Book Stacking Mathematics for Computer Science MIT 6.042J/18.062J Harmonic Sum Integral Method table Albert R Meyer, April 5, 2010 lec 9M.1 Book Stacking Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.10 Book Stacking How far out? One book book center of mass ? overhang Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.11 Book Stacking One book Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.12 Book Stacking balances if Oneofbook center mass over table book center of mass book center of mass 1 2 Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.13 Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.14 1 n books n books 1 1 2 2 n n Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.15 Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. n books April 5, 2010 lec 9M.16 n books 1 1 2 2 center of mass of the whole stack n n Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. center of mass April 5, 2010 lec 9M.17 balances if center of mass over table Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.18 n+1 books 1 2 center of mass of all n+1 books at table edge n n+1 center of mass of top n books at edge of book n+1 -overhang ::= horizontal distance from n-book to (n+1)-book centers of mass overhang Albert R Meyer, Copyright © Albert R. Meyer, 2007. All rights reserved. April 5, 2010 lec 9M.19 Albert R Meyer, April 5, 2010 lec 9M.20 2 n+1 books -overhang 1 2 n center of mass of all n+1 books 1 1/2 1 n n+1 1 = 2 = n + 1 2(n + 1) Albert R Meyer, April 5, 2010 Albert R Meyer, lec 9M.21 Book stacking summary center of mass of top n books April 5, 2010 lec 9M.23 Harmonic Sums Bn ::= overhang of n books B1 = 1/2 Bn+1 = Bn + nth Harmonic number Bn = Hn/2 Bn = Albert R Meyer, April 5, 2010 Albert R Meyer, lec 9M.24 Integral estimate for Hn Hn = area of rectangles > area under 1/(x+1) = 1 x+1 0 1 n 1 x + 1 dx = 0 n+1 1 1 dx = ln(n + 1) x 1 3 1 2 1 lec 9M.25 Integral estimate for Hn 1 1 2 1 3 April 5, 2010 2 3 Albert R Meyer, 4 5 6 April 5, 2010 7 8 lec 9M.26 Albert R Meyer, April 5, 2010 lec 9M.27 3 Book stacking Book stacking for overhang 3, need Bn 3 Hn 6 integral bound: ln(n+1) 6 so ok with n e6-1 = 403 books actually calculate Hn: 227 books are enough. Albert R Meyer, April 5, 2010 Hn as n, so overhang can be as big as desired! Albert R Meyer, lec 9M.30 April 5, 2010 lec 9M.31 Upper bound for Hn CD cases over the edge 1 1 2 1 3 1 2 1 43 cases high --top 4 cases completely off the table --1.8 or 1.9 case-lengths Albert R Meyer, April 5, 2010 lec 9M.33 Upper bound for Hn 0 1 1 3 2 3 Albert R Meyer, 4 5 6 7 8 April 5, 2010 lec 9M.37 Asymptotic bound for Hn ln(n+1) < Hn < 1+ ln(n) Hn ln(n) Albert R Meyer, April 5, 2010 lec 9M.38 Albert R Meyer, April 5, 2010 il lec 9M.39 4 Asymptotic Equivalence ~ Asymptotic Equivalence Example: Def: f(n) ~ g(n) (n2 + n) ~ n2 pf: Albert R Meyer, April 5, 2010 lec 9M.40 Albert R Meyer, April 5, 2010 lec 9M.41 Team Problems Problems 13 Albert R Meyer, April 5, 2010 lec 9M.43 5 MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.