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Unformatted text preview: SECTION 7.4" GENERAL LOCARlTHMIC AND EXPONENTIAL FUNCTIONS ‘iii “7 arrest to three decimal places, the area of the region 88. (a) Use the inequality of Exercise 87(a) to show that, for [by the curves y = e’, y = e”, and x = 1. ‘ x 2 0, x) if f”(x) = 38" + 5 sin x, f(0) = 1, and f’(0) = 2. e" a 1 + x + gxz volume 0f the solid obtained by rotating about thB (b) Use part (a) to improve the estimate of I; exzdx given in w ‘ .fe region bounded by the curves y = e‘, y = 0, x = 0, Exercise 87(1)), ' i [i -1. ’ , _ _ it: i 89. (a) Use mathematical induction to prove that for x 2 O and any j olurne of the solid obtained by rotating about the positive intager n, i v i region bounded by the curves y = e“ , y = 0, z i p ’ x " ' —' ex21+x+——+~- .‘d the inverse function of f. Check your answer by 1 3th f and fel on the same screen, (b) Use part (a) to show that e > 2.7. ‘ 1' g (c) Use part (a) to show that i 1 + e" (x + 3) oz. f(x) = 1 _ X ex , 6 _ lim "7 : 00 t ‘7 H L n r X’qm x '3 + x + e", ﬁnd (f ’1)’(4). for any positive integer k. , E 90. This exercise illustrates Exercise 89(0) [or the ease k = 10. (a) Compare the rates of growth Of f (x) = x10 and g(x) = 6x ‘ i by graphing both f and g in several viewing rectangles. second law 0f exponents [see (7)]’ When does the graph of 9 ﬁnally surpass the graph of f? ‘ _ (b) Find a viewing rectangle that shows how the function y : x 10 , W 1 HMO. gs; e/xb bgavcgfggargex- how that f (x) = e" — (1 + x) is increasing (e) m a num er sue" é ‘> 0.] 2 * ex 10 ._ , ‘ ‘15 that; S in] ex dx s e. x10 > 10 whenever x > N esinx 7 1 X‘ﬁ' lim ’5’)?! third law of exponents [see (7)]. _ tonotol Loootitttnit and Exponential Functions In this section we'use the natural exponential and logarithmic functions to study exponen- tial and logarithmic functions with base [a > 0. ‘ ' ”H tonornl Exponential Functions If a > 0 and’r, is any rational number, then by (4)and (7) in Section 73*, ar = (elnu)r : erlna Thereforeoeven for irrational numbers x, we deﬁne ’ , [It Thus, for instance, _ F r 2v} =rey31n2 m 61.20 z 332 The function f (x) = a” is called the exponential function with base a, Notice that a" is positive for all x because 6" is positive for all x. ...
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