1

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

Mr. Mellina

Chapter 5: Applications of Derivatives Part 2

Sections:

v 5.3 Cont.: Connecting f’ and f’’ with the Graph of f v 5.4 Modeling and Optimization v 5.3 Cont.: PVA

HWSets

SetA(Section5.3)Ch5P2WS#93:#’s1&2.2005AB4(a&bonly)&1989AB5.

SetB(Section5.3)Ch5P2WS#93:#’s5-7&DrawingGraphsofFunctions Worksheet.

SetC(Section5.3)APPractice(1991AB5,2006FormBAB2partaonly),Ch5P2ApplicationsofDerivativesMultipleChoiceWS#’s12-30.

SetF(Section5.3)5.3PVAPracticeWS#’s1-6.

SetG(Section5.3)5.3PVAPracticeWS#’s12-14.

SetD(Section5.4)OptimizationWorksheet#’s1-8.

SetE(Section5.4)Page231,#1,6,9,13,14,16.

2

5.3 Cont. Connecting f’ and f’’ with the Graph of f Topics

v First Derivative Test for Local Extrema v Concavity

v Points of Inflection v Second Derivative Test for Local Extrema v Learning about Functions from Derivatives

Warm Up! Findallpointsofinflectionofthegraphof! " = $%&'.

Formulas to Remember

3

Example1:MultipleChoice

Connecting Graphs

4

5

f’’(x) f’(x) f’(x)

6

Example2:ConnectingGraphsConsiderthegraphof!′,thederivativeof) = !(")definedonthedomain−9 < " < 9.Justifyyourresponses.a. Forwhatvaluesofxdoesfhavearelativeminimum?b. Forwhatvaluesofxdoesfhavearelativemaximum?c. Determinetheopenintervalswherethegraphoffisconcavedownwards.Showthe

analysisthatleadstoyourconclusion.d. Sketchthegraphoffontheinterval

−9, 9 if! 0 = 0.Showtheanalysisthatleadstoyourconclusion.

7

Example3:MultipleChoicePractice1. Giventhegraphof) = 2 " ,estimatethevalueof2’(2). 2. AtwhichpointA,B,C,D,orEonthegraphof) = !(")areboth)′and)′′positive? 3. Giventhegraphofℎ′("),whichofthefollowingstatementsaretrueaboutthegraphof

h?

8

Example4:FreeResponsePracticeThegraphofthefunctionfisshowninthefigure.Justifyyourresponses.a. Estimate!′(0).b. Onwhatopenintervalsisfincreasing?c. Onwhatopenintervalsisfconcavedownwards?d. Whatarethecriticalnumbersoff?e. Sketchthegraphof!′.

9

Example5Letthegraphof!5 " begivenbelow.Finda. Thex-coordinateofeachinflectionpointoff.b. Wherethegraphoffisconcaveupandisconcavedown.2013AB4–omitb.(NoCalculator)2003AB4–partsa-c.(NoCalculator)

10

5.3 Cont. PVA & Rectilinear Motion Topics

v Position, Velocity, Acceleration Applications v Rectilinear Motion

Warm Up! Explainhowthepositionfunction!(#),thevelocityfunction%(#),andtheaccelerationfunction&(#)areallrelated.Example1:UsingGraphsThegraphofthepositionfunction' = !(#)ofaparticlemovingalongalineisgiven.Estimatewhen:(justifyyourresponses).a. v(t)=0

b. a(t)=0?c. Theparticlemovingleft?d. Theparticlemovingright?e. Theparticlespeedingup?f. Theparticleslowingdown?

11

g. v(t)=0

h. a(t)=0?i. Theparticlemovingleft?j. Theparticlemovingright? k. Theparticlespeedingup?l. Theparticleslowingdown? Example2:StudyingMotionAlongaLineAparticleismovingalongthex-axiswiththepositionfunctionprovidedbelow.Findwhen:(justifyyourresponses).

) # = 2#: − 14#= + 22# − 5a. v(t)=0 b. a(t)=0?c. Theparticlemovingleft?d. Theparticlemovingright?e. Theparticlespeedingup?f. Theparticleslowingdown?

12

) # = #: − 3# + 3g. v(t)=0 h. a(t)=0?i. Theparticlemovingleft?j. Theparticlemovingright?k. Theparticlespeedingup?l. Theparticleslowingdown?

) # = 6 − 2# − #=m. v(t)=0 n. a(t)=0?o. Theparticlemovingleft?p. Theparticlemovingright?q. Theparticlespeedingup?r. Theparticleslowingdown?

13

5.4 Modeling and Optimization Topics

v Examples from Mathematics v Examples from Business & Industry

v Examples from Economics

Warm Up! Fora,usetheFirstDerivativeTesttoidentifythelocalextremaof4()).Forb,usetheSecondDerivativeTesttoidentifythelocalextremaofℎ()).a. 4 ) = ): − 6)= + 12) − 8 b. ℎ ) = 2): + 3)= − 12) − 3

Optimization One of the oldest applications of Differential Calculus was to find maximum and minimum values of functions by finding where horizontal tangent lines might occur. We will use both algebraic and graphical methods in this section to solve “max-min” problems in a variety of contexts, but the emphasis will be on the modeling process that both methods have in common.

14

Example1:UsingtheStrategyUsetheinformationtosolvetheproblema. Findtwonumberswhosesumis20andwhoseproductisaslargeaspossible.

Strategy for Solving Max-Min Problems 1. _______________ the problem. Read the problem carefully. Identify the

information you need to solve the problem. 2. Develop a Mathematical __________ of the problem. Draw pictures and label the

parts that are important to the problem. Introduce a variable to represent the quantity to be maximized or minimized. Using that variable, write a function whose extreme value gives the information sought.

3. _________ the Function. Find the domain of the function. Determine what values of the variable make sense in the problem.

4. Identify the ___________ Points and _____________. Find where the derivative is zero or fails to exist.

5. __________ the Mathematical Model. If unsure of the result, support or confirm your solution with another method.

6. ____________ the Solution. Translate your mathematical result into the problem setting and decide whether the result makes sense.

_____________ Equation: is what you are trying to maximize or minimize. _____________ Equation: an equation that helps you get to 1 variable.

15

b. Arectangleistobeinscribedunderonearchofthesinecurve.Whatisthelargestareatherectanglecanhave,andwhatdimensionsgivethatarea?

c. Find2positivenumbersthatsatisfythegivenrequirements:Thesecondnumberisthe

reciprocalofthefirstandthesumisaminimum.

16

d. Ihave100ftoffencetomakearectangulardogpen,whatisthemaximumareaIcanconstruct?

Example2:FabricatingaboxUsetheinformationtosolvea. Aboxwithasquarebasewithnotophasasurfaceareaof108squarefeet.Findthe

dimensionsthatwillmaximizethevolume.[Besuretousecalculus]

17

b. Anopen-topboxistobemadebycuttingcongruentsquaresofsidelengthxfromthecornersofa20-by25-inchsheetoftinandbendingupthesides.Howlargeshouldthesquaresbetomaketheboxholdasmuchaspossible?Whatistheresultingmaximumvolume?

Example3:GraphicalExampleUsetheinformationtosolvea. Whatpointsonthegraph' = 4 − )=aretheclosesttothepoint(0,2)?