We will see what is meant by standard form very shortly

Size: px
Start display at page:

Download "We will see what is meant by standard form very shortly"

Transcription

1 THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible region. We will see wht is ment by stndrd form very shortly More generlly, the bove theorem nd the grphs we sw tell us tht every LP is in exctly one of the following sttes: 1. Fesible with unique optimum solution (cluse b of theorem) 2. Fesible with infinitely mny optim (cluse b of theorem) 3. Fesible, with no optimum solution becuse the objective is unbounded (cluse of theorem) 4. Infesible, nd hence with no optimum solution 2018, Jynt Rjgopl

2 Assume for now tht we hve fesible LP nd tht the objective is bounded (i.e., we re in Sttes 1 or 2). Then the following re true: A. The LP hs t lest one optiml corner point (or extreme point) 1. If in Stte 1, exctly one extreme point is optiml 2. If in Stte 2, t lest two djcent (neighboring) extreme points re optiml B. The number of extreme points is finite C. If the objective function t some extreme point is s good s or better thn t ll of its djcent extreme point, then this extreme point is optiml for the LP This suggest simple lgorithmic pproch to finding the optimum of fesible nd bounded LP 2018, Jynt Rjgopl

3 STEP 0 (Initiliztion): Find n initil extreme point nd mke it the current cndidte (if one cnnot be found the LP is in stte 4, i.e. it is infesible so STOP). STEP 1(Stopping Criterion Check): Is the objective t the current extreme point t lest s good or better thn it is t ll of its djcent (neighboring) extreme points? If so this must be the optiml extreme point (vi C) so STOP. If not, go to Step 2. STEP 2(Itertive Step): One (or more ccurtely, t lest one ) of the djcent extreme points is better so mke it the current cndidte. Then return to Step 1. QUESTIONS: 1) How to find n initil extreme point? 2) Wht is the lgebric chrcteriztion of n extreme point? 3) Wht is the lgebric chrcteriztion of djcent extreme points, i.e., how to move from n extreme point to one of its neighbors (in Step 2)? 2018, Jynt Rjgopl

4 The SIMPLEX Method - Development Every LP is in exctly one of the following sttes: 1. Fesible with unique optimum solution. 2. Fesible with infinitely mny optim. 3. Fesible, with no optimum solution (becuse the objective is unbounded). 4. Infesible, nd hence with no optimum solution. Assume we re in Sttes 1 or 2. Then the following re true: A. The LP hs t lest one optiml corner point (or extreme point). 1) If in Stte 1, exctly one extreme point is optiml. 2) If in Stte 2, t lest two djcent (neighboring) extreme points re optiml. B. The number of extreme points is finite. C. If the objective function t some extreme point is s good s or better thn t ll of its djcent extreme point, then this extreme point is optiml for the LP.

5 It follows then tht we cn use the following itertive lgorithmic procedure: STEP 0 (Initiliztion): Find n initil extreme point nd mke it the current cndidte (if one cnnot be found the LP is in stte 4, i.e. it is infesible so STOP). STEP 1 (Stopping Criterion Check): Is the objective t the current extreme point t lest s good or better thn it is t ll of its djcent (neighboring) extreme points? If so this must be the optiml extreme point (vi C) so STOP. If not, go to Step 2. STEP 2 (Itertive Step): At lest one of the djcent extreme points is better so mke it the current cndidte nd go to Step 1. QUESTIONS: 1) How to find n initil extreme point? 2) Wht is the lgebric chrcteriztion of n extreme point? 3) Wht is the lgebric chrcteriztion of djcent extreme points, i.e., how to move from n extreme point to one of its neighbors (in Step 2)?

6 Stndrd Form of Liner Progrm Three conditions must be met: 1) All constrints must be stted s equlities of the form g k (x)=b k, where g k (x) is liner function of x. 2) The RHS for ech constrint must be nonnegtive, i.e., ll b k 0 3) All vribles must be nonnegtive, i.e., x i 0 for ll i. Thus the LP (with m constrints nd n vribles) looks like Min (or Mx) c 1 x 1 + c 2 x c n x n st 11 x x n x n = b 1 21 x x n x n = b 2 : m1 x 1 + m2 x mn x n = b m x 1, x 2,, x n 0. where b 1, b 2,, b n 0. More compctly, let c=[c 1 c 2... c n ] nd A= : m : m n 2n : mn x1 x 2 x= : x n b1 b 2 nd x= : b m

7 Then the LP my be restted s Min (or Mx) cx, st Ax=b, x 0, where b 0. This finl system of constrint equtions of the LP in stndrd form is clled the AUGMENTED SYSTEM. Consider such n ugmented system of n vribles in m equtions, where m n. Note tht the ugmented system does not include the nonnegtivity conditions! Also note tht this system hs three possible sttes: (1) no solution, (2) exctly one solution, or (3) infinitely mny solutions. Fesible solution: Any solution to the ugmented system tht lso stisfies nonnegtivity is clled fesible solution E.g. Consider the following constrints nd the corresponding ugmented system: x 1 + 2x x 1 + 2x 2 + S 1 = 120 x 1 + x 2 90 x 1 + x 2 + S 2 = 90 x 1 70 x 1 + S 3 = 70 x 2 50 x 2 + S 4 = 50

8 Note tht n=6 nd m=4. 1) An (infesible) solution is [x 1 =70, x 2 =50, S 1 =-50, S 2 =-30, S 3 =0, S 4 =0] 2) A fesible solution is [x 1 =20, x 2 =20, S 1 =60, S 2 =50, S 3 =50, S 4 =30] Bsic Solution: Suppose we fix (n-m) out of the n vribles t zero, nd try to solve the system of m equtions in the remining m vribles. If solution to this exists, then it is clled bsic solution. Solution (1) on the previous pge is n exmple of bsic solution. Bsic Fesible Solution: A bsic solution tht lso stisfies nonnegtivity is clled bsic fesible solution (BFS). An exmple of BFS is [x 1 =20, x 2 =50, S 1 =0, S 2 =20, S 3 =50, S 4 =0]. Note tht (1) is not BFS even though it is bsic solution! Augmented System One Solution No Solution Infinitely mny solutions If it is lso fesible ( 0) Originl LP is infesible then the originl LP hs Fesible Infesible exctly one fesible solution - it is lso optiml! Bsic Nonbsic Bsic Nonbsic

9 FACT: Ech BFS corresponds to n extreme point of the fesible region. In bsic solution, the (n-m) vribles tht re chosen to be fixed t zero re clled the nonbsic vribles nd the remining m vribles re clled bsic vribles. Note tht we cn choose (n-m) out of the n vribles in n n n! = = different wys. Thus we cn get n m m ( n m)! m! mximum of this mny different bsic solutions (some of which will lso be bsic fesible solutions) in prctice ll of these my not exist since in some cses, the resulting m x m system my not hve solution! n 6! In our exmple, n=6 nd m=4 so tht = = 15 n m (6 4)!4! The 15 combintions of nonbsic vribles re: 1) x 1, x 2 2) x 1, S 1 6) x 2, S 1 3) x 1, S 2 7) x 2, S 2 10) S 1, S 2 4) x 1, S 3 8) x 2, S 3 11) S 1, S 3 13) S 2, S 2 5) x 1, S 4 9) x 2, S 4 12) S 1, S 4 14) S 2, S 4 15) S 3, S 4 Try to locte ech of these on the grphicl representtion!

10 X 2 Mx 20X 1 +10X 2 S 3 =0, X 1 =70 st X 1 +2X (Constr. 1, S 1 ) X 1 +X 2 90 (Constr. 2, S 2 ) 90 X 1 70 (Constr. 3, S 3 ) S 2 =0, X 1 +X 2 =90 X 2 50 (Constr. 4, S 4 ) X 1, X II S 1 =S 4 =0 S 2 =S 4 =0 S 3 =S 4 =0 S 4 =0, X 2 =50 III 10 I S 1 =S 2 =0 X * 1 =70 IV S 1 =S 3 =0 X * 2 =20 Fesible Region Opt. Vl. =1600 V S 2 =S 3 =0 VI S 1 =0, X 1 +2X 2 = X 1 1., : BASIC SOLUTIONS (13 of them). Note tht there should be ( 4 6 )=15 of these, but only 13 exist becuse X 1 =70 is prllel to the X 2 -xis nd X 2 =50 is prllel to the X 1 -xis. 2. : BASIC FEASIBLE SOLUTION: (6 of the bove 13, numbered I, II, III, IV, V, VI) 3. : Contour of the objective function corresponding to vlue of : Set of ll Fesible solutions to the LP

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below. Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

Can the Phase I problem be unfeasible or unbounded? -No

Can the Phase I problem be unfeasible or unbounded? -No Cn the Phse I problem be unfesible or unbounded? -No Phse I: min 1X AX + IX = b with b 0 X 1, X 0 By mnipulting constrints nd dding/subtrcting slck/surplus vribles, we cn get b 0 A fesible solution with

More information

Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem

Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA www.olivierdejong.com Solving Flir Furniture

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

How do you know you have SLE?

How do you know you have SLE? Simultneous Liner Equtions Simultneous Liner Equtions nd Liner Algebr Simultneous liner equtions (SLE s) occur frequently in Sttics, Dynmics, Circuits nd other engineering clsses Need to be ble to, nd

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

Chapter 3 Solving Nonlinear Equations

Chapter 3 Solving Nonlinear Equations Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,

More information

Unit 1 Exponentials and Logarithms

Unit 1 Exponentials and Logarithms HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!! Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but... Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

More information

Administrivia CSE 190: Reinforcement Learning: An Introduction

Administrivia CSE 190: Reinforcement Learning: An Introduction Administrivi CSE 190: Reinforcement Lerning: An Introduction Any emil sent to me bout the course should hve CSE 190 in the subject line! Chpter 4: Dynmic Progrmming Acknowledgment: A good number of these

More information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

More information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

More information

19 Optimal behavior: Game theory

19 Optimal behavior: Game theory Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

Theoretical foundations of Gaussian quadrature

Theoretical foundations of Gaussian quadrature Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

MAA 4212 Improper Integrals

MAA 4212 Improper Integrals Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

More information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 4310 Solutions to homework 1 Due 9/1/16 Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Matrix Solution to Linear Equations and Markov Chains

Matrix Solution to Linear Equations and Markov Chains Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before

More information

(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer

(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.

More information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

Chapter 3 Polynomials

Chapter 3 Polynomials Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling

More information

Module 6: LINEAR TRANSFORMATIONS

Module 6: LINEAR TRANSFORMATIONS Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

Problem Set 3

Problem Set 3 14.102 Problem Set 3 Due Tuesdy, October 18, in clss 1. Lecture Notes Exercise 208: Find R b log(t)dt,where0

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Chapter 1 Cumulative Review

Chapter 1 Cumulative Review 1 Chpter 1 Cumultive Review (Chpter 1) 1. Simplify 7 1 1. Evlute (0.7). 1. (Prerequisite Skill) (Prerequisite Skill). For Questions nd 4, find the vlue of ech expression.. 4 6 1 4. 19 [(6 4) 7 ] (Lesson

More information

Summary: Method of Separation of Variables

Summary: Method of Separation of Variables Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004 Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

More information

Orthogonal Polynomials

Orthogonal Polynomials Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

Math 270A: Numerical Linear Algebra

Math 270A: Numerical Linear Algebra Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Numerical Analysis: Trapezoidal and Simpson s Rule

Numerical Analysis: Trapezoidal and Simpson s Rule nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =

More information

Integral points on the rational curve

Integral points on the rational curve Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box

More information

Linearly Similar Polynomials

Linearly Similar Polynomials Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd

More information

Tech. Rpt. # UMIACS-TR-99-31, Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, June 3, 1999.

Tech. Rpt. # UMIACS-TR-99-31, Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, June 3, 1999. Tech. Rpt. # UMIACS-TR-99-3, Institute for Advnced Computer Studies, University of Mrylnd, College Prk, MD 20742, June 3, 999. Approximtion Algorithms nd Heuristics for the Dynmic Storge Alloction Problem

More information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

Reinforcement learning II

Reinforcement learning II CS 1675 Introduction to Mchine Lerning Lecture 26 Reinforcement lerning II Milos Huskrecht milos@cs.pitt.edu 5329 Sennott Squre Reinforcement lerning Bsics: Input x Lerner Output Reinforcement r Critic

More information

Matrices and Determinants

Matrices and Determinants Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

More information

Abstract inner product spaces

Abstract inner product spaces WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

Heteroclinic cycles in coupled cell systems

Heteroclinic cycles in coupled cell systems Heteroclinic cycles in coupled cell systems Michel Field University of Houston, USA, & Imperil College, UK Reserch supported in prt by Leverhulme Foundtion nd NSF Grnt DMS-0071735 Some of the reserch reported

More information

AMATH 731: Applied Functional Analysis Fall Some basics of integral equations

AMATH 731: Applied Functional Analysis Fall Some basics of integral equations AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)

More information

Downloaded from

Downloaded from POLYNOMIALS UNIT- It is not once nor twice but times without number tht the sme ides mke their ppernce in the world.. Find the vlue for K for which x 4 + 0x 3 + 5x + 5x + K exctly divisible by x + 7. Ans:

More information

Elementary Linear Algebra

Elementary Linear Algebra Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

UNIT-2 POLYNOMIALS Downloaded From: [Year] UNIT-2 POLYNOMIALS

UNIT-2 POLYNOMIALS Downloaded From:   [Year] UNIT-2 POLYNOMIALS UNIT- POLYNOMIALS Downloded From: www.jsuniltutoril.weebly.com [Yer] UNIT- POLYNOMIALS It is not once nor twice but times without number tht the sme ides mke their ppernce in the world.. Find the vlue

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

DATA Search I 魏忠钰. 复旦大学大数据学院 School of Data Science, Fudan University. March 7 th, 2018

DATA Search I 魏忠钰. 复旦大学大数据学院 School of Data Science, Fudan University. March 7 th, 2018 DATA620006 魏忠钰 Serch I Mrch 7 th, 2018 Outline Serch Problems Uninformed Serch Depth-First Serch Bredth-First Serch Uniform-Cost Serch Rel world tsk - Pc-mn Serch problems A serch problem consists of:

More information

CHAPTER 4a. ROOTS OF EQUATIONS

CHAPTER 4a. ROOTS OF EQUATIONS CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

Uninformed Search Lecture 4

Uninformed Search Lecture 4 Lecture 4 Wht re common serch strtegies tht operte given only serch problem? How do they compre? 1 Agend A quick refresher DFS, BFS, ID-DFS, UCS Unifiction! 2 Serch Problem Formlism Defined vi the following

More information

USA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year 1/1/5. Alex is trying to oen lock whose code is sequence tht is three letters long, with ech of the letters being one of A, B or C, ossibly reeted. The lock hs three buttons, lbeled A, B nd C. When the

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More

More information

0.1 Chapters 1: Limits and continuity

0.1 Chapters 1: Limits and continuity 1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

For the percentage of full time students at RCC the symbols would be:

For the percentage of full time students at RCC the symbols would be: Mth 17/171 Chpter 7- ypothesis Testing with One Smple This chpter is s simple s the previous one, except it is more interesting In this chpter we will test clims concerning the sme prmeters tht we worked

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Expectation and Variance

Expectation and Variance Expecttion nd Vrince : sum of two die rolls P(= P(= = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(=2) = 1/36 P(=3) = 1/18 P(=4) = 1/12 P(=5) = 1/9 P(=7) = 1/6 P(=13) =? 2 1/36 3 1/18 4 1/12 5 1/9 6 5/36 7 1/6

More information

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information