BusinessCalculusFinalExamReview

1. Evaluatethefollowinglimits:

a. limx→4

6x +1− 5x − 4

b. limx→4

x2 − 2x −8x − 4

2. Determinewhichstatementbestdescribesthecontinuityofthefunction:

f x( ) =x2 −3 for x < −22x + 5 for − 2 ≤ x <17x for x >1

⎧

⎨⎪

⎩⎪

a. f x( ) iscontinuousforallrealvaluesof x .b. f x( ) iscontinuousforallrealvaluesof x except x = −2. c. f x( ) iscontinuousforallrealvaluesof x except x =1. d. f x( ) iscontinuousforallrealvaluesof x except x = −2 and x =1.

3. Firststatethedefinitionofthederivativeandthenuseittoconstructthe

derivativeof f x( ) = 3x2 − 5x + 4

4. Findthederivativeofeachofthefollowing:

a. f x( ) = 7− x + 4x2 + 5x4 b. y = 2x3−4x

5. Usemarginalanalysistopredicttheestimatedprofitsfromthe221stunit

soldifC x( ) =1200+ 5x andR x( ) = −0.04x2 +32x − 200.

6. Findtheequationofthetangentlineto f x( ) = 3x2 − 2x + 5 at x =1 .

7. Useimplicitdifferentiationtofind dydxif y3 − 5x + 6 = 3x2y4 .

8. Amanufacturercansell20,000itemsatapriceof$100.Foreachdropin

priceof$2,themanufacturercansellanadditional1000items.a. Constructtherevenuefunction.b. Howmanyitemsmustbesoldtomaximizerevenue?c. Whatisthemaximumrevenue?

9. Findderivativesofeachofthefollowingfunctions

a. h x( ) = x +3x2 + 2

b. y = 2x3 − 5( )6

c. F x( ) = ln 8x + 2x4( ) d. h x( ) = e4 x

e. f x( ) = x3e−x2

f. h x( ) = ln 4x2 − 2x3( )

10. FindtheAbsoluteMaximumandAbsoluteMinimumoff x( ) = −x3 + 4x2 +3 on −1, 5[ ].

11. A15-ftladderisplacedagainstawall.Theladderisslidingawayfromthe

buildingataconstantrateof6ft/sec.Howfastistheladderslidingdownthewallwhenthebottomoftheladderis8feetfromthebuilding?

12. Howlongwillittakeaninvestmentof$45,000tobecome$100,000ifitearns4.65%compoundedcontinuously?

13. Howmuchmoneyshouldbeinvestedtodayinordertohaveatotalof

$30,000ineightyearsiftheinterestrateis4.8%compoundedcontinuously?

14. Findtheelasticityofdemandforapriceof$75.Theprice-demandequation

is 4x + p = 800. E p( ) =−p ⋅ ʹf p( )f p( )

.

15. Usethefirstderivativetesttoidentifyallrelativeminimumsandmaximums.

Besuretoidentifytheintervalswherethefunctionisincreasingand

decreasing. f x( ) = −x5 + 52x4 + 40

3x3 + 5

16. Evaluatethefollowinglimits.

a. limx→2

x3 −83x − 6

b. limx→1

4 ln x7x − 7

c. limx→0

ex − e−x

x

17. Giventhefunction f x( ) = 54x2 − x4 .Determinetheintervalswherethe

functionisconcaveupandtheintervalswherethefunctionisconcavedown.

18. Findthespecificfunction f x( ) satisfyingthefollowingconditions.ʹf x( ) = 6x2 + x − 2 and f 4( ) = 5.

19. Evaluatetheintegral 4x3 − 4x +3( )−1

3

∫ dx .

20. Evaluatethedefiniteintegrals.

a. x2 x3 +3( )3

−2

1

∫ dx

b. x2ex3+1

−2

1

∫ dx

c. 2xx2 + 42

10

∫ dx

21. Findtheareaboundedby y = x + 5 and y = x2 − 6x +11

22. Timinvestsmoneyina403baccountaccordingtothefunction

f t( ) =15,000e0.02t .Iftheaccountearnscontinuousgainsat5.5%,andthecontributionsaremadefor25years,whatwillbethevalueoftheaccount?

UseFV = erT f t( ) ⋅e−rt dt0

T

∫ androundtothenearestdollar.

23. Therateofchangeoftheincomeproducedbyavendingmachineis 𝑓 𝑡 = 5000𝑒!.!"! .wheretisthetimeinyearssinceinstallation.Findthetotalincomeproducedbythemachineduringthefirstfiveyearsofoperation.

24. Find f −3,2( ) given f x, y( ) = 4xy2

2x + y( )2.

25. Identifyallcriticalpointsandclassifyeachasarelativemin,relativemax,or

saddlepoint. f x, y( ) = x2 + y2 − xy+ x3

26. Giventhefunction f x, y( ) = 2x4 + y2 −12xy withpartialderivativesfx = 8x3 −12y and fy = 2y−12x .Identifyallcriticalpointsandclassifyeachasarelativemin,relativemax,orsaddlepoint.

27. Giventhefunction f x, y( ) = 1yln x .Findthesecond-orderpartial fxy

28. Giventhefunction f x, y( ) = 2x4 − y5 + 4xy6 .

a. Find fx. b. Find fy. c. Find fxx. d. Find fyy. e. Find fxy. f. Find fyx.

29. Giventhefunction f x, y( ) = 2xy−y6x

a. Find fx. b. Find fy. c. Find fxx. d. Find fyy. e. Find fxy. f. Find fyx.

30. UsethemethodofLagrangemultiplierstofindtheminimumandmaximum

of f x, y( ) = 2xy subjectto x2 + y2 =18 .

31. Findtheconsumers’surplusatapricelevelof$8fortheprice-demandequationof𝑝 = 𝐷 𝑥 = 20− 0.05𝑥

32. Findtheproducers’surplusatapricelevelof$20fortheprice-supply

equation𝑝 = 𝑆 𝑥 = 2+ 0.0002𝑥!

33. Findtheequilibriumpriceandthenfindtheconsumers’surplusandproducers’surplusattheequilibriumpricelevelif𝑝 = 𝐷 𝑥 = 20− 0.05𝑥and𝑝 = 𝑆 𝑥 = 2+ 0.0002𝑥!

34. Evaluatethefollowinglimits:

a. limx→∞

7e3x

5e3x +9

b. limx→∞

4ln x3x2 + 2

35. Let f x, y( ) = 2ex2y3

a. Evaluate f x b. Evaluate f y