Sums, Products & Asymptotics
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1 Mathematics for Computer Sciece MIT 6.04J/8.06J Sums, Products & Asymptotics L7-. Sum for Childre Sum for Childre Nie-year old Gauss saw 30 umbers each 3 greater tha the previous oe. (So the story goes.) L7-.3 L7-.4 Sum for Childre st + 30 th = = 555 d + 9 th = ( st +3) + (30 th 3) = rd + 8 th = ( d +3) + (9 th 3) = 555 Sum for Childre Sum of k th term ad (30 k) th term is ivariat! Total = = (st + last) (# terms/) = first Average + last #terms L7-.5 L7-.6
2 Sum for Childre Geometric Series Example: ( ) + = G :: = + x + x x + xg = x + x x x + x + ( + ) L7-.7 L7-.8 Geometric Series Geometric Series G :: = + x + x x + x xg = x + x x + x + x + G xg= x + G :: = + x + x x + xg = x + x G xg= x + G x x = x + x + x + x + L7-.9 L7-.0 Auities The future value of $$. I will promise to pay you $00 i exactly oe year, if you will pay me $X ow. Auities My bak will pay me 3% iterest. If I deposit your $X for a year, I ca t lose if.03 X 00. L7-. L7-.
3 Auities I ca t lose if you pay me: X $00/.03 $97.09 Auities today is worth $.00 i a year $.00 i a year is worth $/.03 today $ i a year is worth $r today, where r = /.03. L7-.3 L7-.4 Auities $ i two years is worth $r today $ i k years is worth $r k today Auities I will pay you $00/year for 0 years If you will pay me $Y ow. I ca t lose if you pay me 00r +00r +00r r 0 =00r(+ r + + r 9 ) = 00r( r 0 )/( r) = $853.0 L7-.5 L7-.6 Class Problems Book Stackig Problems & Rose Rose Rose Rose Rose Rose table L7-.7 All rights reserved. October 6, 00 L7-.8
4 October 6, 00 Book Stackig Book Stackig How far out? Oe book ceter of mass of book? All rights reserved. L7-.9 All rights reserved. October 6, 00 L7-.0 Book Stackig Book Stackig Oe book ceter of mass of book Oe book ceter of mass of book All rights reserved. October 6, 00 L7-. All rights reserved. October 6, 00 L7-. books books ceter of mass All rights reserved. October 6, 00 L7-.3 All rights reserved. October 6, 00 L7-.4
5 October 6, 00 books books Need ceter of mass over table ceter of mass of the whole stack overhag All rights reserved. L7-.5 All rights reserved. October 6, 00 L books ceter of mass of all + books at table edge overhag } ceter of mass of top books at edge of book + overhag ::= Horizotal distace from -book to +-book ceters-of-mass All rights reserved. October 6, 00 L7-.7 L7-.8 Choose origi so ceter of -stack at x = 0. Now ceter of + st book is at x = /, so ceter of +-stack is at ceter of mass of all + books at table edge + books x = 0 + / = + ( + ) ceter of mass of top books at edge of book }+ ( + ) L7-.9 All rights reserved. October 6, 00 L7-.30
6 Book stackig summary B ::= overhag of books B = / B + = B + ( + ) B = H :: = th Harmoic umber B = H / L7-.3 L Estimate H : Itegral Method x+ dx x dx x H l( + ) H L7-.33 L7-.34 Book stackig So H as, ad overhag ca be ay desired size. Book stackig Overhag 3: eed B 3 H 6 Itegral boud: l (+) 6 So ca do with e 6 = 403 books Actually calculate H : 7 books are eough. L7-.35 L7-.36
7 Crossig a Desert Gas depot truck D ::= max distace o tak How big a desert ca the truck cross? L7-.37 L7-.38 Tak of Gas tak + Taks of Gas x x x x x D ::= max distace o tak = L7-.39 L7-.40 So have: If depot at x is taks, ca cotiue with tak Travel. + Taks of Gas (-x) + (-x) If ( x) + ( x) =, the usig tak strategy from positio x, gives D + = D + x pyright Albert R. Meyer, 00. L7-.4 L7-.4
8 ( x) + ( x) = x = + D + = D + + D = dx ( x + ) l( + ) D D Ca cross ay desert! L7-.43 L7-.44 Class Problem Problem 3 L7-.45
Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
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