SalasSV_07_02_ex - EXERCISES 7.2 Estimate the given natural...

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Unformatted text preview: EXERCISES 7.2 Estimate the given natural logarithm on the basis of Table 7.2.1; check your results on a calculator. ln 20. 2. ln 16. ln 1. 6. 4. ln 34 . ln 0. 1. 6. ln 2. 5. ln 7. 2. 8. ln 630. 10. ln 0. 4. ln 2. Interpret the equation ln n = ln mn - ln m in terms of area under the curve y = 1/x. Draw a figure. 12. Given that 0 < x < 1, express as a logarithm the area under the curve y = 1/t from t = x to t = 1. 13. Estimate 1. 3. 5. 7. 9. 11. 1.5 HINT: ln x - ln 1 ln x = ; interpret the limit in terms x-1 x-1 of the derivative of In x. 24. (a) Use the mean-value theorem to show that x-1 ln x x - 1 x for all x > 0. HINT: Consider the cases x 1 and 0 < x < 1 separately. (b) Use the result in part (a) to show that ln x = 1. x1 x - 1 lim 25. (a) Show that for n 2, 1 1 1 1 1 1 + + + < ln n < 1 + + + + . 2 3 n 2 3 n-1 HINT: See the figure. (b) Show that the area of the shaded part is given by 1+ 1 1 1 + + + - ln n. 2 3 n-1 ln 1. 5 = 1 dt . t using the approximation 1 [Lf (P) + Uf (P)] with 2 P = {1 = 8 , 9 , 10 , 11 , 12 = 1. 5}. 8 8 8 8 8 14. Estimate 2.5 ln 2. 5 = 1 dt t using the approximation 1 [Lf (P) + Uf (P)] with 2 P = {1 = 4 , 5 , 6 , 7 , 8 , 9 , 10 = 5 }. 4 4 4 4 4 4 4 2 15. Taking ln 5 1. 61, use differentials to estimate; = (a) ln 5. 2, (b) ln 4. 8, (c) ln 5. 5, 16. Taking ln 10 2. 30, use differentials to estimate: = (a) ln 10. 3, (b) ln 9. 6, (c) ln 11, In Exercises 1722, solve the given equation for x. 17. 19. 21. 22. 23. ln x = 2. 18. ln x = -1. (2 - ln x) ln x = 0. 20. 1 ln x = ln (2x - 1). 2 ln [(2x + 1)(x + 2)] = 2 ln (x + 2). 2 ln (x + 2) - 1 ln x4 = 1. 2 Show that ln x lim = 1. x1 x - 1 As n , this area approaches the number known as Euler's constant. (c) Use geometric reasoning to show that 1 < < 1. (To 2 three decimal places, 0. 577). = y y= 1 x 1 2 3 n 1 n x 388 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS c In Exercises 2628, a function g is given. (a) Show that there is a number r in the indicated interval such that ln r = g(r). HINT: Use the intermediate value theorem. (b) Use a graphing utility to graph ln x and g(x) together. Then use your graphs to find r accurate to four decimal places. 26. g(x) = 2x - 3; 27. g(x) = sin x; [1, 2]. [2, 3]. 28. g(x) = 1 ; x2 [1, 2] ln x ; x = 1 0. 5, 1 0. 1, 1 0. 01, x-1 1 0. 001, 1 0. 0001. x = 0. 5, 0. 1, 0. 01, 0. 001, 0. 0001. 30. lim x ln x; 29. lim x1 x0+ c In Exercises 29 and 30, estimate lim f (x) numerically by evalxa uating f at the indicated values. Then use a graphing utility to zoom in on the graph near x = a to justify your estimate. c 31. (a) Use a graphing utility to draw the graph of f (x) = cos (ln x). (b) Estimate some of the zeros of f . (c) Use a CAS to find a general formula for the zeros of f . c 32. Repeat Exercise 31 with f (x) = ln (cos x). ...
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