CALCULUSTHOMAS'CALCULUSARLYTRANSCENDENTAE8.6JAMESSTEWARTIndeterminate Forms andL'Hopital's RuleERMSONAThe meaninglkess expressions which cannot be evaluated in a consistentwayare ndcterminate foms.Suchas0,co0,ooco,o,andTHEOREM6-LHopitat'sRuteSuppose thit f(e) = g(a) 0, that f andg arediffmtiableneninl/ainingandtatg(x)onifThenx)1)1.Indeterminate FormO/0Han suefaning that the limit on the right side of this equa55Esing EHopitaly RaleTofindfGx)aby IHopital's Rule, continue to differentiate fand g,so long as we still get thefonm O/o at r a,Butas soon as one or the other of these derivatives is diffesent from zero at x α we stop differentiating. L'Hopitals Rule does not applywhen either the mumerator or demominator has a finite nonzero limitEXAMPLE1The following limits involve o/0 indeterminate forms, so we apphyr'Hopital's Rule, In, it must be applied repeatedlyEXAMPLE 2 Be careful to apply FHapital's Rule correcty:(a)im3=sinz lim3.c05) 3 c05.x2S营VI+2V11-!-m-f-0Sng,lim i tand(b) limin40EXAMPLE3In this example the one-sided limits are diffrent.V1+x-1 -x/2(1/2)1 + x)-1/2 1/2Solrglimeiateag(0)lim电国信-国#-82xx0+x(/4)(1 + x)-/org mai fonl=m1cOSx402Soll3r2Y雪罪SgNetlimt iefound目
2016/11/15 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.6 Indeterminate Forms and L’ Hopital’s Rule ^ 1. Indeterminate Form 0/0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The meaningless expressions which cannot be evaluated in a consistent way are indeterminate forms. Such as 0/0, ∞·0,∞-∞,00,and 1∞ ( ) ( )' ( ) f x g x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE4Find the limits of these co/oo forms:asSolutionerator and den(a) The nuinator are discmuousat =w/2,sowe investigate theonesided limits there.To apply IHopitalRule, we canchoose /tobeany openinterval with x = /2 as an endpoint.-, r x 2.IndeterminateFormsseexmtrtimarimar-1 + tan xsecx0/80,00,00-00sec x tan xsecx=, lim,sinx=1lim/I+tanx.limsecxseex_=lseexseexSo limr i+any, and Jima +tans- lim, +tanx1/x一=0002VxVE:A会-EXAMPLE6Fiod the limit ofthisoc-oo formEXAMPLE5Findthe limits ofthese co-0forms(-)(a) m (rsin)(b) J VinxIfxtbenx-nds-1-0SolutionSolutioSimilarly.ifx--o-,then.sin.x-o"and(a) (in)(n)1--VNeither form revealsted oca/asat happensinthelimit.Tofind out,the fraction-1-1/x--1/2/iHipralyRuleThen we apply I'Hopital's Rule to the resultm(-2V) - 0品hnsinx2x9-0.国广Limits that lead to the indeterminate forms , o°, and oo°can sometimes be handled byfirst taking the logarithm of the function, We use I'Hopital's Rule to find the limit of thclogarithm expression and then exponentiate the result to find the original function limitIf lim, In f(x) -- L, thenlim fo) - lin ehroe.3.Indeterminate PowersHere e may be either finite or infinite.目山
2016/11/15 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 2. Indeterminate Forms ∞/∞,∞·0,∞-∞ 2 ( /2) ( /2) ( /2) sec sec tan lim lim lim sin 1 x x x 1 tan sec x x x x x x ( /2) ( /2) / 2 sec sec sec lim lim lim =1 x x 1 tan 1 tan 1 tan x x x x So and x x x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 3. Indeterminate Powers
EXAMPLEApply I"Hopital's Rule to show that lim,or (1 + x)/ = e.EXAMPLE8Find limx/SolutiomThe limit leads to the indeterminate form 1e, We let f(x) = (1 + x)/ andfind lim,-grIn f(x)SincoThe limit leads to the indeterminate form oo, We let f(t) = x/ and findSolutionlim,In f(x).Sincen Ju)= n(I+)/=Im(1+ )In fo) = In x/w = .'Hopital's Rule now applies to giveI'Hopital Rule givesIn(1 + x)m, o im.一1+x-蓝Tm= ↓ = 1.-1 = 0.Therefore, + - m F ee' =Therefore lin / im /() im eh Ao e° = 1.国AEExercisessint?1613.jm1225. _m(--号)x -121. jimo in(se a)Chapter 6sine4()1n327.limApplicationofDefinite Integrals51. lim x/-ne57.m(t+2)2m)6()66. Jim, sinx*Inx6二广6.1OVERVIEWIn Chapter 5 we saw that a continuous function over a closed interval has definite integral, which is the limit of any Riemann sum for the function. We proved thawe could evaluate definite integnals using the Fundimental Theorem of Calculus. We alsoVolumes Using Cross-Sectionsfound that the area under a curve and the area between two curves could be computed adefinite integrals.白百
2016/11/15 3 Exercises 0 1 6 2 1 ln 3 1 2 1 e 1 2 e 3 e 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 6 Application of Definite Integrals Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.1 Volumes Using Cross-Sections
1. The cross-sectionof solid S is the planeregion formed by intersecting S with aplane1.CrossSectionandVolumeofHLuE6.tAmioeSxofCylindrical Solid2. If the cylindrical solid has aknown base area A and height h,thenVolumebase area xheighAhnanetteCtetiedneaTaoinidwHindeandfednwahAAApmosirRda-Ge Sl) hes heigh-4-12.Volumes of Solids ofLopcal sabRevoluton:The Method of SlicingATHEURE63 Atypioat tlissabinthesolidsFIGURE 6.4The solidthin slab inFigure 6.3 is shown enlangod here.liiadby thlecylindnical solid with广eS(xa) beareaA(n)and heighCThe volume , of this cylindrical solid is A(xa) - A.xi, which is approximately the samevolume as that of the slab:Volume ofthe kth slab V, =A(x) A)The volume Vofthe entire solidSis thierefore approximated by the sum ofthese cylindri-cal volumes1-2n-2)Calculating the Volume of a Solid1.Sechheoldypicalss-sectonThis is a Riemann sum for the function 4(x) on [a, b], We expect the approximations from2. Find a formula for A(x), the area of a typical cross-section.these sums to improve as the norm ofthe partition of [a, b] goes to zero. Taking a partitionof [a, b] into n subintervals with [Pf→ 0 gives3,Find the fimits of integration4. negute 4(x)to find the volumega04(x)dsDEFINITIONTheolumeofasotidoCnltegrahross-soctional area A(x)fromxtoxbistheinlegralof.4fromaloV[A)dAAAESRAE
2016/11/15 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 1. Cross Section and Volume of Cylindrical Solid Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. The cross-section of solid S is the plane region formed by intersecting S with a plane. 2. If the cylindrical solid has a known base area A and height h, then Volume=base area X heigh=Ah. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 2. Volumes of Solids of Revoluton: The Method of Slicing typical slab
EXAMPLE2EXAMPLE1Apymid3mhighhsqurehsethatismonsidTherossAcurdwodgescutfromacircularcylinderofadius3bywoplansOne plane is perpendicular to the axis of the cylinder. The second plane cnrosses the firsectice of thepyramid perpendicular to the altitudex m down from the vertex isa squareplane at a 45*sngle at the center ofthe cylinder.Find the volume of the wedge.X m on a side. Find the volume of the pyramid.2V9-2(t) (heighi)(width) (o)(2V9 P)Selution=.2xV9 2A sketck. We draw the pyramid with its alitude along the x-axis.miThe rectangles run fr=. 3, so we haveE-0Xad iswrtheorigindincludeapicalcossctic[aa) d-L2rV9-Pa(-V9-2)2.4 formula for,d(x). The.cross-section.at x is a square x mcte on a side, so jts area ib--(9-FIGURE 6.6 The Enple 2Ax)=1.stioodperpendicdartofhexatisTeThe limisofigynitiThesuans leon the plais fiomx tox33,cross-setios are rextangke-0+f(omJutagrtatelo findhe volam= 18.A-[=[起-号-9m.ExercisesVolumes by SticingEXAMPLE3Caalienis principlesys that solids withequlaltitudes and idenicalFind the volumes of the solids in Elxercises 110.tionsachithvehsewolue (figure7hisfolscross-sectmedistely from thdefinitioa of vodumese the cross-sectioaal area functios A(x1, The solid lies betseen planes perpeadicselar to the x-axis at 0aadtheiteral a,b]aethesmeforothsolidand r 4., The cross-sectioms perpendicular to the axis on thinterval o ≤ x ≤ 4 are squares whose diagonals rn from theparaholay = Vx to the parabolay VrA) ( de 42x dx =[x] = 16A(x) dx =ameylndFIeuBE6.7CooolmtorcwicThesesolikvealchcaeanlodwithsacksofcoin国五2. The solid lies between planes perpendicular lin the x-axisaExercisesThe molid fie hten plins pependicler m te satiExercises1 and LThe cesons-I and x i.The erogs-sectis perpendicular to thes-aticinular dals shae dindesm fiom the pandbetwen these planese squares whose bases run from the semi-to the panhoa y=2~1circke y - -Vi'o the semicinde y = V1?()=(base)=(2y/1-P=4(1)-4-L4=4r-+-(2) 2((22+),bA(x) =dv] M)dx (1-2+)=↓-g+],-2(1-3+)g普白E
2016/11/15 5 Exercises Exercises Exercises 2 2 2 2 A x base x x ( ) ( ) (2 1 ) 4(1 ) 1 2 3 1 1 1 4 16 ( ) 4(1 ) [4 ] 3 3 b a V A x dx x dx x x