calculus for the clueless, calc ii - bob millers

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1、Chapter1—LogarithmsMostofyou,atthispointinyourmathematicaljourney,havenotseenlogsforatleastayear,maybelonger.Thenormalhighschoolcourseemphasizesthewrongareas.Youspendmostofthetimedoingendlesscalculations,noneofwhichyouneedhere.Bytheyear2000,studentswi

2、lldoalmostnologcalculationsduetocalculators.Incaseyoufeeltortured,justrememberyouonlyspentweeksonlogcalculations.Ispentmonths!!!TheBasicLawsofLogs1.Defined,logbx=y(logofxtothebasebisy)ifby=x;log525=2because52=25.2.Whatcanthebasebbe?Itcan'tbenegative,s

3、uchas-2,since(-2)1/2isimaginary.Itcan'tbe0,since0niseitherequalto0ifnispositiveorundefinedifnis0ornegative.balsocan'tbe1since1nalways=1.Thereforebcanbeanypositivenumberexcept1.NoteThebasecanbe21/2,butitwon'tdoyouanygoodbecausethereareno21/2tables.Thet

4、womostcommonbasesare10,becausewehave10fingers,ande,anumberthatoccursalotinmathematicsstartingnow.A.eequalsapproximately2.7.B.Whatisemoreexactly?Onacalculatorpress1,inv,ln.C.log=log10D.ln=loge(Inisthenaturallogarithm).3.Alogyisanexponent,andexponentsca

5、nbepositive,negative,andzero.Therangeisallrealnumbers.4.Sincethebaseispositive,whethertheexponentispositive,zero,ornegative,theanswerispositive.Thedomain,therefore,ispositivenumbers.NoteInordertoavoidgettingtootechnical,mostbookswritelog

6、x

7、,therebyexc

8、ludingonlyx=0.5.logbx+logby=logbxy;log2+log3=log6.6.logbx-logby=logb(x/y);log7-log3=log(7/3).7.logbxp=plogbx;ln67=7ln6isOK.NoteLaws5,6,and7aremostimportant.Ifyoucansimplifyusingtheselaws,abouthalfthebattle(theeasyhalf)isdone.Example1—Writethefollowing

9、assimplerlogswithnoexponents:4lna+5lnb-6lnc-½lnd8.logbb=1sinceb1=b.log77=1.Ine=1.log10=1.9.logb1=0sinceb0=1.log81=log1=ln1=0.10.Logisa1:1function.Thismeansiflogc=logd,c=d.NoteNoteverythingis1:1.Ifx2=y2,x=±y.11.Logisanincreasingfunction.Ifm

10、logn.12.13.isaweirdwayofwritingx.elnx=x.14.logbbx=x;lnex=x.15.Youshouldnowbeabletosolvethefollowingkindsoflogequations:Example2—Solveforx:4·3x+2=28.Divideby4;isolateexponent.Takelogs.Itnowbecomesanelementaryalgebraequation,whichwesolveforx,usi