(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.
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].
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
(b) Findf¢(x)and simplify. Showallsteps.
(c) Useyourf¢(x) tofind allinflection points,allintervalswhereisconcaveupand allintervals where f(x)isconcavedown. Showallwork. Give youranswerin intervalnotation.
Useintervalnotationto estimatewheretheoriginalfunction f(x)is… (a) increasing
(c) Sketch agraph ofwhaty=f¢(x)mightlooklike.
(d) Giveallintervalswheref(x) isCU (e) Giveallintervalswheref(x)isCD
(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.
(b)Given belowisadiagramofacanofCalcu-Colawheretheinnerradiusis 5.2 inchesand thethickness ofthecan is0.08inches. Theheightis15 inchesasshown. Usedifferentialsto estimatethevolumeof
(a) Uselimitsto find anyand all verticaland horizontalasymptotes. Showallwork.
andsimplify. Find allcriticalpoints ofthefunction.Testaround
yourcriticalpointsand includeasign graph tosupportyouranswer.
and simplify. Find allpossibleinflectionpoints.Testaround your
points. Includeasign graph to support youranswer.
(d) Useyourasymptotesand yoursign graphsfrom (b)and (c)tosketch agraphofy=f(x).
=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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