Homework

 

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.]

 

  1. 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].

 

 

  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.

1/3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(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

 

 

f¢()x

 

 

 

  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.

,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!

 

 

 

 

 

  1. 9.         Considerthefunction

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).

 

 

 

 

 

 

  1. 10.       Considerthefunction

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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