Final Exam Spring 2013 - Final Exam Topics The nal exam is comprehensive We have covered the following chapters Chapter 2 Limits Chapter 3 Derivatives

Final Exam Spring 2013 - Final Exam Topics The nal exam is...

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Final Exam Topics The final exam is comprehensive. We have covered the following chap- ters. Chapter 2: Limits Chapter 3: Derivatives Chapter 4: Derivative Applications Chapter 5: Antiderivatives First, let’s consider the overall purpose of each chapter. This will give us a “birds eye” overview of what we’ve done. What was the purpose of Chapter 2? The formal definition of each idea in Calculus uses limits . Without limits, we would not have Calculus. The most fundamental application of limits is to examine the behavior of functions at x -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 more precision than simply observing “ f is undefined there”. The most critical concept here is that a limit may exist at a point where a function is undefined . In fact, the definition of the derivative is motivated by that very notion! Continuity : A function f is continuous at a point p if (1) f ( p ) exists (2) lim x p f ( x ) = L , a real number, and (3) f ( p ) = L . For most functions we work with, discovering where a function is continuous is the same as finding the domain of that function. Examining where a function is discontinuous can reveal where holes, vertical asymptotes, or “jumps” exist. Vertical Asymptotes : If at least one of lim x p + f ( x ) or lim x p - f ( x ) is infinite, then f ( x ) has a vertical asymptote x = p . We abridged the vertical-asymptote analysis procedure by saying if lim x p f ( x ) is of the form “Non-Zero / Zero” then there’s a vertical asymptote at x = p . Looking at the one-sided limits at p is how we determine the behavior of f to either side of p , 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) = limh0f(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+12. “Find limh0f(a+h)-f(a)hforf(x) =xx2+13. “Find the instaneous rate of change off(x) =xx2+1atx=a4. “Find the derivative off(x) =xx2+1atx=a5. “Find the slope off(x) =xx2+1atx=a6. “Find the slope of the tangent line of