Considerthefunctionf(x)x2*sin×(x)3+ 1



(a)     Find the Linearization,L(x), ofthegiven function atx=-2. Showallsteps!Round thecoefficients to 3decimalplaces.















(b)       Find the x-interceptofL(x)asa wayofestimating thex-interceptoff(x).Round thecoefficients to 3decimalplaces.









  1. 2.         Usethederivativeofthefunctionf(x)x=


fromproblem #1above,and useyourTIto


graph thisderivativeontheinterval[-2,3 ]. Accuratelygraph and labelyourderivative. Usethegraph to find criticalpoints. Labelthesepointson yourgraph.Applytheextremevalue theoremto find the absoluteextremaoff(x)on theclosedinterval,[-2,3].



  1. 3.         OpieTimization hasfound that ona6hourtrip,thefunctions(t)0=.0-2t     33.4+7t   02.0-5t

estimateshispositionvs.timerelationship. sgivestheposition inmilesand givesthetimein hourswith in theinterval[0,6hrs.]


(a)       Aretheconditionsforthemeanvaluetheoremsatisfied forOpie’sfunction s(t)?Showthe specificcheckoftheconditions with reasonssupporting yourstatements.







(b)       Writeasentenceto summarize whatconclusion canOpiedrawfromtheMeanValueTheorem and then find thecguaranteed bytheMVT
























  1. 4.         You aregiven thesign graphsforthefirstderivativeand second derivativeofsomeunknown functionf(x). Usethesetwographstogiveapossiblesketchofwhatthegraphofy=f(x) mightlooklike.



  1. 5.         Forsomeunknown function f(x),thefirstderivativeofthefunction isgiven byf¢(x)x=(x   2-) (a)Using thisderivative,find allcriticalpoints,forma sign graph and useittofindallintervals(use intervalnotation) where f(x)isincreasing and allintervalswheref(x)isdecreasin

















(b)       Findf¢(x)and simplify. Showallsteps.

















(c)       Useyourf¢(x) tofind allinflection points,allintervalswhereisconcaveupand allintervals where f(x)isconcavedown. Showallwork. Give youranswerin intervalnotation.




  1. 6.         Belowisthegraphofsomeunknownfunction’sderivative y=



Useintervalnotationto estimatewheretheoriginalfunction f(x)is… (a)       increasing




(b)       decreasing





(c)       Sketch agraph ofwhaty=f¢(x)mightlooklike.















(d)       Giveallintervalswheref(x) isCU (e)       Giveallintervalswheref(x)isCD







  1. 7.         An open boxofmaximumvolumeistobeconstructedfrom asingle, 36inch by 36inch pieceof cardboard,bycuttingequalsized squares(xinchesbyxinches)fromeachcornerand turning up the sides.



(a)       Find afunction V(x)thatdescribesthevolumeofsuchabox,wherexrepresents thelength (and width)ofthesquarecutout.












(b)       UseCalculusonyourfunction V(x)from aboveto find thedimensionsofthebox(rounded to thenearesttenthofan inch;givethelength, width and height), wherethetotalvolumeoftheboxis maximized. Usethe ExtremeValueTheorem to showthatyoursolution isin factoptimal.Writea sentencetosummarizeyourfindings,including theoptimalvolume.




  1. 8.         (a)Given forsomefunctiony=g(x),thatg(6)=13andg¢(6)5=

g(6.04). Showallwork.






















(b)Given belowisadiagramofacanofCalcu-Colawheretheinnerradiusis 5.2 inchesand thethickness ofthecan is0.08inches. Theheightis15 inchesasshown. Usedifferentialsto estimatethevolumeof

themetalneededfortheside ofthesodacan.HintV=pr2h
























  1. 9.         Considerthefunction

f()x =





(a)       Uselimitsto find anyand all verticaland horizontalasymptotes. Showallwork.












(b)       Showallworktofind


andsimplify. Find allcriticalpoints ofthefunction.Testaround


yourcriticalpointsand includeasign graph tosupportyouranswer.

















(c)       Showallworktofind


and simplify. Find allpossibleinflectionpoints.Testaround your


points. Includeasign graph to support youranswer.















(d)       Useyourasymptotesand yoursign graphsfrom (b)and (c)tosketch agraphofy=f(x).







  1. 10.       Considerthefunction


=2on [-2, 2]. Itcaneasilybeshown thattheslopeofthesecant



line ofthisfunction over[-2, 2]iszero,BUTthereisnotangentlineto y=f(x)on [-2, 2]whoseslopeis zero. Explain whytheMEAN VALUETHEOREMisNOTcontradicted here.

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