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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 240 (Fall, 2008)
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.11.6, 2.12.2, 3.23.8, 4.34.5, 5.15.3, 5.5, 6.16.5, 7.17.2, 7.4 Chapter 1: Linear Equations and Matrices DEFINITIONS There are a number of deni...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 520 (Fall, 2008)
%!PS-Adobe-2.0 %Creator: dvips 5.55 Copyright 1986, 1994 Radical Eye Software %Title: algebra.dvi %CreationDate: Tue Aug 25 14:36:13 1998 %Pages: 42 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips -p =7 -l =48 algeb...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 232 (Fall, 2008)
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N. Illinois >> MATH >> 232 (Fall, 2008)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 232 (Fall, 2008)
T w t s 3 9 ` dA 3 C 0#!7fq}pr6I}@#fb0!\"#w{Fp o T5 s |A 5 x 5 \" g}grubeg0D6q x!B66X66i\"g66Ir~igp{6yhr0#i6{6#vuUiWT!g~}4 A \" h 7 A s h \" dA s t 5 5 h \" s t \" 5 \" T A t s | x#}4g0#!7{i\"fz6!\"gv46By6g(#i7fI6bvi96u66esrg0!\"#wqFp (n ...
N. Illinois >> MATH >> 232 (Fall, 2008)
1. (a) Since x (t ) ! t 2 and y(t ) ! t 3 , it must be that t ! \"2 at the point (4, \"8) . The slope is given by dy / dt 3t 2 3t ! ! so when t ! \"2 it is m ! \"3 . The tangent line is dx / dt 2t 2 y ! \"3x # b and \"8 ! \"3 $ 4 # b implies b ! 4 so y ! ...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 444 (Fall, 2008)
2 The linear one-dimensional case. 2.1 Extremizers are not in the interior. Let c ! ! be given and assume that c \" 0 . The collection of nontrivial continuous linear functions is given by fc : ! # ! defined by fc $ x % & cx . All x ! ! are in the int...
N. Illinois >> MATH >> 444 (Fall, 2008)
4 Excel and Solver. 4.1 Spreadsheets. There are many computer programs to solve optimization problems. The most widely available is Microsoft Excel. Excel is an all-purpose spreadsheet-based program. Think of a spreadsheet as a grid of cells where ea...
N. Illinois >> MATH >> 444 (Fall, 2008)
7 The simplex algorithm by example. 7.1 A second pivoting rule. Consider the primal problem x1 + 3x 2 9 max 2x1 + 3x 2 when 2x1 + x 2 8 . x1 , x 2 0 The corresponding initial tableau is given by 1 3 1 0 9 2 1 0 1 8. 2 3 0 0 0 Since bot...
N. Illinois >> MATH >> 444 (Fall, 2008)
9 Matrix form of the simplex algorithm. 9.1 Initial tableau. The primal problem Ax b max cT x when , x 0 with x n , b m , so that A is m n , yields an augmented system of the form x A I = b . y To keep the notation simple, contin...
N. Illinois >> MATH >> 444 (Fall, 2008)
10 Sensitivity. 10.1 Issues. The solution of a linear programming problem yields an optimal feasible point. It is often necessary to be concerned about how changes in the data A, b, c affect the solution. It is particularly troubling if tiny changes ...
N. Illinois >> MATH >> 444 (Fall, 2008)
13 An introduction to nonlinear problems. 13.1 The standard form. The standard form for a nonlinear problem is in these notes given by f1 (x ) 0 max f0 (x ) when f (x ) 0 P and fr : Rn R for all r { 0, , P } .The standard form covers...
N. Illinois >> MATH >> 444 (Fall, 2008)
1. (a) -2 6 8 x 1 + 2x 2 = 8 x1 + 2x 2 = 8 and . The The extreme points of the feasible set are given by x1 + x 2 = 6 2x1 x 2 = 2 12 14 points are ( 4,2 ) and , . The values are 4 and 4 respectively. 5 5 2x1 + x 2 2 x +x 6 1 2...
N. Illinois >> MATH >> 444 (Fall, 2008)
Math 444 Exam 2 1 2 Suppose A = 1 1 . 3 2 What is the value of v if the final tableau is 0 1 1 1 0 5 1 0 0 0 0 0 1 1 2 4 0 1 0 4 2 v ? Name:_ S. S. #:_ Each problem is worth 20 points. Do any five of the six problems. Circle the five yo...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 444 (Fall, 2008)
1. 1 2 T = 1 1 0 , cT T A = cT 1 1 0 1 1 = cT 2 3 = 0 0 3 2 4 v = cT x = 2 3 = 23 5 27 (a) B 1 (b + b ) = B 1b + B 1 b = 3 + B 1 b 0 leads to 27 + b 0 , 1 12 27 1 b 0 , 3 + 1 b 0 , 27 + 2b...
N. Illinois >> MATH >> 444 (Fall, 2008)
1. 81 + 22 1 2 + 4 2 1 2 3 Dual: min 121 + 62 + 83 when 1 + 23 3 1, 2 , 3 0 8 2 1 1 0 0 12 2 Initial tableau: 0 1 1 4 2 0 2 3 0 0 0 1 0 0 0 1 0 6 8 0 . 8 2 Second tableau: 0 1 4 1 2 4 0 0 1 0 1 0 0 0 0 1 0 0 1/ 2 0 1/ 2 3 ...
N. Illinois >> MATH >> 444 (Fall, 2008)
Exam 1 1. Use the simplex algorithm to solve the problem: 2x1 + 3x 2 + x 3 10 x1 + x 2 + 2x 3 8 max 2x1 + 3x 2 + 6x 3 when . 2x 2 + 3x 3 6 x 1, x 2 , x 3 0 Make sure that each step employed is the step suggested by the simplex algo...
N. Illinois >> MATH >> 444 (Fall, 2008)
Exam 2 1. Consider a primal problem with 2 3 1 A = 1 1 2 , 0 2 3 and a final tableau given by 0 3 0 1 1 0 0 1/ 3 2/ 3 1/ 3 0 1 0 0 0 0 2 1 0 2 1 2 / 3 1/ 3 2 / 3 0 4 2 20 . (a) Determine b . The basic variables are y1 , x1 , x 3 and it ...
N. Illinois >> MATH >> 360 (Fall, 2008)
Math 360 Sample Project: River Crossing N a c w b Project: Consider a straight portion of a river of width w as depicted in the map. The project is to determine the most cost-effective configuration of a buried cable connecting a location north o...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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N. Illinois >> MATH >> 211 (Fall, 2008)
MATH 211 FINAL EXAM 5/3/97 NO CALCULATORS! 1. (30 points) Find the derivative of each of these functions. You do not need to simplify your answers. (a) f (x) = x2 + 1 x2 (b) f (x) = (x5 + ln x)4 (a) lim (c) f (x) = x3 1 x2 + 1 (d) f (x) = ex (b...
N. Illinois >> MATH >> 211 (Fall, 2008)
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N. Illinois >> MATH >> 211 (Fall, 2008)
MATH 211 FINAL EXAM 12/9/98 NO CALCULATORS! 1. (32 points) Find the derivative of each of the following functions. You do not need to simplify your answers. x5 (b) f (x) = ln(5 - 6x7 ) (a) f (x) = 3 7x + 6x - 1 2 (c) f (x) = 4e-3x +1 + 7 (d) f (x...
N. Illinois >> MATH >> 211 (Fall, 2008)
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N. Illinois >> MATH >> 211 (Fall, 2008)
E k ` F S ` c T ` y p u ph Q P h fV V Q F h SV s Sph c h fV u Q T S S T h c S h fV u SX 7 `XV P IF E Weeb86Uub F ddY0bsd9QG60idg\'dTbrg\'aUrdig8\'ebQgbVddBUV2bUSig2dbTY5aYd2wgGw ` F` j mtbWh0ibF UTeg86DdF ` k F `& ` F ` s e #b8sF EX T SV...
Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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Carnegie Mellon >> DISK >> 05103113 (Fall, 2009)
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Carnegie Mellon >> TERA >> 05103113 (Fall, 2009)
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