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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.8: The Derivative, as a Function Now lets do what I know youve been itchin to do for the last few classes. Notice that f gives us a rule for assigning, to any given number a , a new value in the set of real numbers. That is, f is really a . Lets just change the a in our formula to an x so we can write f in a more familiar functional notation: Definition. We define the derivative of f ( x ) to be the function f whose value at x is given by f ( x ) = lim h f ( x + h )- f ( x ) h when this limit exists. Because we know how to interpret f ( a ) at any point a , we can sketch a graph of f , even if we have only a graph (and no formula!) for f ( x )! Just use the following simple principles: 1. f ( x ) = 0 whenever the tangent to f ( x )s graph is horizontal! 2. Where does f ( x ) get its steepest, with both negative and positive slope?...
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- Fall '07