Integrated Math 3 Module Honors Limits Introduction to ...msshultis.weebly.com/uploads/1/0/9/3/10930910/2016...9. Cindy is the star runner of her school’s track team. Among other

1

SDUHSDMath3Honors

IntegratedMath3Module8Honors

Limits&IntroductiontoDerivativesReady,SetGoHomework

Solutions

Adaptedfrom

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutorius

©2014MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense.

2

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.1HReady,Set,Go!ReadyTopic:SimplifyingrationalexpressionsSimplifyeachrationalexpression.

1. 2. ⋅

3. 4.

5.

3

SDUHSDMath3Honors

SetTopic:RatesofchangeCardiffKookAcademyisheadedtotheUniversityofUtahforaRoboticstournament.TheRoboticsteamhasdecidedtotakeatraintogettotheirtournamentsotheycanensurethesafekeepingoftheirrobots.TheyneedtocatchanearlytrainsincetheUniversityofUtahis750milesfromhome.6. Thetrainleavesat6:00a.m.Assumethetrainrideisexactly750mileswithnostops.Howmanymiles

perhourmustthetrainaveragefortheCardiffKookAcademyRoboticsteamtogettheUtahby5:00p.m.?Note:Utahis1houraheadofSanDiego.

75mph7. The6:00a.m.trainaverages50milesperhourforthefirsttwohours.Whatspeedmustitaverageforthe

restofthetripfortheCardiffKookAcademyRoboticsteamtoreachtheUniversityofUtahby5:00pm? 81.25mph8. Supposethetrainactuallyaveraged60milesperhourforthewholetrip(whichmeansthetriptook12.5

hoursaltogether).Thatdoesn’tnecessarilymeanthetraintraveledataconstantrateof60mph. Makeupascenarioinwhichthetrainaverage60mphforthetripbuttraveledatleasttwodifferent

speedsalongtheway.Bespecificaboutspeeds,times,anddistances. AnswersmayvaryGoTopic:FeaturesoffunctionsandgraphingfamiliesoffunctionsGrapheachfunctionandidentifytheindicatedfeaturesofthefunction.9. 2 7 Domain: ∞,∞ Range: ∞, Interval(s)ofIncrease: ∞, Interval(s)ofDecrease: , ∞ EndBehavior: As → ∞, → ∞ As → ∞, → ∞

4

SDUHSDMath3Honors

10. 3 1 2 Domain: ∞,∞ Range: ∞,∞ Interval(s)ofIncrease:Approximateanswersgiven. ∞, . ∪ . ,∞ Interval(s)ofDecrease:Approximateanswersgiven. . , . EndBehavior: As → ∞, → ∞ As → ∞, → ∞

11. 2√ 4 5 Domain: , ∞ Range: , ∞ Interval(s)ofIncrease: , ∞ Interval(s)ofDecrease:NA EndBehavior: As → , → As → ∞, → ∞

12. Domain: ∞, ∪ ,∞ Range: ∞, ∪ ,∞ Interval(s)ofIncrease: ∞, ∪ ,∞ Interval(s)ofDecrease:NA EndBehavior: As → ∞, → As → ∞, →

5

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.2HReady,Set,Go!ReadyTopic:ReadingfunctionvaluesfromagraphUsethegraphattherighttoanswereachquestion.1. 2 2 2. 1 3. 4 undefined 4. 1 35. Findthevalue(s)of when 7. & 6. Findthevalue(s)of when 1. 7. Findthevalue(s)of when 3. , ,

6

SDUHSDMath3Honors

SetTopic:Averagespeed8. Rogerisanadventureseekerwholovesthethrillofcliffdiving.Hismostfamousdiveisoffofacliffinto

LakeChamplaininRedRocksPark,Vermont.

Unfortunately,Rogerhassprainedhiswrist.Roger’sdoctorisconcernedthathemightdofurtherdamagetohiswristifhehitsthewateratspeedsgreaterthan58meterspersecond.ThecliffatRedRocksParkis89metershigh.Rogeralwaysbeginshisdivewithajump,soheactuallystartshisfallfromaheightof90meters.Therefore,hisheightabovethelakeisgivenbytheformula

90 10 wheretisthetime(inseconds)fromwhenhebeginstofalltowardsthewaterandistheheight(inmeters)abovethelake.

a. HowhighabovethelakeisRoger1secondafterhebeginshisfall? 80metersb. WhatisthevalueoftwhenRogerhitsthewater? 3secondsc. WhatisRoger’saveragespeedduringthefinalsecondofhisdive? 50meterspersecondd. WhatisRoger’saveragespeedduringthefinalhalf‐secondofhisdive? 55metersperseconde. CanRogerperformhisfamousdivewithoutviolatinghiddoctor’sinstructions?Explain. Nobecausehewouldbegoingcloseto60meterspersecondwhenhehitsthewater.

7

SDUHSDMath3Honors

9. Cindyisthestarrunnerofherschool’strackteam.Amongotherevents,sherunsthatlast400metersofthe1600‐meterrelayrace.

Cindy’scoachstudiedthevideoofoneofherraces.Hecameupwiththeformula 0.1 3 todescribethedistanceCindyhadrunatgiventimesintherace.Inthisformula, givesthenumberofmetersCindyhadrunaftertseconds,withtimeanddistancemeasuredfromthebeginningonher400‐metersegmentoftherace.(AdaptedfromInteractiveMathematicsProgram,Year3)a. HowlongdidittakeCindytofinishherlegoftherelayrace?Explainhowyoufoundyouranswer. 50seconds,find where b. ThecoachphotographedCindyattheinstantshecrossedthefinishline.Thephotoisslightly

blurred,soyouknowCindywasgoingprettyfast,butyoucan’ttellherexactspeedattheinstantthephotowastaken.FindCindy’sspeedatthatinstant.

Approximately13meterspersecondc. FindCindy’sspeedatthreeotherinstantsduringtherace. Answerswillvary.StudentsshouldnotethatCindy’sspeedincreasesthroughouttherace.d. WasthereaninstantwhenCindywasgoingexactly10meterspersecond?Ifso,whenwasthat

instant?

Shewillrunexactly10meterspersecond35secondsintotherace.

8

SDUHSDMath3Honors

GoTopic:SolvingtrigonometricequationsSolveeachtrigonometricequationoverthedomain , .10.4 sin 5 6 11.2 cos 2 1 1 , , , , 12.cos 2 1 3 cos 13.5 sin 2 6 sin 0 , , , . , .

14.2 sec 4 0 15.csc 2 0

, , , , 16.cot sec cot 0 17.sec csc 2 csc ,

9

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.3HReady,Set,Go!ReadyTopic:GraphingpiecewisefunctionsGrapheachpiecewisefunction. 

1.5, 2

2 3, 2 2.

2 1, 24 1, 2

3.

4 3, 4

2| 1| 3, 4 3

3 2, 3

10

SDUHSDMath3Honors

SetTopic:Usingsecantstofindthederivativeofafunctionatapoint.Asecantlineforthegraphofafunctionistheline(orlinesegment)connectingtwopointsonthegraph.Atangentlineisalinethat“justtouches”thegraphatapoint(sometangentlinesmaycrossthecurveelsewhere).Inthefollowingproblems,youwillexplorethesetwogeometricconceptsandtheirconnectionswithderivatives.4. Considerthefunctionfdefinedbytheequation 0.25 .

a. Sketchthegraphofthisfunction.Labelthepoint 2, 1 onyourgraph.

b. Thepointslistedbelowarealsoonyourgraph.Ineachcase,useastraightedgetodrawthesecant

lineconnectingthepointto 2, 1 andfindtheslopeofthatsecantline. i. 0, 0 ii. 1, 0.25 iii. 1.5, 0.5625 iv. 1.9, 0.925

0.5 0.75 0.875 0.975

c. Drawthelinethatistangenttoyourgraphat 2, 1 .Estimatetheslopeofthattangentlineandexplainyourreasoning.

Theslopeappearstobe1.Theslopeofthesecantline,asitapproachesthepoint , ,isapproachingthevalueof1.

d. Findthederivativeofthefunctionfatthepoint 2, 1 . 1

11

SDUHSDMath3Honors

5. Considerthegraphofthefunctionattheright. a. Drawthetangentlinestothegraphatthe

pointsA,B,andC. b. Useyourtangentlinestoestimatethe

derivativeofthefunctionateachofthepoints.

A:approximately B:approximately0.5 C:0GoTopic:SolvingexponentialandlogarithmicequationsSolveeachequation.6. 3 81 7. 2 4 5 . 8. log 3 1 2 9. log 2 1 2

12

SDUHSDMath3Honors

10.5 4 11. log 5 log 2 3 . 12.4 ln 2 3 11 13.2 5 3 0

. .

13

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.4HReady,Set,Go!ReadyTopic:DomainandrangeWritethedomainandrangeofeachfunctioninintervalnotation.1. 2. √ 2 5 Domain: ∞, ∪ ,∞ Domain: , ∞ Range: ∞, ∪ ,∞ Range: , ∞ 3. 4. 2 5 8

Domain: ∞, ∪ , ∪ ,∞ Domain: ∞,∞ Range: ∞,∞ Range: , ∞ 5. 6.

Domain: ∞, ∪ ,∞ Domain: ∞, ∪ , ∪ ,∞ Range: ∞,∞ Range: ∞, ∪ ,∞

14

SDUHSDMath3Honors

Topic:Usinggraphstodeterminethesignsofthederivativefunctions.Whenyouaregraphingafunction,knowingthesignsofthecoordinatesofthepointscanbehelpful.Forinstance,thosesignstellyouwhichquadrantapointisin.Andifoneofthecoordinatesis0,youknowthatthepointisonacoordinateaxis.Inthefollowingquestions,you’llexploresimilarissuesconcerningthesignofthederivativeofafunction.7. Usingthreedifferentcolors,identifywhereonthegraphthefunction’sderivativeispositive,wherethe

derivativeisnegative,andwherethederivativeis0.Rememberthattheslopeofatangentlineatapointonthegraphisthesameasthederivativeatthepoint.

8. Sketchthegraphofafunctionforwhichthederivativeispositiveforallvaluesofx. Answersmayvary.Twosampleanswersprovided.

15

SDUHSDMath3Honors

9. Sketchthegraphofafunctionforwhichthereareexactlytwopointswherethederivativeis0. Answersmayvary.Sampleanswerprovided.

SetTopic:FindinglimitsusingtablesandgraphsUsethegraphattherighttoanswerthefollowingquestionsabout .10. lim

→ 11. lim

→

12. lim

→ doesnotexist 13. 5

14. 3 undefined 15. 2 .

16

SDUHSDMath3Honors

Completethetablesofvaluesforeachfunctiontodeterminethevalueofthelimit.

16. 17.

lim→ . lim

→ .

1.5 0.28571 2.5 . 1.9 0.25641 2.9 . 1.99 0.25063 2.99 . 2 Undefined 3 Undefined

2.01 0.24938 3.01 . 2.1 0.2439 3.1 . 2.5 0.2222 3.5 .

GoTopic:FactoringpolynomialexpressionsFactoreachexpressioncompletely.18. 9 8 19.2 17 21 20.28 16 80 21.5 18 22.64 81 23.16 24 9 24.3 5 2 25.12 2 2

17

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.5HReady,Set,Go!ReadyTopic:FindinglimitsusinggraphsandtablesUsethegraphof attherighttofindthefollowinglimitsandfunctionvalues.1. lim

→ 2. lim

→2

3. lim

→ 4. 6 undefined

5. 3 6. 5 Findthelimitofeachfunctionrepresentedbythetablesbelow.7. lim

→ 8. lim

→

0.5 4.5 4.5 1.09090.8 4.8 4.2 1.03850.9 4.9 4.1 1.01960.999 4.99 4.001 1.00021 Undefined 4 Undefined

1.001 5.001 3.999 0.99981.1 5.1 3.9 0.97961.2 5.2 3.8 0.95831.5 5.5 3.5 0.8889

18

SDUHSDMath3Honors

SetTopic:ComparingderivativesatpointsBelowaretheequationsforthreefunctionswhosegraphsallpastthroughthepoints , and , .

Thefollowingquestionswillhaveyouinvestigatewhetherornotthesefunctionshavethesamederivativesat , and , .9. Drawgraphsofallthreefunctionsonthesamesetofaxes.Plotenoughpointsforeachfunction

(includingnon‐integervaluesofx)togetaccurategraphs.Takeparticularcareinplottingvaluesofxbetween0and1.

10.Basedonyourgraphs,answereachofthefollowingquestionsandexplainyourreasoning. a. Whichofthethreefunctionshasthegreatestderivativeatthepoint 0, 0 ?

b. Whichofthethreefunctionshasthegreatestderivativeatthepoint 1, 1 ? 11.Findtheactualderivativeofeachfunctionatthepoints 0, 0 and 1, 1 .Howdotheycomparewithyour

answerstoquestion10?(Reminder:thederivative, lim → ) Derivativesat 0, 0 : Derivativesat 1, 1 : :1 :1 :0 :2 :0 :3

19

SDUHSDMath3Honors

Topic:Graphingaderivativeusingagivengraph.Letthegraphbelowrepresentsthefunctiondefinedbytheequation .12.Usingthreedifferentcolors,identifywhereonthegraphthefunction’sderivativeispositive,wherethe

derivativeisnegative,andwherethederivativeis0.

13.Sketchthegraphofthederivativeof .

20

SDUHSDMath3Honors

14.Usethegraphbelowtodiscussthecontinuityofthefunction.Statethetypeofdiscontinuity(removableornonremovable)andexplainwhyone(ormore)oftheconditionsforcontinuityisnotmet.

At ,thereisanonremovabledisconituitybecause

→ doesnotexist.

At ,thereisaremovablediscontinuitybecause→ .

GoTopic:Solvingradical,rational,andquadraticequationsSolveeachequation.15. 8 √5 5 3 16. 12 6√ 4 17.√3 √4 1 18. √42 isextraneous

21

SDUHSDMath3Honors

19. 20. 2 21. 22. 6 13 0 isextraneous23. 13 36 0 24.3 16 7 5 , ,

22

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.6HReady,Set,Go!ReadyTopic:EndbehaviorDescribetheendbehaviorofeachfunction.1. 4 8 2. as → ∞, → ∞ as → ∞, →

as → ∞, → ∞ as → ∞, → 3. √ 9 6 4.

as → , → as → ∞, → as → ∞, → ∞ as → ∞, → Topic:Usingareastoconnectdistance,rate,andtime5. Burtontravelsoncruisecontrolat50mphfrom2:00PMuntil5:00PM.

a. Howfarhashetraveled?150milesb. Sketchagraphofthepreviousinformationusingspeedinmphandtimeinhours.Shadethearea

underthegraphinthefirstquadrant.Whatshapeistheshadedfigure? Rectangle

c. Explainwhytheshadedarearepresentsadistanceof150miles. Distanceistheproductoftime(hours)andvelocity(milesperhour)

23

SDUHSDMath3Honors

6. ErinstarteddrivingfromSacramentotoSanFrancisco.Forthefirsthalfhourshedrovethroughresidentialneighborhoodandtraveledataconstantspeedof25milesperhour.Shethengotonthefreeway,onlytoencounterheavytraffic.Shewas,however,abletoslowlyincreaseherspeedataconstantrateuntilshereachedaspeedof75milesperhour,45minutesintohertrip.Shecontinuedatthatspeeduntilshegottoherdestinationafteratotalof2hoursofdriving.Assumeshemadenostopsalongtheway.a. Graphthesituationcarefullyonthegridatright. b. Usingtheunitsfromeachaxis,whataretheresultingunits

whenyoufindthearea? Milesc. HowfardidErintravel? 118.75miles

SetTopic:FindinglimitsusingalgebraicmethodsFindthevalueofeachlimit.

7. lim→ 8. lim

→√

6

9. lim→ 10. lim

→√

2

24

SDUHSDMath3Honors

GoTopic:GraphingtrigonometricfunctionsGraphatleasttwoperiodsofeachtrigonometricfunction.

11. 2 sin 12. tan 2

13. 3 cos 2 1 14. 4 sin 3

15. tan 1 16. 2 cos

25

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.7HReady,Set,Go!ReadyTopic:Evaluatingfunctions

Foreachfunctionbelow,findthevalueof .Anexampleisprovidedforyou.

Example:If ,then 1. 2. √

√ √

3. 4. 2 3

SetTopic:FindinglimitsFindthevaluesofthefollowinglimits.

5. lim→ 6. lim

→√

26

SDUHSDMath3Honors

7. lim→ 8. lim

→√

1

9. lim→

10. lim→

2Usethegraphattherighttofindthevalueofeachlimit.11. lim

→ 12. lim

→

13. lim

→ 14. lim

→ doesnotexist

15. lim

→ 16. lim

→

17. lim

→ 18. lim

→

19. lim

→ ∞ 20. lim

→ ∞

27

SDUHSDMath3Honors

GoTopic:SimplifyingrationalexpressionsSimplifyeachexpressionasmuchaspossible.

21. 22.

23. 24.

25. 26.

28

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.8HReady,Set,Go!ReadyTopic:Estimatingareas1. Theshadedregionbelowhassidesmadeupofthreecurvesandthe ‐axis.Estimatetheareaofthe

region.

Answerswillvary.Approximately27squareunits.Topic:UsingsigmanotationRecall:

2. Findthesum:

4 6

6

3. Rewriteeachinsigma(summation)notation: a. 14 20 26 32 38

or

29

SDUHSDMath3Honors

b. 14 3 20 3 26 3 32 3 38 3

or

SetTopic:FindingderivativesoffunctionsusingthelimitprocessFindthederivativeofeachfunctionusingthelimitdefinition:

→

4. 3 2 5. √ 4

√

6. 7. 6 5 8. 2 5 9. √2 1

√

10.Usethederivativeof 3 2 (question4)tofindwhen hasaslopeof0.Whatfeatureofthe

graphof isatthislocation? ,thisoccursatthevertexoftheparabola

30

SDUHSDMath3Honors

GoTopic:SolvingquadraticandrationalinequalitiesSolveeachquadraticandrationalinequality.Writeyouranswersinintervalnotation.11. 4 3 0 12.5 10 27

, ∞, ∪ ,∞

13. 2 5 12 0 14.4 9

∞, ∪ ,∞ ∞, ∪ ,∞

15. 2 16. 0 , ∞, ∪ ,∞

17. 0 18. 0 , ∪ ,∞ ,

31

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.9HReady,Set,Go!ReadyTopic:Infinitelimits1. Findeachlimitas → ∞(Reminder,thelimitas → ∞issimilartofindingendbehavior):

a. b. c. ⋅ 5

d. 10 e. 6 ⋅

Topic:AreaofcompositeregionsFindtheareaoftheentireshape.Useonlyverticalsegmentsandquadrilateralsinyourfiguredissections.2.

a. 88squareunits b. 56squareunits

32

SDUHSDMath3Honors

SetTopic:DistanceasanareaMichellewastrainingforhernexttriathlon.Accordingtohertrainingschedulesheneededtoride75milesonherbikethisupcomingweekendUnfortunately,theweatherreportiscallingforheavyrainsoMichellewillhavetodoherbikerideonastationarybike.ThestationarybikeMichelleusesonlyshowsthe“speed”ofthebike.3. Michellenoticedthatshewasabletokeepasteadypaceof20mphforthefirsthourshewasonthe

stationarybike.Shewasabletoincreaseherspeedto30mphforthenext45minutes.Shethenslowedto20mphforthenexthour.

a. Drawagraphofthissituation.Besuretolabelandscaletheaxesappropriately.

b. Findtheareaunderthecurveofyourgraph.

62.5squareunitsc. HowfarhasMichellegone?

62.5milesd. Ifshewantstobedonewithherworkoutin15minute,howfastshouldshegoinordertocomplete

her75miletrainingride?Doesthisseemreasonable?

50mph,thisdoesnotseemreasonable

33

SDUHSDMath3Honors

GoTopic:LimitsFindthevaluesofthefollowinglimits.

4. lim→ 5. lim

→ √

6. lim→ 7. lim

→√

Usethegraphattherighttofindthevalueofeachlimit.8. lim

→ 9. lim

→

10. lim

→ 11. lim

→

12. lim

→ 13. lim

→

14. lim

→ ∞ 15. lim

→

16. lim

→ 17. lim

→ ∞

34

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.10HReady,Set,Go!ReadyTopic:SigmanotationRecall:

1. Findthesum:

2 1

4282. Rewriteeachinsigma(summation)notation: a. 2.4 2.8 3.2 3.6 4

. .

or

.

b. 2.4 1 2.8 1 3.2 1 3.6 1 4 1

. .

or

.

35

SDUHSDMath3Honors

Topic:Infinitelimits3. Findeachlimitas → ∞(Reminder,thelimitas → ∞issimilartofindingendbehavior):

a. b. c. ⋅

d. 10 e. 3 ⋅

SetTopic:EstimatingareasusingRiemannsums4. Let √ 5.

a. Graphthefunctionoverthedomain 1, 6 onthegridbelow.

b. Findanapproximationoftheareaunderthecurvebyusing5right‐endpointrectanglesofequal

width. .

36

SDUHSDMath3Honors

5. Youaretryingtofindanapproximationfor 6 .Theregionisdrawnforyouatright.a. Use4right‐endpointrectanglestofindanapproximationforthearea. . b. Expresstheareaapproximationinsigmanotation.

. .

GoTopic:VerifyingtrigonometricidentitiesVerifyeachtrigonometricidentity.6. cot 1 csc cos sin 7. sec csc Answersmayvary Answersmayvary8. 9. 2 sec Answersmayvary Answersmayvary

37

SDUHSDMath3Honors

10. tan sin tan sin 11.csc cot csc cot Answersmayvary Answersmayvary12. cot csc 13. 2 tan Answersmayvary Answersmayvary

38

SDUHSDMath3Honors

Name Limits&IntroductiontoDerivatives 8.11HSetTopic:FindingvaluesoflimitsFindthevaluesofthelimitsusingthegraphattheright.1. lim

→ 7 2. lim

→ 2

3. lim

→ doesnotexist 4. lim

→ 5

5. lim

→ 5 6. lim

→ 5

7. lim

→ 7 8. Lim

→ ∞

9. lim

→ ∞

Findthevaluesofthefollowinglimits.

10. lim→

11. lim→

12. lim→

13. lim→

14. lim→

√ 15. lim→√

16. lim→ 17. lim

→

18.Usethefunction 6, 22, 2

tofindthevaluesofthefollowinglimits:

a. lim

→ 10 b. lim

→ 10 c. lim

→ 10

19.Usethefunction5, 17, 1tofindthevaluesofthefollowinglimits:

a. lim

→ 6 b. lim

→ 8 c. lim

→ doesnotexist

39

SDUHSDMath3Honors

Topic:FindingderivativesusingthelimitprocessFindthederivativeofeachfunction.Thatis,find:

→

20. 3 21. 2 22. 23. √ 4

√

Topic:Averagerateofchangeandinstantaneousrateofchange24.Thefunction describesthevolumeofacube. ismeasuredincubicinches.Thelength,

width,andheightareeachmeasuredininches. a. Findtheaveragerateofrangeofthevolumewithrespecttoxasxchangesfrom5inchesto5.1. 76.51cubicinchesperinch b. Findtheaveragerateofrangeofthevolumewithrespecttoxasxchangesfrom5inchesto5.01

inches. 75.1501cubicinchesperinch c. Findtheinstantaneousrateofchangeofthevolumewithrespecttoxatthemomentwhen 5

inches. 75cubicinchesperinch

40

SDUHSDMath3Honors

25.Aballisthrownstraightupfromarooftop160feethighwithaninitialspeedof48feetpersecond.Thefunction 16 48 160describestheball’sheightabovetheground, ,infeet,tsecondsafteritisthrown.Theballmissestherooftoponitswaydownandeventuallystrikestheground.

a. Whatistheinstantaneousspeedoftheball2secondsafteritisthrown? feetpersecond b. Whatistheinstantaneousspeedoftheballwhenithitstheground? feetpersecondTopic:EstimatingareasusingRiemannsums26.Considerthefunction, 6for0 4.Estimate usingright‐endpoint

rectanglesofwidth1unit.Followthestepsbelowasnecessarytocompletetheproblem:a. Sketchaneatgraphshowingthecurveovertheindicateddomain.Drawinright‐endpointrectangles

fromthex‐axistothecurveshowingawidthof1unitforeachrectangle.Therectanglesshouldbebelowthecurve.

b. Findtheheightofeachrectanglebyusingthe ‐valuesof . Seepointslabeledongraphabovec. Writeanexpressionforthesumoftheareasoftherectangles. ⋅ ⋅ ⋅ ⋅ ⋅ . ⋅ ⋅ . ⋅ d. Estimate .

. 27.Considerthefunction, 6for0 4.Estimate usingright‐endpoint

rectanglesofwidth0.5units.Followthestepsbelowasnecessarytocompletetheproblem:

41

SDUHSDMath3Honors

a. Sketchaneatgraphshowingthecurveovertheindicateddomain.Drawinright‐endpointrectangles

fromthex‐axistothecurveshowingawidthof0.5unitsforeachrectangle.Therectanglesshouldbebelowthecurve.

b. Findtheheightofeachrectanglebyusingthe ‐valuesof . Seepointslabeledongraphabovec. Writeanexpressionforthesumoftheareasoftherectangles. . ⋅ . . ⋅ . ⋅ . . ⋅ . ⋅ . . ⋅ . ⋅ .

. ⋅

. ⋅ . . ⋅ . . ⋅ . . ⋅ . ⋅ . . ⋅ . . ⋅. . ⋅

d. Estimate .

. 28.Whyistheapproximationinquestion27betterthantheoneinquestion26? Morerectanglesgivesabetterapproximation