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34.9 Show that 1
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1 ln x x 1 for x > 0. x
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For x 1, note that 1/t is a decreasing function on [1, x] and, therefore, its minimum on [1, x] is 1/x and its maximum is 1. Then, by Problem 30.3(c),
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x 1 1 (x 1) ln x = dt 1(x 1) x 1 t 1 1 ln x x 1 x
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Note that 1 1/x < ln x < x 1 for x > 1 by Problem 30.11. For 0 < x < 1, 1/t is an increasing function on [x, 1]. By Problem 30.3(c),
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x 1 1 1 1 (1 x) ln x = dt = dt 1(1 x) x t t 1 x
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Hence, 1
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34.10 Find the derivatives of the following functions: (a) ln (4x 1) (b) (ln x)3 (c) ln x x 1 (g) ln |5x 2| ( f ) ln (e) x 2 ln x x+1 (d) ln (ln x) (h) ln (sin2 x) ln x + 1 dx x
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34.11 Find the following antiderivatives. Use Quick Formula II whenever possible. 1 1 x3 (a) dx dx (b) dx (c) 3x 7x 2 x4 1 dx cos 2x 3x 5 + 2x 2 3 (e) (f ) dx (g) dx x ln x 1 sin 2x x3 1 sec2 x ln x dx (i) dx (h) ( j) dx tan x x x(1 x) 34.12 Use logarithmic differentiation to nd y : (x 2)4 3 x + 5 (a) y = x 3 4 x 2 (b) y = x2 + 4 2 1 sin x x x+2 (c) y = (d) y = 3 x 2 (2x + 3)4 34.13 (a) Show that ln x < x. [Hint: Use Problem 34.9.] (b) Show that 2 ln x < . [Hint: Replace x by x in part (a).] x x
CHAP. 34]
THE NATURAL LOGARITHM
(c) Prove: lim
ln x
x + x x 0+
= 0. [Hint: Use part (b).] 1 in part (c).] x
(d) Prove: lim (x ln x) = 0. [Hint: Replace x by (e) Show that lim (x ln x) = + .
x +
34.14 Calculate in terms of ln 2 and ln 3: 2 (b) ln (a) ln (212 ) 9 34.15 Calculate in terms of ln 2 and ln 5: 1 1 (c) ln (a) ln 10 (b) ln 2 5 1 (e) ln 2 ( f ) ln 3 5 (g) ln 20
(d) ln 25 (h) ln 27
34.16 Find an equation of the tangent line to the curve y = ln x at the point (1, 0). 34.17 Find the area of the region bounded by the curves y = x 2 , y = 1/x and x = 1 . 2 34.18 Find the average value of 1/x on [1, 4]. 34.19 Find the volume of the solid obtained by revolving about the x-axis the region in the rst quadrant under y = x 1/2 between x = 1 and x = 1. 4 34.20 Sketch the graphs of: (a) y = ln (x + 1) (b) y = ln 1 x (c) y = x ln x (d) y = ln (cos x) 6 . (a) Find a formula for the velocity v(t) if v(1) = 1.5. t
34.21 An object moves along the x-axis with acceleration a = t 1 + (b) What is the maximum value of v in the interval [1, 9] 34.22 Use implicit differentiation to nd y : (a) y2 = ln(x 2 + y2 ) 34.23 Find lim 1 3+h ln . 3 h 0 h (b) ln xy + 2x y = 1
(c) ln (x + y2 ) = y3
34.24 Derive the formula 34.25 Find: (a) dx 0 4+x
csc x dx = ln |csc x cot x| + C. [Hint: Similar to Problem 34.7.]
x dx
1 1 2x 2
34.26 GC Approximate ln 2 =
(1/x) dx in the following ways:
(a) By the trapezoidal rule [Problem 31.9(a)], with n = 10. (b) By the midpoint rule (Problem 31.34), with n = 10. (c) By Simpson s rule (Problem 31.35), with n = 10. 34.27 GC Use Newton s method to approximate a solution of: (a) ln x + x = 0; (b) ln x = 1/x.
Exponential Functions
35.1 INTRODUCTION Let a be any positive real number. We wish to de ne a function ax that has the usual meaning when x is rational. For example, we want to obtain the results 43 = 4 4 4 = 64
5 2 =
1 1 = 2 25 5
82/3 =
3 8
= 22 = 4