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(a) the area of R; (b) the
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35.19 Let R be the region bounded by the curve y = ex , the x-axis, the y-axis, and the line x = 1. Find the volume of the solid generated by revolving R about the y-axis. 35.20 Find the absolute extrema of y = esin x on the interval [ , ]. [Hint: eu is an increasing function of u.] 35.21 If y = enx , where n is a positive integer, nd the nth derivative y(n) . 35.22 Let y = 2esin x . (a) Find y and y . (b) Assume that x and y vary with time and that y increases at a constant rate of four units per second. How fast is x changing when x = 35.23 The acceleration of an object moving on the x-axis is 9e3t . (a) If the velocity at time t = 0 is four units per second, nd a formula for the velocity v (t). (b) How far does the object move while its velocity increases from four to ten units per second (c) If the object is at the origin when t = 0, nd a formula for its position x(t). 35.24 Find an equation of the tangent line to the curve y = 2ex at the point (0, 2). 35.25 Sketch the graphs of the following functions, indicating relative extrema, in ection points, and asymptotes: ln x 2 (b) y = x ln x (c) y = (a) y = e x x 1 2 x (e) y = 1 ln x (f ) y = + ln x (d) y = e x [Hint: For parts (b) and (c) you will need the results of Problem 34.13(d) and (c).] 35.26 Sketch the graphs of y = 2x and y = 2 x [Hint: ax = ex ln a .] 35.27 For a > 0 and a = 1, de ne loga x = ln x . This function is called the logarithm of x to the base a (log10 x is called the ln a common logarithm of x). Prove the following properties: 1 (a) Dx loga x = (b) aloga x = x (c) loga ax = x x ln a u (e) loga (uv) = loga u + loga v (f ) loga = loga u loga v (d) log ex = ln x v loga x (h) ln x = (g) loga ur = r loga u loga e
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35.28 Show that the only functions f (x) such that f (x) = f (x) are the functions Cex , where C is a constant. [Hint: Let F(x) = f (x)/ex and nd F (x).] 35.29 Find the absolute extrema of f (x) = (ln x)2 /x on [1, e].
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35.30 (a) Prove ex =
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x u x u . Hint: Let y = 1 + . Then, u u x 1 ln y = u ln 1 + = u ln (u + x) ln u = u x u u
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[by the mean-value theorem] x x u u < 1 + if x > 0 or 1 + < <1 u u u u
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where u < u < u + x if x > 0, and u + x < u < u if x < 0. Then either 1 < if x < 0. In either case, (b) Prove e = lim lim
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u = 1. So, lim ln y = x and, therefore, lim y = lim eln y = ex . u u u + u + u + 1 n . [Hint: Use part (a)]. n 1 n
n +
(c) GC Approximate e by nding 1 +
for large values of n (say, n 10 000).
35.31 Show that, for any positive n, lim
xn = 0. x + ex n n ln x e 1 1 x = = . Now apply Problems 34.13(c) and 35.7(d).] [Hint: x = x 1 n ln x/x e ex ex n ln x e
35.32 Evaluate the following de nite integrals: (a)
1 e 1 + ln x
x lim
dx 1
ex dx x 0 e +1
35.33 Evaluate
n + n
e1/n + e2/n + + en/n .
35.34 Show that e > e . [Hint: Prove generally that uv > v u when v > u e. u/e uv eu ev uu By Problem 34.9, v v v v ve ve 1 > ln , > ln + 1 = ln , ev/u > , and so, u u u u u u ev uu > eu v u By (1) and (2), uv eu > eu v u , uv > v u .]
1 since u/e 1 and v > u. So, (1)
35.35 If interest is paid at r percent per year and is compounded n times per year, then P dollars become P 1 + after 1 year. If we let n , then the resulting interest is said to be continuously compounded.