MidtermȱExamȱReviewȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱPre -CalculusȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱInstructor:ȱRezaȱKhademPleaseȱshowȱallȱwork,ȱNWNC,ȱnoȱworkȱnoȱcredit!ȱPleaseȱkeepȱinȱmindȱthatȱalthoughȱcalculatorsȱareȱallowedȱonȱthisȱtestȱthealgebraicȱstepsȱtoȱarriveȱatȱanȱanswerȱMUSTȱbeȱshown.ȱWriteȱtheȱcorrectȱanswerȱinȱtheȱspaceȱprovidedȱforȱeachȱproblem.

Name:ȱ-------------------------------------------ȱȱȱȱȱȱȱDays/Times:ȱ--------------------------------------Formȱaȱpolynomialȱwhoseȱzerosȱandȱdegreeȱareȱgiven.

1) Zeros:ȱ3,ȱmultiplicityȱ2;ȱȱ-3,ȱmultiplicityȱ2;ȱȱdegreeȱ4 1)

Findȱtheȱdomainȱofȱtheȱrationalȱfunction.

2) H(x)ȱ=ȱ xȱ+ȱ4x4ȱ+ȱ4

2)

Solveȱtheȱproblem.3) Aȱdrugȱisȱinjectedȱintoȱaȱpatientȱandȱtheȱconcentrationȱofȱtheȱdrugȱisȱmonitored.ȱTheȱdrugȇs

concentration,ȱC(t),ȱinȱmilligramsȱafterȱtȱhoursȱisȱmodeledȱby

C(t)ȱ=ȱ 5t3t2ȱ+ȱ2

.

Whatȱisȱtheȱhorizontalȱasymptoteȱforȱthisȱfunction?ȱDescribeȱwhatȱthisȱmeansȱinȱpracticalterms.

3)

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Precalculus Midterm Review!

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4) AȱcompanyȱthatȱproducesȱradiosȱhasȱcostsȱgivenȱbyȱtheȱfunctionȱC(x)ȱ= 15xȱ+ 25,000ȱ,whereȱxȱisȱtheȱnumberȱofȱradiosȱmanufacturedȱandȱC(x)ȱisȱmeasuredȱinȱdollars.ȱȱTheaverageȱcostȱtoȱmanufactureȱeachȱradioȱisȱgivenȱby

_C (x)ȱ=ȱ15xȱ+ȱ25,000

x.

Findȱ_C (200).ȱȱ(Roundȱtoȱtheȱnearestȱdollar,ȱifȱnecessary.)

4)

Findȱtheȱverticalȱasymptotesȱofȱtheȱrationalȱfunction.

5) Q(x)ȱ=ȱ xȱ+ȱ4x2ȱ+ȱ1

5)

UseȱtheȱRemainderȱTheoremȱtoȱfindȱtheȱremainderȱwhenȱf(x)ȱisȱdividedȱbyȱxȱ - c.6) f(x)ȱ=ȱx4ȱ+ȱ8x3ȱ+ȱ12x2;ȱȱxȱ+ȱ1 6)

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Precalculus Midterm Review!

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UseȱtheȱFactorȱTheoremȱtoȱdetermineȱwhetherȱxȱ- cȱisȱaȱfactorȱofȱf(x).7) f(x)ȱ=ȱx3ȱ+ȱ2x2ȱ-ȱ6xȱ+ȱ8;ȱȱȱxȱ+ȱ4 7)

UseȱtheȱRationalȱZerosȱTheoremȱtoȱfindȱallȱtheȱrealȱzerosȱofȱtheȱpolynomialȱfunction.ȱUseȱtheȱzerosȱtoȱfactorȱfȱoverȱtherealȱnumbers.

8) f(x)ȱ=ȱx4ȱȱ-ȱ3x2ȱ-ȱ4 8)

Findȱtheȱrealȱsolutionsȱofȱtheȱequation.9) x3ȱ+ȱ8x2ȱ+ȱ19xȱ+ȱ12ȱ=ȱ0 9)

10) 2x3ȱ-ȱ13x2ȱ+ȱ22xȱ-ȱ8ȱ=ȱ0 10)

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Precalculus Midterm Review!

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Findȱaȱboundȱonȱtheȱrealȱzerosȱofȱtheȱpolynomialȱfunction.11) f(x)ȱ=ȱx4ȱ-ȱ8x2ȱ-ȱ9 11)

UseȱtheȱIntermediateȱValueȱTheoremȱtoȱdetermineȱwhetherȱtheȱpolynomialȱfunctionȱhasȱaȱzeroȱinȱtheȱgivenȱinterval.12) f(x)ȱ=ȱ4x3ȱ+ȱ5x2ȱ+ȱ2xȱ+ȱ8;ȱȱ[-2,ȱ-1] 12)

Informationȱisȱgivenȱaboutȱaȱpolynomialȱf(x)ȱwhoseȱcoefficientsȱareȱrealȱnumbers.ȱFindȱtheȱremainingȱzerosȱofȱf.13) Degreeȱ3;ȱȱzeros:ȱ4,ȱ2ȱ-ȱi 13)

14) Degreeȱ4;ȱȱzeros:ȱȱ8ȱ-ȱ5i,ȱȱ3i 14)

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Precalculus Midterm Review!

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Useȱtheȱgivenȱzeroȱtoȱfindȱtheȱremainingȱzerosȱofȱtheȱfunction.15) f(x)ȱ=ȱx4ȱ-ȱ32x2ȱ-ȱ144;ȱȱzero:ȱ-2i 15)

16) f(x)ȱ=ȱx3ȱ-ȱ3x2ȱ-ȱ5xȱ+ȱ39;ȱȱzero:ȱ-3 16)

Findȱtheȱdomainȱofȱtheȱcompositeȱfunctionȱf ȱHȱg.

17) f(x)ȱ=ȱ 2xȱ+ȱ6

;ȱȱȱȱg(x)ȱ=ȱxȱ+ȱ3 17)

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Precalculus Midterm Review!

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Graphȱtheȱfunctionȱasȱaȱsolidȱlineȱorȱcurveȱandȱitsȱinverseȱasȱaȱdashedȱlineȱorȱcurveȱonȱtheȱsameȱaxes.18) f(x)ȱ=ȱx3ȱ+ȱ3

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

18)

Theȱfunctionȱfȱisȱone-to-one.ȱFindȱitsȱinverse.19) f(x)ȱ=ȱ6x2ȱ-ȱ3,ȱxȱLȱ0 19)

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Precalculus Midterm Review!

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Useȱtransformationsȱtoȱgraphȱtheȱfunction.ȱȱDetermineȱtheȱdomain,ȱrange,ȱandȱhorizontalȱasymptoteȱofȱtheȱfunction.20) f(x)ȱ=ȱ4-xȱ+ȱ3

x-6 -4 -2 2 4 6

y6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y6

4

2

-2

-4

-6

20)

21) f(x)ȱ=ȱ-2x+3ȱ+ȱ4

x-10 -5 5 10

y

10

5

-5

-10

x-10 -5 5 10

y

10

5

-5

-10

21)

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Precalculus Midterm Review!

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Solveȱtheȱequation.

22) 133xȱ+ȱ4

ȱ=ȱ9xȱ-ȱ5 22)

23) exȱ-ȱ3ȱ=ȱ 1e2

xȱ+ȱ623)

Changeȱtheȱlogarithmicȱexpressionȱtoȱanȱequivalentȱexpressionȱinvolvingȱanȱexponent.24) log1/4 16ȱ=ȱ-2 24)

Useȱaȱcalculatorȱtoȱevaluateȱtheȱexpression.ȱȱRoundȱyourȱanswerȱtoȱthreeȱdecimalȱplaces

25) eȱlogȱ60ȱ+ȱlnȱ2logȱ2ȱ+ȱlnȱ50

25)

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Precalculus Midterm Review!

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Findȱtheȱdomainȱofȱtheȱfunction.26) f(x)ȱ=ȱln(5ȱ-ȱx) 26)

27) f(x)ȱ=ȱlog3(81ȱ-ȱx2) 27)

Graphȱtheȱfunction.28) f(x)ȱ=ȱ log2 (xȱ-ȱ2)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

28)

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Precalculus Midterm Review!

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Solveȱtheȱequation.29) log8ȱ(x2ȱ-ȱ7x)ȱ=ȱ1 29)

30) 2ȱ+ȱ8ȱlnȱxȱ=ȱ11 30)

31) lnȱ xȱ+ȱ2ȱ=ȱ2 31)

32) e xȱ+ȱ5ȱ=ȱ7 32)

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Precalculus Midterm Review!

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33) 13ȱ log2 (xȱ+ȱ6)ȱ=ȱ log8 (3x) 33)

34) log3 (xȱ-ȱ1)ȱ+ȱ log3 (xȱ-ȱ7)ȱ=ȱ3 34)

Useȱaȱgraphingȱcalculatorȱtoȱsolveȱtheȱequation.ȱRoundȱyourȱanswerȱtoȱtwoȱdecimalȱplaces.35) log3ȱxȱ+ȱlog6ȱxȱ=ȱ2 35)

36) exȱ-ȱlnȱxȱ=ȱ3 36)

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Precalculus Midterm Review!

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Solveȱtheȱproblem.37) TheȱsizeȱPȱofȱaȱsmallȱherbivoreȱpopulationȱatȱtimeȱtȱ(inȱyears)ȱobeysȱtheȱfunction

P(t)ȱ=ȱ600e0.14tȱifȱtheyȱhaveȱenoughȱfoodȱandȱtheȱpredatorȱpopulationȱstaysȱconstant.Afterȱhowȱmanyȱyearsȱwillȱtheȱpopulationȱreachȱ3000?

37)

38) Howȱlongȱwillȱitȱtakeȱforȱtheȱpopulationȱofȱaȱcertainȱcountryȱtoȱdoubleȱifȱitsȱannualȱgrowth

rateȱisȱ2%?ȱRoundȱtoȱtheȱnearestȱyear.38)

39) Susanȱpurchasedȱaȱpaintingȱinȱtheȱyearȱ2000ȱforȱ$4000.ȱAssumingȱanȱexponentialȱrateȱofinflationȱofȱ2.6%ȱperȱyear,ȱhowȱmuchȱwillȱtheȱpaintingȱbeȱworthȱ8ȱyearsȱlater?

39)

40) Theȱhalf-lifeȱofȱsilicon-32ȱisȱ710ȱyears.ȱIfȱ70 gramsȱisȱpresentȱnow,ȱhowȱmuchȱwillȱbepresentȱinȱ700ȱyears?ȱ(Roundȱyourȱanswerȱtoȱthreeȱdecimalȱplaces.)

40)

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Precalculus Midterm Review!

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41) Anȱendangeredȱspeciesȱofȱfishȱhasȱaȱpopulationȱthatȱisȱdecreasingȱexponentially(Aȱ=ȱA0ekt).ȱTheȱpopulationȱ8ȱyearsȱagoȱwasȱ1600.ȱToday,ȱonlyȱ900ȱofȱtheȱfishȱareȱalive.Onceȱtheȱpopulationȱdropsȱbelowȱ100,ȱtheȱsituationȱwillȱbeȱirreversible.ȱWhenȱwillȱthishappen,ȱaccordingȱtoȱtheȱmodel?ȱ(Roundȱtoȱtheȱnearestȱwholeȱyear.)

41)

42) Aȱthermometerȱreadingȱ79°Fȱisȱplacedȱinsideȱaȱcoldȱstorageȱroomȱwithȱaȱconstanttemperatureȱofȱ37°F.ȱIfȱtheȱthermometerȱreadsȱ73°Fȱinȱ13ȱminutes,ȱhowȱlongȱbeforeȱitreachesȱ57°F?ȱAssumeȱtheȱcoolingȱfollowsȱNewtonȇsȱLawȱofȱCooling:

ȱȱUȱ=ȱTȱ+ȱ(U0ȱ-ȱT)ekt.(Roundȱyourȱanswerȱtoȱtheȱnearestȱwholeȱminute.)

42)

43) TheȱlogisticȱgrowthȱmodelȱP(t)ȱ=ȱ 11001ȱ+ȱ23.44e-0.356t

ȱrepresentsȱtheȱpopulationȱofȱa

bacteriumȱinȱaȱcultureȱtubeȱafterȱtȱhours.ȱWhatȱwasȱtheȱinitialȱamountȱofȱbacteriaȱinȱthepopulation?

43)

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Precalculus Midterm Review!

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