ch01_01 - 74 CHAPTER 1 . Limits and Continuity 1-2 1.1 A...

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74 CHAPTER 1 . . Limits and Continuity 1-2 1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE In this section, we approach the boundary between precalculus mathematics and the calculus by investigating several important problems requiring the use of calculus. Recall that the slope of a straight line is the change in y divided by the change in x . This fraction is the same regardless of which two points you use to compute the slope. For example, the points (0, 1), (1, 4), and (3, 10) all lie on the line y = 3 x + 1. The slope of 3 can be obtained from any two of the points. For instance, m = 4 1 1 0 = 3o r m = 10 1 3 0 = 3 . 0.96 0.98 1.00 1.02 1.04 1.90 1.95 2.00 2.05 2.10 y x FIGURE 1.3 y = x 2 + 1 In the calculus, we generalize this problem to find the slope of a curve at a point. For instance, suppose we wanted to find the slope of the curve y = x 2 + 1at the point (1, 2). You might think of picking a second point on the parabola, say (2, 5). The slope of the line through these two points (called a secant line; see Figure 1.2a) is easy enough to compute. We have m sec = 5 2 2 1 = 3 . However, using the points (0, 1) and (1, 2), we get a different slope (see Figure 1.2b): m sec = 2 1 1 0 = 1 . y 0.5 1 1.5 2 2.5 ± 0.5 ± 2 2 4 6 x y 0.5 1 1.5 2 2.5 ± 0.5 ± 2 2 4 6 x FIGURE 1.2a Secant line, slope = 3 FIGURE 1.2b Secant line, slope = 1 For curves other than straight lines, the slopes of secant lines joining different points are generally not the same, as seen in Figures 1.2a and 1.2b. If you get different slopes using different pairs of points, then what exactly does it mean for a curve to have a slope at a point? The answer can be visualized by graphically zooming in on the specified point. Take the graph of y = x 2 + 1 and zoom in tight on the point (1, 2). You should get a graph something like the one in Figure 1.3. The graph looks very much like a straight line. In fact, the more you zoom in, the straighter the curve appears to be and the less it matters which two points are used to compute a slope. So, here’s the strategy: pick several points on the parabola, each closer to the point (1, 2) than the previous one. Compute the slopes of the lines through (1, 2) and each of the points. The closer the second point gets to (1, 2), the closer the computed slope is to the answer you seek.
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1-3 SECTION 1.1 . . A Brief Preview of Calculus 75 Forexample, the point (1.5, 3.25) is on the parabola fairly close to (1, 2). The slope of the line joining these points is m sec = 3 . 25 2 1 . 5 1 = 2 . 5 . The point (1.1, 2.21) is even closer to (1, 2). The slope of the secant line joining these two points is m sec = 2 . 21 2 1 . 1 1 = 2 . 1 . Continuing in this way, observe that the point (1.01, 2.0201) is closer yet to the point (1, 2). The slope of the secant lines through these points is m sec = 2 . 0201 2 1 . 01 1 = 2 . 01 . The slopes of the secant lines (2.5, 2.1, 2.01) are getting closer and closer to the slope of the parabola at the point (1, 2). Based on these calculations, it seems reasonable to say that the slope of the curve is approximately 2.
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This note was uploaded on 10/03/2010 for the course CHE 10 taught by Professor Toupadakis during the Spring '08 term at UC Davis.

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ch01_01 - 74 CHAPTER 1 . Limits and Continuity 1-2 1.1 A...

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