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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 “point-slope” 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