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Unformatted text preview: (3/19/08) Section 2.3 Tangent lines, rates of change, and derivatives Overview: The derivative was developed in the seventeenth century for determining tangent lines to curves and the velocity of moving objects in cases that could not be handled with geometry and algebra alone. The ancient Greeks had used reasoning similar to that in modern plane geometry to study tangent lines to circles and other special curves. This approach could not be used, however, to find tangent lines to most curves. We will see in this section how tangent lines can be found as the limiting positions of secant lines and how instantaneous velocity and other instantaneous rates of change can be found as limits of average velocities and average rates of change. This leads to the definition of the derivative as the slope of a tangent line and as an instantaneous rate of change. Topics: Tangent lines, derivatives, and instantaneous rates of change Predicting a derivative by calculating difference quotients Finding exact derivatives Equations of tangent lines The x f-formulation of the definition Finding a rate of change from a tangent line A secant-line program Tangent lines, derivatives, and instantaneous rates of change Euclid (c. 300 BC) defined a tangent line to a circle at a point P to be the line that intersects the circle at only that point (Figure 1). The tangent line is outside the circle except at the point of tangency. This implies that the tangent line is perpendicular to the radius at P because P is the closest point on the tangent line to the center O of the circle and consequently is at the foot of the perpendicular line from O to the tangent line. This property enables us to find the slope of the tangent line: if, for example, the center of the circle is the origin in an xy-plane, as in Figure 1, and P has coordinates ( h,k ), then the slope of the radius OP is k/h , and consequently the slope of the perpendicular tangent line is- h/k . x y P ( h,k ) h k x y y = f ( x ) a f ( a ) Tangent line at P Tangent line at x = a FIGURE 1 FIGURE 2 The use of the term tangent in tangent line comes from the Latin tangere , to touch. 89 p. 90 (3/19/08) Section 2.3, Tangent lines, rates of change, and derivatives The graph y = f ( x ) in Figure 2 does not have a similar geometric property that could be used to find its tangent lines. Instead, we find a tangent line as a limit of secant lines. As we saw in the last section, the slope of the secant line through the points at x = a and x = b for b negationslash = a (Figure 3) equals the rise f ( b )- f ( a ) from the point at x = a to the point at x = b , divided by the corresponding run b- a : [Slope of the secant line] = f ( b )- f ( a ) b- a . (1) x y y = f ( x ) a f ( a ) b f ( b )- f ( a ) b- a FIGURE 3 [Slope of the secant line] = f ( b )- f ( a ) b- a ....
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