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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.7: Derivatives: Instantaneous Rates of Change and Tan gent Lines So what’s next? Recall that calculus is all about quantities in change , dynamic quantities. In the applications we’ll soon be dealing with, we’ll see that the tangent line to the graph of a function f at a given point ( a, f ( a )) (or, more to the point, its slope) is very important in both theory and practice. Let’s try to get a grip on this important concept. Definitions. Given a function f which defines a curve y = f ( x ), and a point a , we can define the through the points P = ( a, f ( a )) and Q = ( x, f ( x )) for any point x near a in f ’s domain; its slope is easily seen to be m PQ = . As we let x approach a , we end up taking the “limit” of the secant lines, giving us the line , the line passing through the point ( a, f ( a )) and with slope m = lim x → a . Example. Find the equation of the tangent line to the graph of the function y = f ( x ) = x 2 , at the point (1 , 1). (Recall that to do this we need to find the slope of the line, and then we can use the “pointslope” formula for the line.) How about at the point (2 , 4)? (3 , 9)? Notice a trend? Notice that what’s going on is that we’re really defining the slope of the...
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This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.
 Fall '07
 BAHLS
 Derivative

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