Unformatted text preview: SECTION 7.4“ GENERAL LOGARITHMIC AND EXPONENTIAL FUNCTIONS iii? 467 88. (a) Use the inequality of Exercise 87(a) to show that, for bythecurvesy=e",y=63",andx= . . x20,
ff”(x) = 3e" + 5 sin x, f(0) = 1, and f’(0) = 2. (3" 21+ x + %x2
Tvolume of the solid obtained by rotating about the (b) Use part (a) to improve the estimate of I; e‘zdx given in
egion bounded by the curves y = e", y = 0, x = 0, Exercise 87(b),
,* 1.
89. (a) Use mathematical induction to prove that for x 2 O and any
«Volume of the solid obtained by rotatingzabout the positive integer n,
region bounded by the curves y = e“ , y = 0, 2
x = l. x L . . . x_"
e 2 1 + x + 2! + + n!
the inverse function of f. Check your answer by
. . f and fer on the same screen. (b) Use part (a) to show that e > 2.7.
1 (c) Use part (a) to show that ‘
 + e"
(x + 3) 82. f(x) = X » e" ,
1 _ 6 lim 7 = 00
n n n n r a r x—m x
‘ + x + e", ﬁnd (f "1)’(4). for any positive integer k.
5”” — 1 90. This exercise illustrates Exercise 89(c) for the case k = 10. . (a) Compare the rates of growth of f (x) = x10 and g(x) = .e’r
_, " by graphing both f and 'g in several viewing rectangles.
3' second law 0f exponents [see (7)]‘ When does the graph of 9 ﬁnally surpass the graph off?
(b) Find a viewing rectangle‘that shows how the function
__ I 10 ~ ,
that e" 2 1 + x ifx 2 0. :9“;— e /"b bﬁavefliﬁrtlarge x‘
how that f(x) = e" — (1 + x) is increasing (c) in a num er {suck a third law of exponents [see (7)]. ex 4 1 2
gsfe‘dxse. x 10 > 1010 ’ whenever x > N ‘ tenernl Logarithmic nnn Exponential Functions In this section we use the natural exponential and logarithmic functions to study exponen
tial and logarithmic functions with base [a > 0. ' L ' '  Eenernl Exponential Functions If a > 0 and r, is any rational number, then by (4)‘and (7) in Section 73*,
ar =(e1nu)r = erlna Therefore, even for irrational numbers x, we deﬁne ' III a Thus, for instance,
2%3 = em as e120 z 3.32 The function f (x) = a” is called the exponential function with base a. Notice that ax is
positive for all x because e" is positive for all x. ...
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