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.]

- 2. Usethederivativeofthefunctionf(x)x=

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].

- 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

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

- 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.

1/3.

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

- 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

*f*¢()*x*

- 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.

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

g(6.04). Showallwork.

,usedifferentialstoestimate

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

themetalneededfortheside ofthesodacan.Hint*V*=*p**r*2*h*

5.2in.

0.08in.

15in.

Calcu-Cola

YUM!

- 9. Considerthefunction

*f*()*x *=

3*x*

4*x*2+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).

- 10. Considerthefunction

*f*()*x*

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

*x*2

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

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