Homework - Algorithms - Summer 2008 | CMSC 351, Assignments of Algorithms and Programming

Download Homework - Algorithms - Summer 2008 | CMSC 351 and more Assignments Algorithms and Programming in PDF only on Docsity! Summer 2008 CMSC 351: Homework 0 Clyde Kruskal Problem 1. Use mathematical induction to show that (a) n∑ i=1 i(i + 1) = n(n + 1)(n + 2) 3 (b) n∑ i=0 2i = 2n+1 − 1 Problem 2. See bottom of page 53 of CLRS (bottom of page 34 in CLR) and/or the bottom of this sheet. (a) Assume bx = a. What is x (in terms of a and b)? (b) Using only part (a), show that logc(ab) = logc a + logc b. (c) Show that alogb n = nlogb a Problem 3. Differentiate the following functions: (a) ln(x2 + 5) (b) lg(x2 + 5) (c) 1ln(x2+5) Problem 4. Integrate the following functions: (a) 1 x (b) 13x+7 (c) ln x [HINT: Use integration by parts.] (d) x ln x [HINT: Use integration by parts.] (e) x lg x lg n = log2 n ln n = loge n lgk n = (lg n)k lg lg n = lg(lg n) For all real a > 0, b > 0, c > 0, and n, a = blogb a logc(ab) = logc a + logc b logb a n = n logb a logb a = logc a logc b logb(1/a) = − logb a logb a = 1 loga b alogb n = nlogb a