5128_Ch08_434-471

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55. (b) V ln 2 A(x)dx ln 2 (p/4) e 2x dx 55. Each cross section of the solid infinite horn shown in the figure cut by a plane perpendicular to the x-axis for x ln 2 is a circular disc with one diameter reaching from the x-axis to the curve y e x . y 2 y = e x 1 0 ln 2 (a) Find the area of a typical cross section. A(x) (p/4) e 2x (b) Express the volume of the horn as an improper integral. (c) Find the volume of the horn. p/2 56. Normal Probability Distribution Function In Section 7.5, we encountered the bell-shaped normal distribution curve that is the graph of 1 ( ) 2 f x 1 x m 2 s , 2 the normal probability density function with mean m and standard deviation s. The number m tells where the distribution is centered, and s measures the “scatter” around the mean. From the theory of probability, it is known that f x dx 1. In what follows, let m 0 and s 1. (a) Draw the graph of f. Find the intervals on which f is increasing, the intervals on which f is decreasing, and any local extreme values and where they occur. (b) Evaluate n f x dx for n 1, 2, 3. n (c) Give a convincing argument that f x dx 1. (Hint: Show that 0 f x e x2 for x 1, and for b 1, e x2 dx→0 as b→.) b 57. Approximating the Value of 1 e x2 dx (a) Show that 6 e x2 dx e 6x dx 4 10 17 . 1 1 31. 0 1 e x e x on [1, ), converges because 1 1 e x dx converges 34. 1 1 32. 0 x 3 1 x 3 on [1, ), converges because 1 1 x 3 dx converges 33. 0 1 x 2 cos x on [p, ), diverges because 1 dx diverges x p x 6 x Section 8.4 Improper Integrals 469 (b) Writing to Learn Explain why 1 e x2 dx 6 e x2 dx with error of at most 4 10 17 . (c) Use the approximation in part (b) to estimate the value of 1 e x2 dx. Compare this estimate with the value displayed in Figure 8.19. (d) Writing to Learn Explain why 0 e x2 dx 3 e x2 dx with error of at most 0.000042. Extending the Ideas 58. Use properties of integrals to give a convincing argument that Theorem 6 is true. 59. Consider the integral f n 1 x n e x dx 0 where n 0. (a) Show that 0 x n e x dx converges for n 0, 1, 2. (b) Use integration by parts to show that f n 1 nfn. (c) Give a convincing argument that 0 x n e x dx converges for all integers n 0. 60. Let f x x sin t dt. 0 t (a) Use graphs and tables to investigate the values of f x as x→. (b) Does the integral 0 sin xx dx converge? Give a convincing argument. 61. (a) Show that we get the same value for the improper integral in Example 5 if we express 1 dxx 1 2 1 dxx 2 1 0 and then evaluate these two integrals. 1 , 1 dxx 2 (b) Show that it doesn’t matter what we choose for c in (Improper Integrals with Infinite Integration Limits, part 3) f x dx c f x dx f x dx. dx x 4 2 1 dx 0 x 4 21 1 dx 0 x 4 2 1 dx 1 x and 4 1 1 0 x 4 1 1 x2 on [1, ), converges because 1 1 x 2 dx converges c
470 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals Quick Quiz for AP* Preparation: Sections 8.3 and 8.4 You may use a graphing calculator to solve the following problems. 1. Multiple Choice Which of the following functions grows faster than x 2 as x→? E (A) e x (B) ln(x) (C) 7x 10 (D) 2x 2 3x (E) 0.1x 3 2. Multiple Choice Find all the values of p for which the integral converges dx . C x 1 p1 (A) p 1 (B) p 0 (C) p 0 (D) p 1 (E) diverges for all p 3. Multiple Choice Find all the values of p for which the 1 dx integral converges x p. B 1 0 (A) p 1 (B) p 0 (C) p 0 (D) p 1 (E) diverges for all p 4. Free Response Consider the region R in the first quadrant under the curve y 2ln (x) . x 2 (a) Write the area of R as an improper integral. (b) Express the integral in part (a) as a limit of a definite integral. (c) Find the area of R. Chapter 8 Key Terms Absolute Value Theorem for Sequences (p. 440) arithmetic sequence (p. 436) binary search (p. 456) common difference (p. 436) common ratio (p. 437) comparison test (p. 464) constant multiple rule for limits (p. 439) convergence of improper integral (p. 459) convergent sequence (p. 439) difference rule for limits (p. 439) divergence of improper integral (p. 459) divergent sequence (p. 439) explicitly defined sequence (p. 435) finite sequence (p. 435) geometric sequence (p. 437) grows at the same rate (p. 453) grows faster (p. 453) grows slower (p. 453) improper integral (pp. 459) indeterminate form (p. 444) infinite sequence (p. 435) l’Hôpital’s Rule, first form (p. 444) l’Hôpital’s Rule, stronger form (p. 445) limit of a sequence (p. 439) nth term of a sequence (p. 435) product rule for limits (p. 439) quotient rule for limits (p. 439) recursively defined sequence (p. 435) Sandwich Theorem for Sequences (p. 440) sequence (p. 435) sequential search (p. 456) sum rule for limits (p. 439) terms of sequence (p. 435) transitivity of growing rates (p. 455) value of improper integral (p. 459) Chapter 8 Review Exercises The collection of exercises marked in red could be used as a Chapter Test. In Exercises 1 and 2, find the first four terms and the fortieth term of the given sequence. 1. a n (1) n n 1 for all n 1 12, 35, 23, 57; a 40 41/43 n 3 3, 6, 12, 24; 2. a 1 3, a n 2a n1 for all n 2 a 40 3(2 39 ) 3. The sequence 1, 1/2, 2, 7/2, … is arithmetic. Find (a) the common difference, (b) the tenth term, and (c) an explicit rule for the nth term. (a) 3/2 (b) 25/2 (c) a n 3n 5 2 4. The sequence 1/2, 2, 8, 32, … is geometric. Find (a) the common ratio, (b) the seventh term, and (c) an explicit rule for the nth term. (a) 4 (b) 2048 (c) a n (1) n1 (2 2n3 ) In Exercises 5 and 6, draw a graph of the sequence with given nth term. 5. a n 2n1 (1) , n 2 n n 1, 2, 3, … 6. a n (1) n1 n 1 n In Exercises 7 and 8, determine the convergence or divergence of the sequence with given nth term. If the sequence converges, find its limit. 7. a n 3 n2 1 2n2 converges, 3/2 8. a n (1) n 3 n 1 diverges 1 n 2 In Exercises 9–22, find the limit. 9. lim t ln 1 2t t→0 t 2 sin x 11. lim 2 x→0 1 x cos x 13. lim x→ x 1x 1 14. lim x→ ( 15. lim c os r 0 16. lim r→ ln r 17. lim x→1 ( 1 1 x 1 ln ) 1/2 18. lim x 10. lim t an 3t 3/5 t→0 tan 5t 12. lim x 11x 1/e x→1 x 1 3 x ) u→p2 ( x→0 ( e 3 u p 2 ) sec u x 1 1 x ) 1 1 9. The limit doesn’t exist.
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