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Name: _________________ Period: ______ Date: ________________ AP Calc AB

Mr. Mellina

Chapter 7: Differential Equations and Mathematical Modeling

Sections:

v 4.3 Derivatives of Inverse Functions v 7.2 Antidifferentiation by Substitution

v 7.1 Slope Fields and Differential Equations v 7.4 Exponential Growth and Decay

v 7.5 Logistic Growth

HWSets

SetA(Section4.3)Page175,#’s1-11odd,27,28,37,&38.

SetB(Section7.2)Page342,#’s1-6,17-23odd.

SetC(Section7.2)Page342&343,#’s25,27,31-41odd.

SetD(Section7.2)Page343,#’s53-65odd.

SetE(Section7.1)Page331,#’s1-9odd,29-33odd,41-46.

SetF(Section7.4)Pagepg.361,#’s1-9odd.

SetG(Section7.5)Page#’s1-21odd.

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4.3 Derivatives of Inverse Functions Topics

v Derivatives of Inverse Functions v Derivative of Inverse Trig Functions

Warm Up! Verifythatthefollowingtwofunctionsareinversesofeachother.

a. ! " = 2"% − 1,( " = )*+,

- b. (2,-3)ison! " .Nameapoint

on!.+ " Example1:InvestigatingInversesForthefollowingusethefunction! " = 2" + 3.Findthefollowing:a. Theslopeof!(") b. !.+(")c. Theslopeof!.+(") d. Arelationshipbetweentheslopes

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Exploration:FindingaDerivativeonanInverseGraphGeometricallyLet! " = "3 + 2" − 1.Sincethepoint(1,2)isonthegraphoff,itfollowsthatthepoint(2,1)isonthegraphof!.+.Canyoufind

4!.+4" 2 ,

thevalueof6789

6) at2,withoutknowingaformulafor!.+?a. Graph! " = "3 + 2" − 1andsketchitonthecoordinateplaneprovided.Afunction

mustbeone-to-onetohaveaninversefunction.Isthisfunctionone-to-one?

b. Find!′ " .Howcouldthisderivativehelpyoutoconcludethatfhasaninverse?

c. Reflectthegraphoffacrosstheline< = "toobtainagraphof!.+.d. Sketchthetangentlinetothegraphof!.+atthepoint(2,1).CallitL.e. ReflectthelineLacrosstheline< = ".AtwhatpointisthereflectionofLtangentto

thegraphoff?f. WhatistheslopeofthereflectionofL?

g. WhatistheslopeofL?

h. Whatis6789

6) 2

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Example2:DerivativesofInversesGiven! " ,verifythat(0,-1)isonthegraph.Thenfind !.+ =(−1).a. ! " = "3 + 2" − 1 Example3:DerivativesofInverse

Given! " ,find6789

6) ,

a. ! " = "% + 2" − 1.

Derivatives of the Inverse Function at a point (a, b)… this implies the point (b, a) is on the original function. To find the derivative of !.+ at the point (a, b) we find the _______________ of the derivative of f as the point (b, a)

4!.+4" ?

@=

**In other words, if f and g are inverses, then their derivatives at the inverse points are reciprocals.**

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Example4:DerivativesofInverseTrigFunctions

Find6A6)usingimplicitdifferentiation.

a. < = sin.+ " b. < = tan.+ "

Example5:DerivativesofInverseTrigFunctionsFindthederivativeofthefunctiongiven.

a. ! G = sin.+ G, b. < = tan.+ " − 1

Derivatives of Inverse Trigonometric Functions 6

6) [sin.+ "] = 6

6) [cos.+ "] =

6

6) [tan.+ "] =

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7.2 Integration by Substitution Topics

v Substitution in Indefinite Integrals v Substitution in Definite Integrals

Warm Up! Find6A6).a. < = 3L)

, 4G b. < = 3L)M 4G c. < = "% − 2", + 3 N

d. < = sin, 4" − 5 e. < = ln cos "Example1:AntiderivativesfromDerivativeRulesCompletefrommemory. ! ! ! ! "R sec, "

".+ = +) csc, "

T) +

+.)Usin V" +

)U*+cos V"

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Example2:RememberingtheChainRuleUsethechainruletodifferentiate!(").a. ! " = 3"N − 5 +, Example3:ConnectingChainRuletoAnti-DerivativesTellwhetherornoteachantiderivativeisgoingtoundoachainrule.a. ", − 1 % 2"4" b. 3", "% + 24" c. " "% − 5 4" Example4:IndefiniteIntegrationwithSubstitutionIntegratea. "% "N + 24"

U-Substitution Method 1. Let u be the “_________ function” and du =W′(")4". Be sure to list this!! 2. Find ∫ 3. Re-substitute so expression in the terms of the original function.

What to look for: A function & its derivative or constant multiple of its derivative **It is important to note, that with substitution, the goal is to substitute ALL values of the integrand with either u or du. Any “extras” must be accounted for, or substitution will NOT work!

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b. sin, 3" cos(3") 4"c. )

)U*, 4"

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d. ", 5 + 2"%4"e. cot 7" 4"

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Example5:DefiniteIntegrationwithSubstitutionEvaluatea. UsingMethod1. " 1−",4"+

M

Definite Integrals with Substitution Indefinite Integrals needed a “+C” at the end of every antiderivative… Definite Integrals have limits. If you change the variables, the limits still refer to the ______________ variable. How will you decide to deal with those limits? You have two choices…

1. __________ the limits in terms of the of the original variable and integrate like you did for the indefinite integrals. Once you have returned all variables _________ to the original letter, you can plug in the upper and lower limits.

2. Using the rule for the change of variables, change the ___________ with the same rule, then you never need to return to the original.

**The limits must match the variable being used, or there must be some notation to indicate that the limits being used are different from the variable being used!**

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b. UsingMethod2. [\] )N*% ]^_ ) 4"

`.`

c. Usingamethodofyourchoice. ab_- c[\]U c 4d

efM

When substitution doesn’t work, you sometimes to “massage” the problem into a form that will work using some algebraic techniques. When integrating, be sure to follow the guidelines

1. Memorized integrals 2. U-substitution 3. Change with algebra

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Example6:UsingAlgebraicTechniquesLongDivision–Whenthenumeratorhasadegreegreaterthanorequaltothedenominatora. )U.+

)U*+ 4"Example7:UsingAlgebraicTechniquesExpand–Whenthe“inside”doesn’thaveaderivativeontheoutside,tryexpandingthefunction.a. sin " + cos " , 4"

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Example8:UsingAlgebraicTechniquesCompletetheSquare–Usefulwhenyouhavea",andxterminthedenominatorbutnoxterminthenumerator.a. ,6)

)U.g)*+M Example9:UsingAlgebraicTechniquesSeparatethenumerator–whenyouhavemorethanoneterminthenumeratora. %)*,

+.)U 4"

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Example10:IntegrationwithTrigIntegratea. tan " 4" b. cot " 4"c. tan, " 4" d. cot, " 4"e. 6)

N.)U

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7.1 Differential Equations & Slope Fields Topics

v Differential Equations v Slope Fields

v Euler’s Method

Warm Up! FindtheconstantC.a. < = 2 sin " − 3 cos " + hand< = 4when" = 0

Example1:FindingGeneralSolutionsFindthegeneralsolutiontothefollowingdifferentialequation.(separateandintegrate)a. 6A

6) =%)UA

Differential Equations A differential equation is an equation that relates a function (or relation) with its __________________. Just like in Algebra, when you want to solve an equation, you use an inverse operation. To “undo’ a derivative we find ______________________. Recall, that a function can have many antiderivatives, all of which vary by a ______________. The ___________ of a differential equation is the order of the highest derivative involved in the equation.

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Example2:FindingParticularSolutionsFindtheparticularsolutiontothefollowingdifferentialequation.(separateandintegrate)a. 6A

6) = sin "if< 0 = 2b. 6A

6) =,)

,A.)UAif< 1 = 2

Finding the ______________ solution to a differential equation involves finding a unique equation that satisfies some ______________ conditions or ______________ values. In other words, instead of finding a family of functions (or relation), you are finding the function (or relation)

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Example3:SketchingSlopeFieldsOnthediagramprovided,plottheslopefieldofthedifferentialequationprovided.a. 6A

6) = 2< b. Supposethatyouknowthatthepoint(0,-1)isonaparticularsolutionofthedifferential

equationabove.Byfollowingslopes,drawonthediagramsabovewhatyouthinktheparticularsolutionlookslike.**Thegraphshouldfollowthepatternoftheslopefield,butmaygobetweenthepointsratherthanthroughthem**

c. Solvethedifferentialequation6A6) = 2<fromthepreviousexamplebyfirstseparating

thevariables.Findtheparticularsolutionthatcontainsthepoint(0,-1).Doesyoursolutionmakesensewhencomparedtothegraphoftheslopefield?

A Graphical Look at Differential Equations A ___________ field (or direction field) for the first order differential equation 6A6) = !(", <) is a plot of short line segments with slope !(", <) for a lattice of points (", <) in the plane.

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Example4:MatchingSlopeFieldsMatchthedifferentialequationwiththeappropriateslopefield.Thenusetheslopefieldtosketchthegraphoftheparticularsolutionthroughthepoint(3,2).(Allslopefieldsareshowninthewindow[-6,6]by[-4,4].)a. 6A

6) = "b. 6A

6) = <c. 6A

6) = " − <d. 6A

6) = < − "e. 6A

6) = − A) f. 6A

6) = − )A

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Example5Sketchanapproximatesolutionofthedifferentialequationoftheslopefield,passingthroughthegivenpoint.Example6

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7.4 Exponential Growth and Decay Topics

v Exponential Growth v Exponential Decay

Warm Up! Rewritetheequationinexponentialformorlogarithmicform.a. lnj = k b. Tl = 4Solvetheequationc. ln " + 3 = 2 d. 2m*+ = 3m Example1Therateofchangeofyisproportionaltoy.Whent=0,y=2andwhent=2,y=4.Whatisthevalueofywhent=3.

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Example2Waterflowscontinuouslyfromalargetankatarateproportionaltotheamountofwaterremaininginthetank.Therewasinitially10,000cubicfeetofwaterinthetankandattimet=4hour,8000cubicfeetofwaterremained.a. Whatisthevalueofkintheequation.b. Tothenearestcubicfoot,howmuchwaterremainedinthetankattimet=8hour?Example3Ifasubstancedecomposesatarateproportionaltotheamountofthesubstancepresent,andiftheamountdecreasesfrom40gto10gin2hr,thentheconstantofproportionalityisa. -ln2 b. -0.5 c. -0.25 d. ln(0.25) e. ln(0.125)