Old Final Exam - Washington State University · Question 2. (6 points each) Compute the following...
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Math 171 Fall 2014 Final Exam December 15, 2014 Last Name: ___________________________________________________________ First Name: ___________________________________________________________ Student ID: ___________________________________________________________ Section: ___________________________________________________________ Remember to show all of your work and provide all necessary explanations for full credit. Good luck and have a good break! Question 1 2 3 4 5 6 7 8 9 10 Total Worth 24 12 14 20 20 20 5 5 10 20 150 Score
Transcript of Old Final Exam - Washington State University · Question 2. (6 points each) Compute the following...
Math171Fall2014FinalExam
December15,2014
LastName: ___________________________________________________________
FirstName: ___________________________________________________________
StudentID: ___________________________________________________________
Section: ___________________________________________________________
Remembertoshowallofyourworkandprovideallnecessaryexplanationsforfullcredit.Goodluckandhaveagoodbreak!
Question 1 2 3 4 5 6 7 8 9 10 Total
Worth 24 12 14 20 20 20 5 5 10 20 150
Score
Question1.(4pointseach)Computethefollowingderivatives.
A. !!"
5𝑥! − 2 − 𝑥! + !!!
B. !!"
sin𝜃 − cos𝜃 + tan 𝜃 − sec𝜃
C. !!"
𝑒! + ln 𝑥 − tan!! 𝑥 + sin!! 𝑥
D. !!"
𝑥! + 4𝑥! 𝑒!
E. !!"
!"# !!!"#!!!!!
F. !!"
ln 𝑒!! − tan 𝑥
Question2.(6pointseach)Computethefollowingindefiniteintegrals.
A.
B.
Question3.(7pointseach)Computethefollowingdefiniteintegrals.Youarenotrequiredtosimplifyyouranswers.
A.
B.
!(2𝑥! − 3+ 𝑥!! − 𝑥!!) 𝑑𝑥
!𝑥! sec!(𝑥!) 𝑑𝑥
! (𝑥! + sin 𝑥) 𝑑𝑥!/!
!
!(ln 𝑥)!
𝑥 𝑑𝑥!
!
Question4.(4pointseach)Considerthefunction𝑓 𝑥 = !!𝑥! − !
!𝑥! − !
!𝑥! + 𝑥 − 1.
Forreference,𝑓! 𝑥 = 𝑥 + 1 𝑥 − 1 !and𝑓!! 𝑥 = 𝑥 − 1 (3𝑥 + 1).Forfullpointsoneachpart,justificationmustbeprovided.
A. Listthecriticalpoint(s)of𝑓.
B. Onwhatinterval(s)isthefunction𝑓decreasing?
C. Foreachcriticalpoint,statewhetheritisalocalminimum,alocalmaximumorneither.
D. Onwhatinterval(s)is𝑓concaveup?
E. Listtheinflectionpoints(s)of𝑓.
Question5.(20points)Drawagraphof𝑦 = 𝑓 𝑥 onthegridbelowthatpassesthroughthepoints −1,0 , 0,1 , 1,0 , and 2,−1 indicatedbyblackdotsonthegrid.Yourgraphfor𝑓mustbecontinuousandsatisfythefollowingsigndatafor𝑓′and𝑓′′.Alsoassume𝑓! 0 and𝑓!! 0 donotexist.
Interval 𝑓!(𝑥) 𝑓!! 𝑥 −∞,−1 Negative Positive−1,0 Positive0,1 Negative Negative(1,2) Positive(2,∞) Positive
Question6.(20points)Arectangularsheetofcardboardofwidth4𝑤andheightℎ(ininches)canbefoldedintoquartersandjoinedattheendstomakea“squaretube”ofvolume𝑉 = 𝑤!ℎasdrawnbelow:
Themanufacturerofthecardboardsheetinsiststhatℎbenomorethan16inchesand𝑤benomorethan4inches.Whatarethevaluesofℎand𝑤sothatthevolumeofthe
squaretubeis16 in!andthequantity𝑃 = 𝑤 + !!ℎisminimized?
Forfullcredit,youmustjustifythatyouhaveinfactminimized,ratherthanmaximized,thevalueof𝑃.
ℎ
𝑤 𝑤 𝑤 𝑤
ℎ
𝑤𝑤
Question7.(5points)Uselinearapproximationtoestimateln 1.05 .(NOTON2015EXAM)
Question8.(5points)Anobject’spositionfunctionis𝑠 𝑡 anditsvelocityfunctionis𝑣 𝑡 = 𝑠! 𝑡 .If𝑠 1 = 3and𝑠 4 = 15,themeanvaluetheoremguaranteesthatthevelocityoftheobjectmustbewhatvalueatsometimebetween𝑡 = 1and𝑡 = 4?Forfullcredit,explicitlyshowthecomputationthatproducesyourresult.
Question9.(10points)Findtheabsoluteminimumandabsolutemaximumvaluesalongwiththeirlocationsattainedby𝑓 𝑥 = 𝑥! − 2𝑥! + 4ontheinterval[−1,2].
Theabsolutemaximumis___occurringat𝑥 =___________
Theabsoluteminimumis___occurringat𝑥 =___________
Question10.(4pointseach)Computethefollowinglimits.AnyusesofL’Hopital’srulemustbejustifiedforfullcredit.
A.
B.
C.
D.
E.
lim!→!
𝑥3 − 8
𝑥2 − 3𝑥 + 2
lim!→!
𝑥3
𝑥3 − 3
lim!→!
𝑒𝑡 − 𝑡 − cos 𝑡
𝑡2
lim!→!
𝑥2 + 3𝑥 + 2
𝑒𝑥
lim!→!!
(2𝑥)3𝑥
