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Chapter 01

# Chapter 01 - 5E-FM.qk 11:09 AM Page 1 CA L C U L U S...

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C A L C U L U S 5E-FM.qk 1/19/06 11:09 AM Page 1

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A Preview of Calculus By the time you finish this course, you will be able to explain the formation and location of rainbows, compute the force exerted by water on a dam, analyze the population cycles of predators and prey, and calculate the escape velocity of a rocket. 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 2
Calculus is fundamentally different from the mathematics that you have studied previously. Calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the sub- ject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems. The Area Problem The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illus- trates this process for the special case of a circle with inscribed regular polygons. Let be the area of the inscribed polygon with sides. As increases, it appears that becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century B . C .) used exhaustion to prove the familiar formula for the area of a circle: We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate as the limit of these sums of areas of rectangles. A A A r 2 . A lim n l A n A n n n A n A¡™ A∞ FIGURE 2 A 3 FIGURE 3 1 n 1 0 x y (1, 1) 1 0 x y (1, 1) 1 4 1 2 3 4 0 x y 1 (1, 1) FIGURE 4 1 0 x y y=≈ A (1, 1) The Preview Module is a numerical and pictorial investigation of the approximation of the area of a circle by inscribed and circumscribed polygons. FIGURE 1 A=A¡+A™+A£+A¢+A∞ A™ A∞ 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 3

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4 ❙ ❙ ❙ ❙ A PREVIEW OF CALCULUS The area problem is the central problem in the branch of calculus called integral cal- culus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.
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Chapter 01 - 5E-FM.qk 11:09 AM Page 1 CA L C U L U S...

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