SECTION 14.4 The Tangent Problem; The Derivative 961 Figure 13 x y y 5 f(x) Tangent line to f at P P Figure 14 P Tangent line This definition, however, does not work in general. Look at Figure 15. The lines L1 and L2 intersect the graph at only one point P , but neither just touches the graph at P . Further, the tangent line LT shown in Figure 16 touches the graph of f at P but also intersects the graph elsewhere. So how should we define the tangent line to the graph of f at a point P ? Figure 15 x y L2 L1 P Figure 16 x y LT P 1 Find an Equation of the Tangent Line to the Graph of a Function The tangent line LT to the graph of a function ( ) = y f x at a point P necessarily contains the point P . To find an equation for LT using the point-slope form of the equation of a line, we need to find the slope mtan of the tangent line. Suppose that the coordinates of the point P are ( ) ( ) c f c , . Locate another point ( ) ( ) = Q x f x , on the graph of f. The line containing P and Q is a secant line. The slope msec of the secant line is ( ) ( ) = − − m f x f c x c sec Now look at Figure 17. As we move along the graph of f from Q toward P , we obtain a succession of secant lines. The closer we get to P , the closer the secant line is to the tangent line L . T The limiting position of these secant lines is the tangent line L . T Therefore, the limiting value of the slopes of these secant lines equals the slope of the tangent line. Also, as we move from Q toward P , the values of x get closer to c . Therefore, ( ) ( ) = = − − → → m m f x f c x c lim lim x c x c tan sec DEFINITION Tangent Line The tangent line to the graph of a function ( ) = y f x at a point P is the line containing the point ( ) ( ) = P c f c , and having the slope ( ) ( ) = − − → m f x f c x c lim x c tan (1) provided the limit exists. Need to Review? Secant lines and their slopes are discussed in Section 2.3, pp. 94 – 95 . Figure 17 Secant lines x y Q 5 (x, f(x)) Q1 5 (x1, f(x1)) LT P 5 (c, f(c)) y 5 f(x) c
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