Posted: September 18th, 2017
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.
[10pts.]
s2in×(x)3+
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.
1/3.
(c) Useyourf¢(x) tofind allinflection points,allintervalswhereisconcaveupand allintervals where f(x)isconcavedown. Showallwork. Give youranswerin intervalnotation.
Useintervalnotationto estimatewheretheoriginalfunction f(x)is… (a) increasing
(b) decreasing
(c) Sketch agraph ofwhaty=f¢(x)mightlooklike.
(d) Giveallintervalswheref(x) isCU (e) Giveallintervalswheref(x)isCD
f¢()x
(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.
g(6.04). Showallwork.
,usedifferentialstoestimate
(b)Given belowisadiagramofacanofCalcu-Colawheretheinnerradiusis 5.2 inchesand thethickness ofthecan is0.08inches. Theheightis15 inchesasshown. Usedifferentialsto estimatethevolumeof
themetalneededfortheside ofthesodacan.HintV=pr2h
5.2in.
0.08in.
15in.
Calcu-Cola
YUM!
f()x =
3x
4x2+1
(a) Uselimitsto find anyand all verticaland horizontalasymptotes. Showallwork.
(b) Showallworktofind
f¢()x
andsimplify. Find allcriticalpoints ofthefunction.Testaround
yourcriticalpointsand includeasign graph tosupportyouranswer.
(c) Showallworktofind
f¢()x
and simplify. Find allpossibleinflectionpoints.Testaround your
points. Includeasign graph to support youranswer.
(d) Useyourasymptotesand yoursign graphsfrom (b)and (c)tosketch agraphofy=f(x).
f()x
=2on [-2, 2]. Itcaneasilybeshown thattheslopeofthesecant
x2
line ofthisfunction over[-2, 2]iszero,BUTthereisnotangentlineto y=f(x)on [-2, 2]whoseslopeis zero. Explain whytheMEAN VALUETHEOREMisNOTcontradicted here.
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