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Unformatted text preview: (3/19/08) CHAPTER 1: LIMITS AND CONTINUITY The main tools of calculusthe derivative and the integralare defined by using limits, and many of the basic results of calculus use the related concept of continuity. We discuss finite limits in Sections 1.1 and 1.2 and continuity in Section 1.3. Infinite limits are covered in Section 1.4, and formal definitions of finite and infinite limits are presented in Section 1.5. Section 1.1 Finite limits Overview: This and the next section contain background material on the types of limits that will be used to define derivatives in Chapter 2. Topics: A first look at instantaneous velocity One-sided finite limits A function without a limit Two-sided finite limits Limits of sums, products, powers, and quotients Limits of polynomials and rational functions A first look at instantaneous velocity Suppose that a ball is dropped and there is no air resistance, so the only force on it is the downward force of gravity. The ball falls h = 16 t 2 feet in t seconds (Figure 1). The graph of this height function is the half parabola in Figure 2. We want to determine how fast the ball is falling at t = 1 by finding its instantaneous velocity at that moment. h = 16 t 2 feet Ball Initial position t 1 h (feet) 16 50 h = 16 t 2 16 t 2 t (seconds) FIGURE 1 FIGURE 2 To do this, we first find a formula for the balls average velocity in the interval from time 1 to a slightly later time t . Let us denote this average velocity by A ( t ). At time 1 the ball has fallen 16 feet and at time t it has fallen 16 t 2 feet (Figure 2), so from time 1 to time t it travels 16 t 2- 16 feet. This takes t- 1 seconds, so the balls average velocity is A ( t ) = Distance traveled Time taken = 16 t 2- 16 t- 1 feet second . (1) We estimate the balls instantaneous velocity at time 1 by calculating this average velocity for times t very close to 1, as in the following table. 1 p. 2 (3/19/08) Section 1.1, Finite limits Table 1. The balls average velocity from time 1 to time t t 1 . 1 1 . 01 1 . 001 1 . 0001 1 . 00001 1 . 000001 A ( t ) = 16 t 2- 16 t- 1 33 . 6 32 . 16 32 . 016 32 . 0016 32 . 00016 32 . 000016 The average velocity (1) is not defined for t = 1, where the numerator and denominator are both zero. The calculations in Table 1 suggest that it gets closer to 32 as t gets closer to 1. In fact, it does approach 32 as t approaches 1, as is suggested by its graph in Figure 3. We say that 32 is the limit of the average velocity and that the balls instantaneous velocity at t = 1 is 32 feet per second. This approach to studying velocity will be the basis of the definition of the derivative in Chapter 2. t 1 A (feet per second) A = A ( t ) 16 t 2- 16 t- 1 t (seconds) 32 FIGURE 3 In the next section we will present a special technique that can be used to find limits of functions like the average velocity (1) . Meanwhile, in this section, we will explain limit concepts with examples where the limits can be determined directly from the formulas for the functions.where the limits can be determined directly from the formulas for the functions....
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