Chapter Gauss Quadrature Rule of Integration


 Kerry Lynch
 1 years ago
 Views:
Transcription
1 Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to solve emples o pproimte integrls. Wht is integrtion? Integrtion is the proess o mesuring the re under untion plotted on grph. Why would we wnt to integrte untion? Among the most ommon emples re inding the veloity o ody rom n elertion untion, nd displement o ody rom veloity untion. Throughout mny engineering ields, there re wht sometimes seems like ountless pplitions or integrl lulus. You n red out some o these pplitions in Chpters 7.A7.G. Sometimes, the evlution o epressions involving these integrls n eome dunting, i not indeterminte. For this reson, wide vriety o numeril methods hs een developed to simpliy the integrl. Here, we will disuss the Guss qudrture rule o pproimting integrls o the orm where I d is lled the integrnd, lower limit o integrtion upper limit o integrtion 7..
2 7.. Chpter 7. Figure Integrtion o untion. Guss Qudrture Rule Bkground: To derive the trpezoidl rule rom the method o undetermined oeiients, we pproimted d Let the right hnd side e et or integrls o stright line, tht is, or n integrted orm o d So d But rom Eqution, we wnt d. d to give the sme result s Eqution or Hene rom Equtions nd,
3 Guss Qudrture Rule 7.. Sine nd re ritrry onstnts or generl stright line Multiplying Eqution y nd sutrting rom Eqution gives 6 Sustituting the ove ound vlue o in Eqution gives 6 Thereore d 7 Derivtion o twopoint Guss qudrture rule The twopoint Guss qudrture rule is n etension o the trpezoidl rule pproimtion where the rguments o the untion re not predetermined s Method : nd, ut s unknowns nd. So in the twopoint Guss qudrture rule, the integrl is pproimted s d I There re our unknowns,, nd. These re ound y ssuming tht the ormul gives et results or integrting generl third order polynomil,. Hene d d 8 The ormul would then give d 9
4 7.. Chpter 7. Equting Equtions 8 nd 9 gives Sine in Eqution, the onstnts,,, nd re ritrry, the oeiients o,,, nd re equl. This gives us our equtions s ollows. Without proo see Emple or proo o relted prolem, we n ind tht the ove our simultneous nonliner equtions hve only one eptle solution Hene d We n derive the sme ormul y ssuming tht the epression gives et vlues or the individul integrls o Method :, d, d, d nd d. The reson the ormul n lso e
5 Guss Qudrture Rule 7.. derived using this method is tht the liner omintion o the ove integrnds is generl third order polynomil given y. These will give our equtions s ollows d d d d These our simultneous nonliner equtions n e solved to give single eptle solution Hene d 6 Sine two points re hosen, it is lled the twopoint Guss qudrture rule. Higher point versions n lso e developed. Higher point Guss qudrture ormuls For emple d 7 is lled the threepoint Guss qudrture rule. The oeiients, nd, nd the untion rguments, nd re lulted y ssuming the ormul gives et epressions or integrting ith order polynomil d. Generl n point rules would pproimte the integrl
6 7..6 Chpter 7. d n n 8 Arguments nd weighing tors or npoint Guss qudrture rules In hndooks see Tle, oeiients nd rguments given or n point Guss qudrture rule re given or integrls o the orm n g d g 9 i i i Tle Weighting tors nd untion rguments used in Guss qudrture ormuls Points Weighting Ftors Funtion Arguments
7 Guss Qudrture Rule So i the tle is given or g d integrls, how does one solve d? The nswer lies in tht ny integrl with limits o [, ] n e onverted into n integrl with limits [, ]. Let mt I, then t I, then t suh tht m m Solving the two Equtions simultneously gives m Hene t d dt Sustituting our vlues o nd d into the integrl gives us d d Emple For n integrl d, show tht the twopoint Guss qudrture rule pproimtes to where d
8 7..8 Chpter 7. Solution Assuming the ormul d gives et vlues or integrls, d d d d d, d d, nd d. Then E. E. E. E. E. Multiplying Eqution E. y nd sutrting rom Eqution E. gives E.6 The solution to the ove eqution is,, or/nd, or/nd. I. is not eptle s Equtions E.E. redue to,,, nd. But sine, ut onlits with. II. is not eptle s Equtions E.E. redue to,,, nd. Sine, then or hs to e zero ut this violtes. III. is not eptle s Equtions E.E. redue to,,, nd. I, then
9 Guss Qudrture Rule 7..9 gives nd tht violtes. I, then tht violtes. Tht leves the solution o s the only possile eptle solution nd in t, it does not hve violtions see it or yoursel E.7 Sustituting E.7 in Eqution E. gives E.8 From Equtions E. nd E.8, E.9 Equtions E. nd E.9 gives E. Sine Eqution E.7 requires tht the two results e o opposite sign, we get Hene Emple d For n integrl d, derive the onepoint Guss qudrture rule. Solution The onepoint Guss qudrture rule is E. d E. d nd d Assuming the ormul gives et vlues or integrls, d d E. the other eqution eomes Sine,
10 7.. Chpter 7. E. Thereore, onepoint Guss qudrture rule n e epressed s d E. Emple Wht would e the ormul or d i you wnt the ove ormul to give you et vlues o, d tht is, liner omintion o nd. Solution I the ormul is et or liner omintion o nd, then d d E. Solving the two Equtions E. simultneously gives 6 6 E. So d 6 6 E. Let us see i the ormul works. Evlute d using EqutionE.
11 Guss Qudrture Rule 7.. d [ ] [ ] The et vlue o d is given y d 6. Any surprises? Now evlute d using Eqution E. d 6. The et vlue o d is given y d [ ] 6 9 Beuse the ormul will only give et vlues or liner omintions o nd not work etly even or simple integrl o d. Do you see now why we hoose s the integrnd or whih the ormul gives us et vlues? d, it does Emple Use twopoint Guss qudrture rule to pproimte the distne overed y roket rom t 8 to t s given y ln 9.8t dt t 8 Also, ind the solute reltive true error.
12 7.. Chpter 7. Solution First, hnge the limits o integrtion rom [ 8, ] to [, ] t dt d 8 9 d using Eqution gives Net, get weighting tors nd untion rgument vlues rom Tle or the two point rule, Now we n use the Guss qudrture ormul 9 d [ 9 9 ] [ ] [.69.8] [ ] 8. m sine.69 ln ln The solute reltive true error,, is True vlue 6. m t % t Emple Use threepoint Guss qudrture rule to pproimte the distne overed y roket rom t 8 to t s given y ln 9.8t dt t 8 Also, ind the solute reltive true error.
13 Guss Qudrture Rule 7.. Solution First, hnge the limits o integrtion rom [ 8, ] to [, ] t dt d 8 9 d The weighting tors nd untion rgument vlues re nd the ormul is sine using Eqution gives 9 d [ ] [ ] [ ] 6. m.79 ln ln ln The solute reltive true error, t, is True vlue 6. m t 6..%
14 7.. Chpter 7. INTEGRATION Topi Guss qudrture rule Summry These re tetook notes o Guss qudrture rule Mjor Generl Engineering Authors Autr Kw, Mihel Ketelts Dte August, We Site
Gauss Quadrature Rule of Integration
Guss Qudrture Rule o Integrtion Mjor: All Engineering Mjors Authors: Autr Kw, Chrlie Brker http://numerilmethods.eng.us.edu Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00 http://numerilmethods.eng.us.edu
More informationGauss Quadrature Rule of Integration
Guss Qudrture Rule o Integrtion Computer Engineering Mjors Authors: Autr Kw, Chrlie Brker http://numerilmethods.eng.us.edu Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00 http://numerilmethods.eng.us.edu
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationFormula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx
Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint
More informationIntegration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationTime in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.
4. Are under Curve A cr is trveling so tht its speed is never decresing during 1second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37
More informationNumerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1
Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multisegmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule
More informationChapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationAP CALCULUS Test #6: Unit #6 Basic Integration and Applications
AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationChapter Direct Method of Interpolation More Examples Electrical Engineering
Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o
More informationChapter Direct Method of Interpolation More Examples Civil Engineering
Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationCHAPTER 6b. NUMERICAL INTERPOLATION
CHAPTER 6. NUMERICAL INTERPOLATION A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering y Dr. Irhim A. Asskk Spring ENCE  Computtion s in Civil Engineering II Deprtment o Civil
More informationMA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES
MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These
More information] dx (3) = [15x] 2 0
Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 09765840 Volume 6, Numer (04), pp. 89 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationReference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant 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 information50 AMC Lectures Problem Book 2 (36) Substitution Method
0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III)  Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III  Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationThe Double Integral. The Riemann sum of a function f (x; y) over this partition of [a; b] [c; d] is. f (r j ; t k ) x j y k
The Double Integrl De nition of the Integrl Iterted integrls re used primrily s tool for omputing double integrls, where double integrl is n integrl of f (; y) over region : In this setion, we de ne double
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationSimpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for rightngled tringles
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More informationSolving Radical Equations
Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
More informationReview 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 informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section  4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More informationMT Integral equations
MT58  Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,
More information6.1 Definition of the Riemann Integral
6 The Riemnn Integrl 6. Deinition o the Riemnn Integrl Deinition 6.. Given n intervl [, b] with < b, prtition P o [, b] is inite set o points {x, x,..., x n } [, b], lled grid points, suh tht x =, x n
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More information2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.
SECTION 81 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationGeneralization of 2Corner Frequency Source Models Used in SMSIM
Generliztion o 2Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationMat 210 Updated on April 28, 2013
Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 0506 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationLesson 2.1 Inductive Reasoning
Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,
More informationy z A lefthanded system can be rotated to look like the following. z
Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between
More informationCALCULUS STUDY MATERIAL. B.Sc. MATHEMATICS III SEMESTER UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
CALCULUS STUDY MATERIAL BS MATHEMATICS III SEMESTER UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY PO MALAPPURAM, KERALA, INDIA  67 65 5 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationQuadrature Rules for Evaluation of Hyper Singular Integrals
Applied Mthemticl Sciences, Vol., 01, no. 117, 53955 HIKARI Ltd, www.mhikri.com http://d.doi.org/10.19/ms.01.75 Qudrture Rules or Evlution o Hyper Singulr Integrls Prsnt Kumr Mohnty Deprtment o Mthemtics
More informationAP Calculus AB Unit 4 Assessment
Clss: Dte: 004 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationEigenvectors and Eigenvalues
MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.
More informationMath 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 informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationPolynomials 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 information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationChapter Direct Method of Interpolation More Examples Chemical Engineering
hter 5. Direct Method of Interoltion More Exmles hemicl Engineering Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t
More information1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang
nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationFor a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs7.2ADV.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationK 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:
Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to
More informationSIMPLE NONLINEAR GRAPHS
S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationx dx does exist, what does the answer look like? What does the answer to
Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl
More information