HH_Sec_9_1 - n ≥ 2000 d n is 29 for leap years n when n 4...

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(9/1/08) Math 10B. Lecture Examples. Section 9.1. Sequences Example 1 A piece of meat at 30 C is put in a freezer at time n = 0. The temperature of the freezer is 0 C, and the temperature of the meat n hours later is T = 30 n + 1 (Figure 1). Does the sequence braceleftbigg 30 n + 1 bracerightbigg 0 as n → ∞ converge? If so, what is its limit? n 10 20 T C 10 20 30 FIGURE 1 Answer: lim n →∞ 30 n + 1 = 0 converges and its limit is 0. (The temperature of the meat approaches the temperature of the freezer as n → ∞ .) Example 2 Figure 2 shows the graph of the population P = 1000(2 n/ 6 ) on day n of a colony of bacteria that consists of 1000 bacteria at n = 0 (a) How long does it take for the population to double? (b) Does the sequence braceleftBig 1000(2 n/ 6 ) bracerightBig n =0 converge? n 10 20 P 5,000 10,000 FIGURE 2 Answer: (a) The population doubles every 6 days. (b) braceleftbig 1000(2 n/ 6 ) bracerightbig n =0 diverges. Lecture notes to accompany Section 9.1 of Calculus by Hughes-Hallett et al. 1
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Math 10B. Lecture Examples. (9/1/08) Section 9.1, p. 2 Example 3 Figure 3 shows the graph of the the number of days
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Unformatted text preview: n ≥ 2000: d n is 29 for leap years n when n/ 4 is an integer and is 28 other years. What happens to the sequence { d n } ∞ n =2000 as n → ∞ ? n y 30 27 28 29 2000 2004 2008 2012 FIGURE 3 Answer: { d n } ∞ n =2000 diverges. Example 4 Does the sequence b e 1 / √ n B ∞ n =1 converge or diverge? If it converges, give its limit. Answer: The sequence converges and its limit is 1. (The table below shows that the limit is approached relatively slowly.) n 1 10 100 1000 10,000 100,000 e 1 / √ n . = 2 . 7183 1 . 3719 1 . 1052 1 . 0321 1 . 0101 1 . 0010 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 10.1: Examples 1–5 ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course....
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