Fig. 35-1 35.9 Prove: (a) ex is continuous (b) ex is differentiable, and Dx ex = ex in .NET

Drawing Denso QR Bar Code in .NET Fig. 35-1 35.9 Prove: (a) ex is continuous (b) ex is differentiable, and Dx ex = ex

Fig. 35-1 35.9 Prove: (a) ex is continuous (b) ex is differentiable, and Dx ex = ex
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(a) Let > 0. To prove continuity at x, we must show that there exists > 0 such that |u x| < implies |eu ex | < . Let 1 be the minimum of /2 and ex /2. Since ln is increasing and continuous, the range of ln on ex 1 , ex + 1 is an interval (c, d) containing x. Let > 0 be such that x , x + is included within (c, d). Then, for any u, if |u x| < , it follows that |eu ex | < 1 < . (b) The proof will consist in showing that ex+h ex = ex h h 0 lim ex eh ex ex+h ex eh 1 eh 1 = = ex , it will suf ce to show that lim = 1. h h h h h 0 Let k = eh 1. Then eh = 1 + k and, therefore, h = ln (1 + k), by Theorem 35.1. Since ex is continuous and e = 1, k 0 as h 0. Hence, Because eh 1 k = lim h h 0 k 0 ln 1 + k 1 = lim k 0 ln 1 + k ln 1 k 1 = ln 1 + k ln 1 lim k k 0 lim But lim
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[since ln 1 = 0]
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ln 1 + k ln 1 eh 1 is the derivative Dx ln x = 1 at x = 1; that is, it is equal to 1. Hence, lim = 1. x k h k 0 h 0
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EXPONENTIAL FUNCTIONS
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[CHAP. 35
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35.10 (a) Evaluate: (i) e2 ln 3 ; (ii) ln e2 . (b) Solve for x: (i) ln x 2 = 5; (ii) ln (ln x + 1) = 3; (iii) ex 6e x = 5.
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(a) (i) By Property (VIII) and Theorem 35.1, e2 ln 3 = (eln 3 )2 = 32 = 9. (ii) By Theorem 35.1, ln e2 = 2. (b) (i) ln x 2 = 5 x 2 = e5 x = e5/2 ii ln ln x + 1 = 3 ln x + 1 = e3 ln x = e3 1
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3 x = ee 1
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ex 6e x = 5 e2x 6 = 5ex e2x 5ex 6 = 0 ex 6 ex + 1 = 0 or ex + 1 = 0 ex + 1 > 0, since ex > 0 eu = b implies u = ln b ex 6 = 0 ex = 6 x = ln 6 multiply by ex
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Supplementary Problems
35.11 Evaluate the following expressions: (a) e ln x (e) eln (x 1) (b) ln e x ex ( f ) ln x (c) e4
ln x
(g) eln (2x)
3e (h) ln 3 e
ln x
35.12 Calculate the derivatives of the following functions: (a) e x ( f ) ex ln x (b) e1/x (g) x (c) ecos x (h) x (d) tan ex (i) ln e2x ex x (j) ex e x (e)
35.13 Evaluate the following antiderivatives: (a) (d) (g) ( j) e3x dx ecos x sin x dx x x dx x 2 2x dx
(b) (e) (h) (k)
e x dx 32x dx ex e2x dx x 3 e x dx
(c) (f) (i)
ex ex 2dx ex dx e2x x + 1 dx e
35.14 Use implicit differentiation to nd y : (a) ey = y + ln x (d) x 2 + exy + y2 = 1 (b) tan ey x = x 2 (e) sin x = ey (c) e1/y + ey = 2x
CHAP. 35]
EXPONENTIAL FUNCTIONS
35.15 Use logarithmic differentiation to nd y : x (a) y = 3sin x (b) y = ( 2)e (d) y = ln x
ln x
(c) y = x ln x
(e) y2 = x + 1 (x + 2)
35.16 Solve the following equations for x: (a) e3x = 2 (b) ln x 3 = 1 (c) ex 2e x = 1 (d) ln ln x = 1 (e) ln x 1 = 0 (a) the area of R;
35.17 Consider the region R under the curve y = ex , above the x-axis, and between x = 0 and x = 1. Find: (b) the volume of the solid generated by revolving R about the x-axis. 35.18 Consider the region R bounded by the curve y = ex/2 , the y-axis, and the line y = e. Find: volume of the solid generated by revolving R about the x-axis.
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