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Unformatted text preview: Math 135 Business Calculus Spring 2009 Class Notes 1.4 Differentiation Using Limits of Difference Quotients In the previous section, we saw that a secant line joining the points ° x,f ( x ) ¢ and ° x + h,f ( x + h ) ¢ on the graph of a function y = f ( x ) has slope m secant = f ( x + h ) − f ( x ) h . This difference quotient also represents the average rate of change of f ( x ) over the interval [ x,x + h ]. In this section, we’ll see that by taking the limit of this difference quotient as h → 0, we obtain the slope of the tangent line to the graph or the instantaneous rate of change of f ( x ) at x . TANGENT LINES A tangent to a curve is sometimes described as a line that touches the curve in exactly one point. For a circle, results from geometry tell us that a tangent line to a point on the circle is perpendicular to a radius intersecting the circle at that point, as shown in the figure below on the left. P L L M For more complicated curves, the above description is inadequate. The above figure on the right displays two lines L and M passing through a point P on a curve. The line M intersects only once, but it certainly does not look like what is thought of as a tangent. In contrast, the linebut it certainly does not look like what is thought of as a tangent....
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