, 7 = 0when& = 1.

Example4:ApplyingEuler’sMethodUseEuler’sMethodwithincrementsof∆& = −0.1giventoapproximatethevalueof7when& = 1.7.Comparetheapproximationwiththeexactvalue.a. !"

!# = 2 − &, 7 = 1when& = 2.b. !"

!# = 1 + 7, 7 = 0when& = 2.

Clio did 2dxdy 2

I O 0.1 O z dy Z

1 I 0.2 O I o 2 1 1,0 2 dip 2dxI Z o 4 O I 0.2dry _21 3 o 6

dy 2

YU 3 O 6CI z o 4 It _2dx

can't find exact value dy 2dy 2x y is nota separablediffegg

toy dydy o dxlu9 it ci.zaaSo2XydxdyTalkin od.es toys z

2 I 0.1 o dy 0.01 dye ozi 9 I o l o ol1 8 99 o l o oz Jdy 2 x dXI 7 97

Y 2 12 2 c 4 2 12 2 I

yo 7 Ko 97 yet77 955zcz z 2 te exact

answer1 4 z 1CI

dy

x.oy.gdxdydxle.es's Etta o.j o

YFElao.o.jodis 3 Ey o da O632 O 0.1 0.3 o i o i oi 9 o l o.o 4 0 3 dy 0.07 dy 0.0631 8 o 37 o l o 063I 7 0.433 JYty fdxlnlltyl x 2

tn lty1 xtcex2 1

yyC17 K 0.433 y Ze cin111 21 C2 ya73 11.948

6

Example5

t coA I z

dy_ 32

y de dy dy YOdfqfcz.ie 4A 2A

2 6 2 4143 dyz 2AI34 4 4 6 3

dy _4 43f 4 4At4

36

o 4AtI 63

4 4143

12 44 48 4AA Z

7

6.2 – Growth & Decay Topics

• Use separation of variables to solve a simple differential equation. • Use exponential functions to model growth and decay in applied problems.

Warm Up! Therateofchangeofavariable7isproportionaltothevalueof7.Writeadifferentialequationthatmodelsthissituationandsolveforthegeneralsolution.

dye Kydx

y fkdxIn lyl Kx te

e Kxtc ly l

y Ice Kx

8

Example1:UsinganExponentialGrowthModelTherateofchangeof7isproportionalto7.When@ = 0, 7 = 2,andwhen@ = 2, 7 = 4.Whatisthevalueof7when@ = 3?

Y Cekt

wheat _ocy 2 Inztno y 2e

2 cec 3

y zg ze 5 657L

y zektwhen E 2 y 44 2 e Zt

2 eatnz ZEf Cuz

9

Example2:PopulationGrowthAnexperimentalpopulationoffruitfilesincreasesaccordingtothelawofexponentialgrowth.Therewere100fliesaftertheseconddayoftheexperimentand300fliesafterthefourthday.Approximatelyhowmanyflieswereintheoriginalpopulation?Example3:Newton’sLawofCoolingLet7representthetemperature(in°F)ofanobjectinaroomwhosetemperatureiskeptataconstant60°F.Theobjectcoolsfrom100°Fto90°Fin10minutes.Howmuchlongerwillittakeforthetemperatureoftheobjecttodecreaseto80°F?

Newton’s Law of Cooling The rate of change of temperature of an object is proportional to the difference between object’s temperature and the temperature of the surrounding medium.

Y cekt

2K 4kOO Ce 300 Ce

C too e 2K

C too e2 tins

300 tooe c e c 33.53 e 2k

When C o the originalIn 3 2 K aunt of fliess was approx 33Ic n3

did key 60dt

La Goen

60dig Kdty 60 when y 80

Inly cool Kttc go 4oetoint treeseton It

y cekt t 60C In Iwhen t o y too when E to y aoem foin Z

too ct 60 90 40 e lot Go40 f 24.09 minI a e lot

f In Ia t itwillrequireabout 1409y 40eat two moreminutes for the objectto cool to atemp of80 F

10

ExtraPracticeExample1Writeandfindthegeneralsolutionofthedifferentialequationthatmodelstheverbalstatement.a. TherateofchangeofJwithrespectto@isinverselyproportionaltothesquareof@.b. TherateofchangeofKwithrespectto@isproportionalto25 − @.Example2Writeandfindthegeneralsolutionofthedifferentialequationthatmodelstheverbalstatement.Evaluatethesolutionatthespecifiedvalueoftheindependent.a. TherateofchangeofLisproportionaltoL.When@ = 0,L = 250,andwhen @ = 1, L = 400.WhatisthevalueofLwhen@ = 4?b. TherateofchangeofKisproportionaltoK.When@ = 0, K = 5000,andwhen

@ = 1, K = 4750.WhatisthevalueofKwhen@ = 5?

dude JdQJkE2dtQ 1 C

t

DPdI k 25

t

DP11425 E ItP 25kt Kt tc or P kzC2s t 2tc

IN 400 250 eE kN

NacektK In E

c 250 N zgoehst

N zsoekt NL4y zsoehF4 1638.4

DP 4 so_soooekE KPpercent

k m S in fo

c sooo p soooekEdt

p soooekt PCs 5000e 0 3868.90g

11

Example3Thenumberofbacteriainacultureisincreasingaccordingtothelawofexponentialgrowth.Thereare125bacteriainthecultureafter2hoursand350bacteriaafter4hours.a. Findtheinitialpopulationb. Writeanexponentialgrowthmodelforthebacteriapopulation.Let@representthe

timeinhours.c. Usethemodeltodeterminethenumberofbacteriaafter8hours.d. Afterhowmanyhourswillthebacteriacountbe25,000?

B cekt

125 Ce 350 Ce

C iz se 2k

350 Cizre 2k euxc zge

2 Echl D

K f In 14gc 6 44.64

approx 45 bacteria t

OB

qzset.nlSt

TyeK 5B 8 625 2744

25,000 62g Itn F te14

t 12.29 hr

12

6.3 – Separation of Variables and Logistic Growth

Topics• Recognize and solve differential equations that can be solved by separation of variables

• Logistic growth as a reasonable model for population growth. • Solving a logistic differential equation.

• Real-world applications of logistic growth. • The general logistic formula.

Warm Up! TherateofchangeofthenumberofcoyotesL(@)inapopulationisdirectlyproportionalto650 − L @ ,where@isthetimeinyears.When@ = 0,thepopulationis300,andwhen@ = 2,thepopulationhasincreasedto500.Findthepopulationwhen@ = 3.Astheyearscontinuetoincrease,doesthenumberofcoyotesapproachaspecificvalue?Doyouknowwhatthisnumberiscalledincontextoftheproblem?

🐺

ddNq k 650 N www.t o.N 300 N 650 350t

me300 650 C

when C 3godY_n dt c 350

I

In1650 NI Kttc N 650 3505kt N 552 coyotes

In1650 Nl Ktt C whent z N 5

650 N eKttc 500 650 350 e

Zk

Kt650 N Ce k in 3 424N 650 ee

let

as C D N 650

the carryingcapacity

13

Example1:ABReview–SeparableDifferentialEquationsFindthegeneralsolutionofthedifferentialequation.a. !"

!# =N0#>,"O b. 2 + & 7P = 37

c. 77P = −8 cos Q& d. 7 ln & − &7P = 0Example2:ABReview–SeparableDifferentialEquationsFindtheparticularsolutionofthedifferentialequation.a. 77P − 2/# = 0, 7 0 = 6 b. 7 1 + &, 7P − & 1 + 7, = 0, 7 0 = 3

fzy3dyf.de a DX µg d

y1 6 13 3 c

y 12 2331 c Inlyl 3lnl2txltC

y I y

ccztxpfydyf 8cos.CI dx yinx xdy.nod x

y2 E sinCtx tc yinxxdydxyziysinctxdtcfufE.IE nxxdx JtYy IJ si c Ickx2tC Inly

Y cetanxjz

y f 2e o yCitx2ddy XCityfydyf2e dx y

y2 2e tcm

I Iydy _dxduty infity21 lz1n ltX2lt C

36 2e tctty c Cctv18 21 c

c 16 11 3 toc 4

ye 2e tl6ty 2 4 Cta

y2 4e t32

Y 4 32Y FFF

posbe initcond is

14

Example3:DerivingtheGeneralSolutionSolvethelogisticdifferentialequation.a. !"

!; = R7 1 − "S

How Populations Grow We have showed that when the rate of change of a population is directly proportional to the size of the population, the population grows exponentially. This seems like a reasonable model for population growth in the short term, but populations in nature cannot sustain exponential growth for very long. Available food, habitat, and living space are just a few of the constraints that will eventually impose limits on the growth of any real-world population.

fyu.tty fkdtLeft ee

et

I c elet

partialfractions

tf y dy fkdtEy ce t

y IInlyI In IL yl Kt t c e att 1In IL yl Inlyl Kttc

n LII Kttc

general formI k 1 elet

y L1 be let

15

Example4:MatchingLogisticEquationswiththeirGraphs1. 7 = <,

<=TUV2. 7 = <,

<=3TUV3. 7 = <,

<=W>TUV

4. 7 = <,

<=TU>VExample5:UsingaLogisticEquationAstategamecommissionreleases40elkintoagamerefuge.After5years,theelkpopulationis104.Thecommissionbelievesthattheenvironmentcansupportnomorethan4000elk.ThegrowthrateoftheelkpopulationXis

YXY@ = RX 1 − X

4000 ,40 ≤ X ≤ 4000

where@isthenumberofyears.a. Writeamodelfortheelkpopulationintermsof@.b. Graphtheparticularsolutionandusethemodeltoestimatetheelkpopulationafter15

years.d. Findthelimitofthemodelas@ → ∞

D

A x ocy 3

B x ocy 8

C approaches 2Faster

sina.e.name

1 4000 PCO _40 p 53 104

p 4000 40 40002 10440007Itb 1 99e 5k1 be b 4000to I K 0.1943610169

db 99p 4000t

p 4000 age0.194T this is approximated1 99e ice

PUSS 628.538 approx629 elk

as 1 a P 4000

carryingcapacity

16

Example6:WeightGainAcalfthatweighs60poundsatbirthgainsweightattherate

Y]Y@ = R 1200 − ] ,

where]istheweightinpoundsandtisthetimeinyears.a. Findthegeneralsolutionofthedifferentialequation.b. UseagraphingutilitytographtheparticularsolutionsforR = 0.8, 0.9and1.c. Theanimalissoldwhenitsweightreaches800pounds.Findthetimeofsaleforeachof

themodelsinpart(b).d. Whatisthemaximumweightoftheanimalforeachofthemodelsinpart(b)?

w o _60duw

dt60 1200 C

c 1140In11200 w KttcIn11200 w Kttc Kt

Kt Wct 1200 1140 e1200 w Ce

KtW 1200 Ce

TTT

800 1200 114 e Ktfor k 0 8 t l 3 ly

k 0.9 t l 16 yr1 1 o t l 05 yr

finew 1200lb1 so

17

Example7:UsingaLogisticEquationThelogisticequationmodelsthegrowthofapopulation.Usetheequationto

(a) FindthevalueofR(b) Findthecarryingcapacity(c) Findtheinitialpopulation(d) Determinewhenthepopulationwillreach50%ofitscarryingcapacity(e) WritealogisticequationthathasthesolutionK @ .

1. K @ = ,<__<=,`TUa.bcd

a K 0.75

b L 2100

c P o 2100 70

d 1050 2100129e o est

11 29e o ist zo 75T Ie 29

f ln 4.4897 yro 75

e ddPz o Isp c Io Pcos 70

18

ExtraPracticeFornumbers1-4:Findthelogisticequationthatpassesthroughthegivenpoint.1. !"

!; = 7 1 − "3N , 0, 4 2. !"

!; = 4.27 1 − ",< , 0, 9

3. !"

!; ='"e −

"><e_ , 0, 8 4. !"

!; =3",_ −

"><N__ , 0, 15

K l L _36 K 4.2 1 21

y Iet y I

0.40,9

4 361tb 92158 it be

q 21tb

y 36118e t b I3

y 211143e4.2T

EE Eye E EE sci Eo1 43 L 120 1 3

20 L 240

y y2401t beOEt because

or8 oils8 120

15 240it bb 14 Itb

b is

Y 1141 240J 1 1getshost

19

Fornumber5:Aconservationorganizationreleases25Floridapanthersintoagamepreserve.After2years,thereare39panthersinthepreserve.TheFloridapreservehasacarryingcapacityof200panthers.5a. Writealogisticequationthatmodelsthepopulationofpanthersinthepreserve.5b. Findthepopulationafter5years.5c. Whenwillthepopulationreach100?5d. Writealogisticdifferentialequationthatmodelsthegrowthrateofthepanther

population.Thenrepeatpart(5b)usingEuler’sMethodwithastepsizeofℎ = 1.Comparetheapproximationwiththeexactanswer.

5e. Atwhattimeisthepantherpopulationgrowingmostrapidly?Explain.

zoo ky 114

k IIIay

20 2,39 ZKEI 1 be 3g 200g

0125 it e 2K y 2001 get'Y3 t25 it e 2K 201

its 39

b e 2K 232g

y 5 69.695 70 panthers

zootoo

It7eI4EaJt

1 7.370 years

TR pgfntgf ko.fi E

If tin PEI Io

P increases most rapidly where P 20 coo

corresponds to t 7 37 years