Calculus: 7th Edition, A SummaryJamesStewartAPreviewofCalculusTheAreaProblemTheoriginof calculusgobackatleast2500yearstotheancientGreeks, whofoundareasusingthemethodofexhaustion.TondtheareaofacircleA,letAnbetheareaoftheinscribed polygon with n sides. As n increases, it appears that Anbecomes closer and closertotheareaofthecircle. AlthoughtheGreeksthemselvesdidnotuselimitsexplicitly,A =limnAnTheareaproblemisthecentralprobleminthebranchofcalculuscalledintegral calculus.TheTangentProblemConsidertheproblemof tryingtondanequationof thetangentlinet toacurvewithequationy=f(x)atagivenpointP. Sinceweknowthatthepointpliesonthetangentline,wecanndtheequationoftifweknowitsslopem. Theproblemisthatweneedtwopointstocomputetheslopeandweknowonlyonepoint,P,ont.Thetangentproblemhasgivenrisetothebranchofcalculuscalleddierential calculus,whichwasnotinventeduntilmorethan2000yearsafterintegralcalculus. Themainideasbehind dierential calculus are due to the French mathematician Pierre Fermat (1601-1665)andwere developedbythe Englishmathematicians JohnWallis (1616-1703), Isaac Bar-row(1630-1677), andIsaacNewton(1642-1727)andtheGermanmathematicianGottfriedLeibniz(1646-1716).VelocityLet d = f(t) be the distance traveled by a car during a period of time t. The average velocityduringatimeintervala t b:averagevelocity =changeinpositiontimeelapsed=f(b) f(a)b aTo nd the velocity at the instant t = a, we can compute the average velocity over a shortertimeintervalaveragevelocity =f(t) f(a)t awheretisevaluatedatavaluesmallerthanb. Astheintervalgetssmallerandsmaller,thevelocityvwhent = aisthelimitingvalueofthisaveragevelocityastapproachesaandiscalledtheinstantaneousvelocityattimea.v= limtaf(t) f(a)t aThisisthesameastheslopeofthetangentlinetothecurved = f(t)atpointP(a, f(a)).1TheLimitofaSequenceA sequence {an} is a set of numbers written in a denite order. For instance, the sequence

1, 12, 13, 14, . . .

can be described by giving the formula an= 1/n. The terms of the sequence becomes closerandcloserto0asnincreases. Wesaythatthelimitofthesequenceis0anddenotelimn1n= 0Ingeneral,thenotationlimnan= LisusedifthetermsanapproachthenumberLasnbecomeslarge.TheSumofaSeriesA series is the sum of the terms of a sequence. For any sequence {an}, the associated seriesisdenedastheorderedformalsum

n=0an= a0 + a1 + a2 + Thesequenceof partial sums {sk}associatedtoaseries

anisdenedforeachkasthesumofthesequence {an}froma0toak.sk=k

n=0an= a0 + a1 + a2 + + akThen,theinniteseriescanbedenotedusingthenotationoflimits:

n=0an=limksk21. FunctionsandLimits1.1FourWaystoRepresentaFunctionFunctionsarisewheneveronequantitydependsonanother. Afunctionf isarulethatassignstoeachelementxinasetDexactlyoneelement, calledf(x), inasetE. ThesetDiscalledthedomainof thefunction. Therangeof f, setE, isthesetof all possiblevaluesof f(x)asxvariesthroughoutthedomain. Asymbol thatrepresentsanarbitrarynumberinthedomainofafunctionfiscalledanindependentvariable. Asymbolthatrepresentsanumberintherangeoffiscalledadependentvariable.RepresentationsofFunctionsTherearefourpossiblewaystorepresentafunction:1. verbally(byadescriptioninwords)2. numerically(byatableofvalues)3. visually(byagraph)4. algebraically(byanexplicitformula)Its oftenuseful togofromonerepresentationtoanother togainadditional insight intothefunction. Butcertainfunctionsaredescribedmorenaturallybyonemethodthanbyanother.The graph of a function is a curve in the xy-plane. The question is: Which curves in thexy-plane are graphs of functions?The verticallinetest allows us to answer this question.Acurveinthexy-planeisthegraphof afunctionof xif andonlyif novertical lineintersectsthecurvemorethanonce.PiecewiseDenedFunctionsFunctions that are dened by dierent formulas in dierent parts of their domains are calledpiecewisedenedfunctions. An example of such function is the absolute value function:|x| =

x x 0x x < 0SymmetryA function fis called an evenfunction if f(x) = f(x) for every number x in its domain.Thegraphofanevenfunctionissymmetricwithrespecttothey-axis.Afunctionf iscalledanoddfunctionif f(x)= f(x)foreverynumberxinitsdomain. Thegraphof anoddfunctionissymmetricabouttheorigin. If wealreadyhavethegraphoffforx 0, wecanobtaintheentiregraphbyrotatingthisportionthrough180abouttheorigin.3IncreasingandDecreasingFunctionsAfunctionfiscalledincreasingonanintervalIiff(x1) < f(x2)wheneverx1< x2inI.ItiscalleddecreasingonIiff(x1) > f(x2)wheneverx1< x2inI.1.2MathematicalModels: ACatalogofEssentialFunctionsAmathematical model isamathematical descriptionof areal-worldphenomenon. Weformulate amodel byadaptingaphysical lawor principle. If there is nophysical lawor principle tohelpus formulateamodel, we construct anempirical model, whichisbasedentirelyoncollecteddata. Amathematical model is never acompletelyaccuraterepresentationofaphysicalsituationitisanidealization. Agoodmodelsimpliesrealityenoughtopermit mathematical calculations but is accurate enoughtoprovide valuableconclusions.There are manydierent types of functions that canbe usedtomodel relationshipsobservedintherealworld.LinearModelsThegraphof alinearfunctiony=f(x)=mx + bisaline. Acharacteristicfeatureoflinearfunctionsisthattheygrowataconstantrate.PolynomialsAfunctionPiscalledapolynomialifP(x) = anxn+ an1xn1+ an2xn2+ + a2x2+ a1x + a0wherenis anonnegativeinteger andthenumbers a0, a1, . . . , anareconstants calledthecoecients of the polynomial. The domain of any polynomial is R. If the leading coecientan = 0,thenthedegreeofthepolynomialisn.PowerFunctionsAfunctionoftheformf(x) = xa,whereaisaconstant,iscalledapowerfunction.RationalFunctionsArationalfunctionfisaratiooftwopolynomials:f(x) =P(x)Q(x)where Pand Q are polynomials. The domainconsists of allvalues of x such that Q(x) = 0.AlgebraicFunctionsAfunctionfiscalledanalgebraicfunctionif itcanbeconstructedusingalgebraicop-erations startingwithpolynomials. Anyrational functionis automaticallyanalgebraicfunction.4TrigonometricFunctionsIncalculustheconventionof trigonometricfunctionsisthatradianmeasureisalwaysused (except when otherwise indicated). For example, when we use the function f(x) = sin x,itisunderstoodthatsin xmeansthesineoftheanglewhoseradianmeasureisx.Animportantpropertyof thesineandcosinefunctionsisthattheyareperiodicfunc-tions. Theperiodicnatureof thesefunctionsmakesthemsuitableformodelingrepetitivephenomenasuchastides,vibratingsprings,andsoundwaves.ExponentialFunctionsTheexponentialfunctionsarethefunctionsoftheformf(x) = ax,wherethebaseaisapositiveconstant.LogarithmicFunctionsThelogarithmicfunctionsf(x) = loga x,wherethebaseaisapositiveconstant,aretheinversefunctionsoftheexponentialfunctions.NewFunctionsfromOldFunctionsNewfunctions canbe obtainedbyshifting, stretching, andreectingthe graphs of oldfunctions.TransformationsofFunctions1. Translations(Shifts): Supposec > 0. Toobtainthegraphofy= f(x) + c,shiftthegraphofy= f(x)adistancecunitsupwardy= f(x) c,shiftthegraphofy= f(x)adistancecunitsdownwardy= f(x c),shiftthegraphofy= f(x)adistancecunitstotherighty= f(x + c),shiftthegraphofy= f(x)adistancecunitstotheleft2. StretchingandReecting: Supposec > 1. Toobtainthegraphofy= cf(x),stretchthegraphofy= f(x)verticallybyafactorofcy= (1/c)f(x),shrinkthegraphofy= f(x)verticallybyafactorofcy= f(cx),shrinkthegraphofy= f(x)horizontallybyafactorofcy= f(x/c),stretchthegraphofy= f(x)horizontallybyafactorofcy= f(x),reectthegraphofy= f(x)aboutthex-axisy= f(x),reectthegraphofy= f(x)aboutthey-axisCombinationsofFunctionsTwofunctions f andg canbe combinedtoformnewfunctions f+g, f g, fg, andf/g. Anotherwayofcombiningtwofunctionstoobtainanewfunction,calledcomposite5functionf g, uses theimageof afunctionas thepreimageof another function. Thecompositionoffandgisdenedby(f g)(x) = f(g(x))Thedomainoff gisthesetofallxinthedomainofgsuchthatg(x)isinthedomainoff. Inotherwords,(f g)(x)isdenedwheneverbothg(x)andf(g(x))aredened.1.4TheTangentandVelocityProblemsSeepages44-48oftheprimarytextbook.1.5TheLimitofaFunction1 Denition Suppose f(x) is dened when x is near the number a. (This means that fisdenedonsomeopenintervalthatcontainsa,exceptpossiblyataitself.) Thenwewritelimxaf(x) = Landsaythelimitoff(x),asxapproachesa,equalsLifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesucientlyclosetoabutnotequaltoa.Noticethephrasebutx = ainthedenitionoflimit. Thismeansthatinndingthelimitof f(x)asxapproachesa, weneverconsiderx=a. Theonlythingthatmattersishowfisdenedneara.One-SidedLimits2 DenitionWewritelimxaf(x0 = Land say the left-hand limit of f(x) as x approaches a is equal to L if we can make the valuesoff(x)arbitrarilyclosetoLbytakingxtobesucientlyclosetoaandxlessthana.Similarly,ifwerequirethatxbegreaterthana,wegettheright-handlimitoff(x)asxapproachesaisequaltoLandwewritelimxa+f(x) = LBycomparingDenition1withthedenitionsofone-sidedlimits,wendthefollowing.3 Theoremlimxaf(x) = L ifandonlyif limxaf(x) = L and limxa+f(x) = L6InniteLimits4 DenitionLetf beafunctiondenedonbothsidesof a, exceptpossiblyataitself.Thenlimxaf(x) = meansthatthevaluesoff(x)canbemadearbitrarilylargebytakingxsucientlyclosetoa,butnotequaltoa.Thisdoesnotmeanthatweareregarding asanumber. Nordoesitmeanthatthelimitexists. Itsimplyexpressestheparticularwayinwhichthelimitdoesnotexist. Asimilar sort of limit, for functions that become large negative as x gets close to a, is dened.5 Denition. Letfbedenedonbothsidesofa,exceptpossiblyataitself. Thenlimxaf(x) = means that the values of f(x) can be made arbitrarily large negative by taking x sucientlyclosetoa,butnotequaltoa.6 Denition. Thelinex = aiscalledavertical asymptoteofthecurvey= f(x)ifatleastoneofthefollowingstatementsistrue:limxaf(x) = limxaf(x) = limxa+f(x) = limxaf(x) = limxaf(x) = limxa+f(x) = 1.6CalculatingLimitsUsingtheLimitLawsLimit Laws. Suppose that c is aconstant andthe limits of f(x) andg(x) exist as xapproachesa. Then1. limxa[f(x) + g(x)] =limxaf(x) + limxag(x)2. limxa[f(x) g(x)] =limxaf(x) limxag(x)3. limxa[cf(x)] = c limxaf(x)4. limxa[f(x) g(x)] =limxaf(x) limxag(x)5. limxaf(x)g(x)=limxa f(x)limxa g(x)if limxag(x) = 0Recall 3 Theoremfromtheprevioussection.1 Theorem.limxaf(x) = L ifandonlyif limxaf(x) = L =limxa+f(x)When computing one-sided limits, we use the fact that the Limit Laws also hold for one-sidedlimits.7Thenexttwotheoremsgivetwoadditionalpropertiesoflimits.2 Theorem. Iff(x) g(x)whenxisneara(exceptpossiblyata)andthelimitsoffandgbothexistasxapproachesa,thenlimxaf(x) limxag(x)3 TheSqueezeTheorem. Iff(x) g(x) h(x)whenxisnearaandlimxaf(x) =limxah(x) = Lthenlimxag(x) = L1.7ThePreciseDenitionofaLimit2 Denition. Let fbe a function dened on some open interval that contains the numbera, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, andwewritelimxaf(x) = Lifforeverynumber > 0thereisanumber> 0suchthatif 0 < |x a| < then |f(x) L| < 3 DenitionofLeft-HandLimit.limxaf(x) = Lifforeverynumber > 0thereisanumber> 0suchthatif a < x < a then |f(x) L| < 3 DenitionofRight-HandLimit.limxa+f(x) = Lifforeverynumber > 0thereisanumber> 0suchthatif a < x < a + then |f(x) L| < See Examples 3 and 4 on pages 76-77 for ways to prove that limit statements are true usingthe,denition. AsExample4shows,itisnotalwayseasytoprovethem.8InniteLimits6 Denition. Let fbe a function dened on some open interval that contains the numbera,exceptpossiblyataitself. Thenlimxaf(x) = meansthatforeverypositivenumberMthereisapositivenumbersuchthatif 0 < |x a| < then f(x) > M7 Denition. Let fbe a function dened on some open interval that contains the numbera,exceptpossiblyataitself. Thenlimxaf(x) = meansthatforeverynegativenumberNthereisapositivenumbersuchthatif 0 < |x a| < then f(x) < N1.8ContinuityThe limit of a function as x approaches a can often be found simply by calculating the valueofthefunctionata. Functionswiththispropertyarecalledcontinuousat a.1 Denition. Afunctionfiscontinuousatanumberaiflimxaf(x) = f(a)Noticethatthedenitionimplicitlyrequiresthreethingsiffiscontinuousata:1. f(a)isdened(thatis,aisinthedomainoff2. limxaf(x)exists3. limxaf(x) = f(a)2 Denition. Afunctioniscontinuousfromtherightatanumberaiflimxa+f(x) = f(a)andfiscontinuousfromtheleftataiflimxaf(x) = f(a)3 Denition. Afunctionf iscontinuousonaninterval if itiscontinuousateverynumberintheinterval. (Iffisdenedonlyononesideofanendpointoftheinterval, we9understandcontinuous at theendpoint tomeancontinuous fromtheright or continuousfromtheleft.)Buildingupfromthedenitionsabove,wecanndthatthefollowingtheoremtrue.4 Theorem. If fand g are continuous at a and c is a constant, then the following functionsarealsocontinuousata:1. f+ g 2. f g 3. cf4. fg 5. fgifg(a) = 08 Theorem. Iffiscontinuousatbandlimxa g(x)=b,thenlimxa f(g(x))=f(b). Inotherwords,limxaf(g(x)) = f

limxag(x)

9 Theorem. If g is continuous at aandf is continuous at g(a), thenthecompositefunctionf ggivenby(f g)(x) = f(g(x))iscontinuousata.10 Theorem. Supposethatfiscontinuousontheclosedinterval[a, b]andletNbeanynumberbetweenf(a)andf(b), wheref(a) =f(b). Thenthereexistsanumbercin(a, b)suchthatf(c) = N.102. Derivatives11