hwk0 - 1 3 x +7 (c) ln x [HINT: Use integration by parts.]...

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Fall 2011 CMSC 351: Homework 0 Clyde Kruskal Do this homework, but do not hand in. Problem 1. Use mathematical induction to show that ( a ) n s i =1 i ( i + 1) = n ( n + 1)( n + 2) 3 ( b ) n s i =0 2 i = 2 n +1 - 1 Problem 2. See bottom of this sheet. (a) Assume b x = a . What is x (in terms of a and b )? (b) Using only part (a), show that log c ( ab ) = log c a + log c b . (c) Show that a log b n = n log b a Problem 3. Di±erentiate the following functions: (a) ln( x 2 + 5) (b) lg( x 2 + 5) (c) 1 ln( x 2 +5) Problem 4. Integrate the following functions: (a) 1 x (b)
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Unformatted text preview: 1 3 x +7 (c) ln x [HINT: Use integration by parts.] (d) x ln x [HINT: Use integration by parts.] (e) x lg x lg n = log 2 n ln n = log e n lg k n = (lg n ) k lg lg n = lg(lg n ) For all real a > 0, b > 0, c > 0, and n , a = b log b a log c ( ab ) = log c a + log c b log b a n = n log b a log b a = log c a log c b log b (1 /a ) =-log b a log b a = 1 log a b a log b n = n log b a...
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