Study Guide for Prelim 1 - 1 Study Guide for Prelim 1 Math 192 Spring 1997 Written by Don Allers Revised by Sean Carver 1 Improper Integrals b Idea A

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1Study Guide for Prelim 1Math 192 - Spring 1997Written by Don Allers; Revised by Sean Carver1Improper IntegralsIdea:A proper integral is an integralRbaf(x)dx,whereaandbare both finite andf(x) is continuous andfinite on [a, b]. For proper integrals, thefundamental theorem of calculustells us that ifFis an antiderivativeoff(i.e. if F’(x) = f(x)) thenRbaf(x)dx=F(b)-F(a).We will define an improper integral as a limit ofproper integrals.CommentThe integralRbaf(x)dxisimproperif: (i)aorbis infinite, or (ii)f(x) is not finite on all of[a, b]. The fundamental theorem of calculus does not apply directly to this class of integrals. Indeed, thedefinition of a definite integral given in Chapter 4 explicitly assumes that the integral is proper. Withoutthe definitions below, an improper integral would be an expression with no meaning.SolutionMethod:1. Ifais finite, andf(x) is continuous on [a,] we defineRaf(x)dx= limc→∞Rcaf(x)dx.2. Ifbis finite, andf(x) is continuous on [-∞, b] we defineRb-∞f(x)dx= limc→-∞Rbcf(x)dx.3. Ifa, bare finite, andf(x) is continuous on (a, b] we defineRbaf(x)dx= limca+Rbcf(x)dx.4. Ifa, bare finite, andf(x) is continuous on [a, b) we b defineRbaf(x)dx= limcb-Rcaf(x)dx.5. If the integral is improper at multiple points on the domain [a, b], we divide this domain intosubintervals [a, c1],[c1, c2], . . . ,[cn-1, cn],[cn, b] so that each subinterval contains only one problempoint at one of its endpoints, and no problem points in its interior.Then defineRbaf(x)dx=Rc1af(x)dx+Rc2c1f(x)dx+. . .Rcncn-1f(x)dx+Rbcnf(x)dx,applying the above rules to each piece.Eg:(a)Z10xlnx dx,(b)Z0dxx2+ 1,(c)Z-∞2x dx(x2+ 1)2,(d)Z1-1dxx2/3.2SequencesIdea:Given a sequence of numbers{an}=a1, a2, a3, . . ., the question is whether or not these numbers areconvergingto somelimita. We say that{an}converges toa, writtenana, asn→ ∞, if the distance|an-a|between the sequence points andagoes to zero asnincreases.Extend the Sequence to a Real FunctionIdea:Compute the limit of a sequence as the limit of a functionf(x) asx→ ∞.When?Use this method if the expression defining the sequenceancan be considered as a realfunction in the variablenwhose limit asn→ ∞can be computed easily.Method:To compute the limit of the sequence, take take the limit of its defining expression as youwould take the limit of a real function. Remember L’Hopital’s Rule. (The theorem behind this methodis the following: Iff(n) =anand limx→∞f(x) =Lthen limn→∞an=L.
22SEQUENCESContinuous Function TheoremIdea:It would be nice if we could move a limit inside a function, that is, if we could saylimn→∞f(an) =f( limn→∞an).