Chapter 02 - 5E-02(pp 064-073 1:24 PM Page 64 CHAPTER 2 The...

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Limits and Rates of Change C H A P T E R 2 The idea of a limit is illustrated by secant lines approaching a tangent line. 5E-02(pp 064-073) 1/17/06 1:24 PM Page 64
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In A Preview of Calculus (page 2) we saw how the idea of a limit underlies the various branches of calculus. It is there- fore appropriate to begin our study of calculus by investi- gating limits and their properties. |||| 2.1 The Tangent and Velocity Problems In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object. The Tangent Problem The word tangent is derived from the Latin word tangens , which means “touching.” Thus, a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once as in Figure 1(a). For more complicated curves this defini- tion is inadequate. Figure l(b) shows two lines and passing through a point on a curve . The line intersects only once, but it certainly does not look like what we think of as a tangent. The line , on the other hand, looks like a tangent but it intersects twice. To be specific, let’s look at the problem of trying to find a tangent line to the parabola in the following example. EXAMPLE 1 Find an equation of the tangent line to the parabola at the point . SOLUTION We will be able to find an equation of the tangent line as soon as we know its slope . The difficulty is that we know only one point, , on , whereas we need two points to compute the slope. But observe that we can compute an approximation to by choosing a nearby point on the parabola (as in Figure 2) and computing the slope of the secant line . We choose so that . Then m PQ x 2 1 x 1 Q P x 1 PQ m PQ Q x , x 2 m t P m t P 1, 1 y x 2 y x 2 t (a) (b) t P C t l FIGURE 1 C t C l C P t l 65 Locate tangents interactively and explore them numerically. Resources / Module 1 / Tangents / What Is a Tangent? FIGURE 2 x y 0 y=≈ t Q {x, ≈} P (1, 1) 5E-02(pp 064-073) 1/17/06 1:25 PM Page 65
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66 ❙ ❙ ❙ ❙ CHAPTER 2 LIMITS AND RATES OF CHANGE For instance, for the point we have The tables in the margin show the values of for several values of close to 1. The closer is to , the closer is to 1 and, it appears from the tables, the closer is to 2. This suggests that the slope of the tangent line should be . We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing and Assuming that the slope of the tangent line is indeed 2, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through as Figure 3 illustrates the limiting process that occurs in this example. As approaches along the parabola, the corresponding secant lines rotate about and approach the tangent line t .
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