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Final Exam TopicsThe final exam is comprehensive. We have covered the following chap-ters.•Chapter 2: Limits•Chapter 3: Derivatives•Chapter 4: Derivative Applications•Chapter 5: AntiderivativesFirst, let’s consider the overall purpose of each chapter. This will giveus a “birds eye” overview of what we’ve done.What was the purpose of Chapter 2?The formal definition of each idea in Calculus useslimits. Without limits,we would not have Calculus.The most fundamental application of limits is to examine the behaviorof functions atx-values where the function encounters a domain problem:a hole, a jump, exploding to infinity, or infinite oscillation.Using limits,we were able to describe function behavior at these locations with moreprecision than simply observing “fis undefined there”. The most criticalconcept here is thata limit may exist at a point where a functionis undefined. In fact, the definition of the derivative is motivated by thatvery notion!Continuity: A functionfis continuous at a pointpif (1)f(p) exists (2)limx→pf(x) =L, a real number, and (3)f(p) =L. For most functions wework with, discovering where a function is continuous is the same as findingthe domain of that function. Examining where a function is discontinuouscan reveal where holes, vertical asymptotes, or “jumps” exist.Vertical Asymptotes: If at least one of limx→p+f(x) or limx→p-f(x)is infinite, thenf(x) has a vertical asymptotex=p.We abridged thevertical-asymptote analysis procedure by saying if limx→pf(x) is of the form“Non-Zero / Zero” then there’s a vertical asymptote atx=p. Looking atthe one-sided limits atpis how we determine the behavior offto eitherside ofp, if this information is needed.
Horizontal Asymptotes: If at least one of limx→∞f(x) or limx→-∞f(x)is a particular real numberL, theny=Lis a horizontal asymptote off(x).f(x) may have no, one, or two distinct horizontal asymptotes. Examiningthese limits is how we determine the presence or lack of horizontal asymp-totes.The Derivative:The derivative off(x) was defined using a limit:f0(x) = limh→0f(x+h)-f(x)h.We have moved beyond finding derivativesthis way. It will not be essential for you to be able to employ this definitionon the final exam (since we can find derivatives in other ways), but it isimportant to recognize the limit definition of a derivative if you see it. Andit is important to understand that, for a functionfand a pointawherefis differentiable,f0(a) is telling us theslopeoffata, which is theinstanta-neous rate of changeat that point.It’s worthwhile to mention the variety of ways we have of asking fora derivative.For illustration’s sake, we’ll need a function to serve as anexample.Let’s usef(x) =xx2+1.Let’s suppose we’re interested in thederivative at a particular pointx=a.1. “Findf0(a) forf(x) =xx2+1”2. “Find limh→0f(a+h)-f(a)hforf(x) =xx2+1”3. “Find the instaneous rate of change off(x) =xx2+1atx=a”4. “Find the derivative off(x) =xx2+1atx=a”5. “Find the slope off(x) =xx2+1atx=a”6. “Find the slope of the tangent line of