Numerical Integration Formulas
|
|
- Tracey Ball
- 6 years ago
- Views:
Transcription
1 Numerical Itegratio Formulas Berli Che Departmet o Computer Sciece & Iormatio Egieerig Natioal Taiwa Normal Uiversity Reerece: 1. Applied Numerical Methods with MATLAB or Egieers, Chapter 19 & Teachig material
2 Chapter Objectives (1/) Recogizig that Newto-Cotes itegratio ormulas are based o the strategy o replacig a complicated uctio or tabulated data with a polyomial that is easy to itegrate Kowig how to implemet the ollowig sigle applicatio Newto-Cotes ormulas: Trapezoidal rule Simpso s 1/3 rule Simpso s 3/8 rule Kowig how to implemet the ollowig composite Newto-Cotes ormulas: Trapezoidal rule Simpso s 1/3 rule NM Berli Che
3 Chapter Objectives (/) Recogizig that eve-segmet-odd-poit ormulas like Simpso s 1/3 rule achieve higher tha epected accuracy Kowig how to use the trapezoidal rule to itegrate uequally spaced data Uderstadig the dierece betwee ope ad closed itegratio ormulas NM Berli Che 3
4 Itegratio Itegratio: I b d a is the total value, or summatio, o () d over the rage rom a to b: NM Berli Che 4
5 Newto-Cotes Formulas The Newto-Cotes ormulas are the most commo umerical itegratio schemes Geerally, they are based o replacig a complicated uctio or tabulated data with a polyomial that is easy to itegrate: I b d d a b a where () is a th order iterpolatig polyomial NM Berli Che 5
6 Newto-Cotes Eamples The itegratig uctio ca be polyomials or ay order - or eample, (a) straight lies or (b) parabolas The itegral ca be approimated i oe step or i a series o steps to improve accuracy NM Berli Che 6
7 The Trapezoidal Rule The trapezoidal rule is the irst o the Newto-Cotes closed itegratio ormulas; it uses a straight-lie approimatio or the uctio: b I d I a b (a) b a ad a b a I b a a b NM Berli Che 7
8 Error o the Trapezoidal Rule A estimate or the local trucatio error o a sigle applicatio o the trapezoidal rule is: E t 1 1 b a 3 where is somewhere betwee a ad b This ormula idicates that the error is depedet upo the curvature o the actual uctio as well as the distace betwee the poits Error ca thus be reduced by breakig the curve ito parts NM Berli Che 8
9 Trapezoidal Rule: A Eample Eample 19.1 NM Berli Che 9
10 Composite Trapezoidal Rule Assumig +1 data poits are evely spaced, there will be itervals over which to itegrate The total itegral ca be calculated by itegratig each subiterval ad the addig them together: 1 I d d d d 0 I I h 1 0 i i NM Berli Che 10
11 Composite Trapezoidal Rule: A Eample Eample 19. NM Berli Che 11
12 MATLAB Program NM Berli Che 1
13 Simpso s Rules Oe drawback o the trapezoidal rule is that the error is related to the secod derivative o the uctio More complicated approimatio ormulas ca improve the accuracy or curves - these iclude usig (a) d ad (b) 3rd order polyomials The ormulas that result rom takig the itegrals uder these polyomials are called Simpso s rules NM Berli Che 13
14 Simpso s 1/3 Rule Simpso s 1/3 rule correspods to usig secod-order polyomials. Usig the Lagrage orm or a quadratic it o three poits: Itegratio over the three poits simpliies to: I I h 3 0 b a 4 0 d h b a NM Berli Che 14
15 Error o Simpso s 1/3 Rule A estimate or the local trucatio error o a sigle applicatio o Simpso s 1/3 rule is: E t b a 5 where agai is somewhere betwee a ad b This ormula idicates that the error is depedet upo the ourth-derivative o the actual uctio as well as the distace betwee the poits Note that the error is depedet o the ith power o the step size (rather tha the third or the trapezoidal rule) Error ca thus be reduced by breakig the curve ito parts NM Berli Che 15
16 Simpso s 1/3 Rule: A Eample Eample 19.3 NM Berli Che 16
17 Composite Simpso s 1/3 Rule Simpso s 1/3 rule ca be used o a set o subitervals i much the same way the trapezoidal rule was, ecept there must be a odd umber o poits Because o the heavy weightig o the iteral poits, the ormula is a little more complicated tha or the trapezoidal rule: NM Berli Che 17 a b I h I h h h I d d d d I j j i i i i j j i i i i eve, 1 odd, 1 0 eve, 1 odd, a b h
18 Composite Simpso s 1/3 Rule: A Eample Eample 19.4 NM Berli Che 18
19 Simpso s 3/8 rule correspods to usig thirdorder polyomials to it our poits. Itegratio over the our poits simpliies to: I I 3h 8 Simpso s 3/8 Rule b a 3 h d 0 3 b a Simpso s 3/8 rule is geerally used i cocert with Simpso s 1/3 rule whe the umber o segmets is odd NM Berli Che 19 3
20 Simpso s 3/8 Rule: A Eample (1/) Eample 19.5 NM Berli Che 0
21 Simpso s 3/8 Rule: A Eample (/) NM Berli Che 1
22 Higher-Order Formulas Higher-order Newto-Cotes ormulas may also be used - i geeral, the higher the order o the polyomial used, the higher the derivative o the uctio i the error estimate ad the higher the power o the step size As i Simpso s 1/3 ad 3/8 rule, the eve-segmet-oddpoit ormulas have trucatio errors that are the same order as ormulas addig oe more poit. For this reaso, the eve-segmet-odd-poit ormulas are usually the methods o preerece NM Berli Che
23 Itegratio with Uequal Segmets Previous ormulas were simpliied based o equispaced data poits - though this is ot always the case The trapezoidal rule may be used with data cotaiig uequal segmets: 1 I d d d d 0 0 I NM Berli Che 3
24 Itegratio Code or Uequal Segmets NM Berli Che 4
25 MATLAB Fuctios MATLAB has built-i uctios to evaluate itegrals based o the trapezoidal rule z = trapz(y) z = trapz(, y) produces the itegral o y with respect to. I is omitted, the program assumes h=1 z = cumtrapz(y) z = cumtrapz(, y) produces the cumulative itegral o y with respect to. I is omitted, the program assumes h=1 NM Berli Che 5
26 Multiple Itegrals Multiple itegrals ca be determied umerically by irst itegratig i oe dimesio, the a secod, ad so o or all dimesios o the problem T (, y) y y 7 NM Berli Che 6
Computation Sessional. Numerical Differentiation and Integration
CE 6: Egieerig Computatio Sessioal Numerical Dieretiatio ad Itegratio ti di ad gradiet commad di() Returs te dierece betwee adjacet elemets i. Typically used or uequally spaced itervals = gradiet(, ) Determies
More informationCHAPTER 6c. NUMERICAL INTERPOLATION
CHAPTER 6c. NUMERICAL INTERPOLATION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig y Dr. Irahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil
More informationCHAPTER 6d. NUMERICAL INTERPOLATION
CHAPER 6d. NUMERICAL INERPOLAION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig by Dr. Ibrahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil ad
More information18.01 Calculus Jason Starr Fall 2005
Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 8 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayaavaam Road 5758 QUESTION BANK (DESCRIPTIVE) Subject with Code : (6HS6) Course & Brach: B.Tech AG Year & Sem: II-B.Tech&
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationx x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula
NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 17 Numerical Integration Formulas PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationTaylor Polynomials and Approximations - Classwork
Taylor Polyomials ad Approimatios - Classwork Suppose you were asked to id si 37 o. You have o calculator other tha oe that ca do simple additio, subtractio, multiplicatio, or divisio. Fareched\ Not really.
More informationCS475 Parallel Programming
CS475 Parallel Programmig Dieretiatio ad Itegratio Wim Bohm Colorado State Uiversity Ecept as otherwise oted, the cotet o this presetatio is licesed uder the Creative Commos Attributio.5 licese. Pheomea
More informationCS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
More informationTAYLOR AND MACLAURIN SERIES
Calculus TAYLOR AND MACLAURIN SERIES Give a uctio ( ad a poit a, we wish to approimate ( i the eighborhood o a by a polyomial o degree. c c ( a c( a c( a P ( c ( a We have coeiciets to choose. We require
More information5 3B Numerical Methods for estimating the area of an enclosed region. The Trapezoidal Rule for Approximating the Area Under a Closed Curve
5 3B Numerical Methods for estimatig the area of a eclosed regio The Trapezoidal Rule for Approximatig the Area Uder a Closed Curve The trapezoidal rule requires a closed o a iterval from x = a to x =
More informationRepresenting Functions as Power Series. 3 n ...
Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.
More informationTECHNIQUES OF INTEGRATION
7 TECHNIQUES OF INTEGRATION Simpso s Rule estimates itegrals b approimatig graphs with parabolas. Because of the Fudametal Theorem of Calculus, we ca itegrate a fuctio if we kow a atiderivative, that is,
More informationSome Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations
Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 8 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayaavaam Road 5758 QUESTION BANK (DESCRIPTIVE Subject with Code : (6HS6 Course & Brach: B.Tech AG Year & Sem: II-B.Tech& I-Sem
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More informationSolving third order boundary value problem with fifth order block method
Matematical Metods i Egieerig ad Ecoomics Solvig tird order boudary value problem wit it order bloc metod A. S. Abdulla, Z. A. Majid, ad N. Seu Abstract We develop a it order two poit bloc metod or te
More informationA NUMERICAL METHOD OF SOLVING CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS BASED ON A LINEAR APPROXIMATION
U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 7 ISSN -77 A NUMERICAL METHOD OF SOLVING CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS BASED ON A LINEAR APPROXIMATION Cristia ŞERBĂNESCU, Marius BREBENEL A alterate
More informationMath 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals
Math 1314 Lesso 16 Area ad Riema Sums ad Lesso 17 Riema Sums Usig GeoGebra; Defiite Itegrals The secod questio studied i calculus is the area questio. If a regio coforms to a kow formula from geometry,
More informationError for power series (Day 2) YOU MAY USE YOUR CALCULATOR TO COMPUTE FRACTIONS AND OTHER SIMPLE OPERATIONS
AP Calculus BC CHAPTE B WOKSHEET INFINITE SEQUENCES AND SEIES Name Seat # Date Error or power series (Day ) YOU MAY USE YOU CALCULATO TO COMPUTE FACTIONS AND OTHE SIMPLE OPEATIONS a) Approimate si usig
More informationModification of Weerakoon-Fernando s Method with Fourth-Order of Convergence for Solving Nonlinear Equation
ISSN: 50-08 Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 Modiicatio o Weerakoo-Ferado s Method with Fourth-Order o Covergece or Solvig Noliear Equatio
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationTopic 9 - Taylor and MacLaurin Series
Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
More informationTaylor Series (BC Only)
Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor
More informationExample 2. Find the upper bound for the remainder for the approximation from Example 1.
Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute
More informationSection A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics
X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationContinuous Random Variables: Conditioning, Expectation and Independence
Cotiuous Radom Variables: Coditioig, Expectatio ad Idepedece Berli Che Departmet o Computer ciece & Iormatio Egieerig Natioal Taiwa Normal Uiversit Reerece: - D.. Bertsekas, J. N. Tsitsiklis, Itroductio
More informationComputational Methods CMSC/AMSC/MAPL 460. Quadrature: Integration
Computatioal Metods CMSC/AMSC/MAPL 6 Quadrature: Itegratio Ramai Duraiswami, Dept. o Computer Siee Some material adapted rom te olie slides o Eri Sadt ad Diae O Leary Numerial Itegratio Idea is to do itegral
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationAreas and Distances. We can easily find areas of certain geometric figures using well-known formulas:
Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio
More informationMaclaurin and Taylor series
At the ed o the previous chapter we looed at power series ad oted that these were dieret rom other iiite series as they were actually uctios o a variable R: a a + + a + a a Maclauri ad Taylor series +
More informationPC5215 Numerical Recipes with Applications - Review Problems
PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationFourier Series and Transforms
Fourier Series ad rasorms Orthogoal uctios Fourier Series Discrete Fourier Series Fourier rasorm Chebyshev polyomials Scope: wearetryigto approimate a arbitrary uctio ad obtai basis uctios with appropriate
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 4 Solutions [Numerical Methods]
ENGI 3 Advaced Calculus or Egieerig Facult o Egieerig ad Applied Sciece Problem Set Solutios [Numerical Methods]. Use Simpso s rule with our itervals to estimate I si d a, b, h a si si.889 si 3 si.889
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationCalculus with Analytic Geometry 2
Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,
More informationUsing An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically
ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationNumerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems
More informationf t dt. Write the third-degree Taylor polynomial for G
AP Calculus BC Homework - Chapter 8B Taylor, Maclauri, ad Power Series # Taylor & Maclauri Polyomials Critical Thikig Joural: (CTJ: 5 pts.) Discuss the followig questios i a paragraph: What does it mea
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationSYDE 112, LECTURE 2: Riemann Sums
SYDE, LECTURE : Riema Sums Riema Sums Cosider the problem of determiig the area below the curve f(x) boud betwee two poits a ad b. For simple geometrical fuctios, we ca easily determie this based o ituitio.
More information9.3 Power Series: Taylor & Maclaurin Series
9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0
More informationDECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan
Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed
More informationMTH Assignment 1 : Real Numbers, Sequences
MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationCalculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)
Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics
More informationFIR Filter Design: Part I
EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some
More informationINTEGRATION BY PARTS (TABLE METHOD)
INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary
More informationRoot Finding COS 323
Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy
More informationlecture 3: Interpolation Error Bounds
6 lecture 3: Iterpolatio Error Bouds.6 Covergece Theory for Polyomial Iterpolatio Iterpolatio ca be used to geerate low-degree polyomials that approimate a complicated fuctio over the iterval [a, b]. Oe
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationA widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α
Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows
More information2 ) 5. (a) (1)(3) + (1)(2) = 5 (b) {area of shaded region in Fig. 24b} < 5
Odd Aswers: Chapter Four Cotemporary Calculus PROBLEM ANSWERS Chapter Four Sectio 4.. (a) ()() + (8)(4) = 5 (b) ()() ()(8) = 76. bh + b(h h) = bh + bh bh = b ( h + H ) 5. (a) ()() + ()() = 5 (b) {area
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationPhys. 201 Mathematical Physics 1 Dr. Nidal M. Ershaidat Doc. 12
Physics Departmet, Yarmouk Uiversity, Irbid Jorda Phys. Mathematical Physics Dr. Nidal M. Ershaidat Doc. Fourier Series Deiitio A Fourier series is a expasio o a periodic uctio (x) i terms o a iiite sum
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationNewton s Method. Video
SECTION 8 Newto s Method 9 (a) a a Sectio 8 (, ( )) (, ( )) Taget lie c Taget lie c b (b) The -itercept o the taget lie approimates the zero o Figure 60 b Newto s Method Approimate a zero o a uctio usig
More informationLyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name:
Lyma Memorial High School Hoors Pre-Calculus Prerequisite Packet 2018 Name: Dear Hoors Pre-Calculus Studet, Withi this packet you will fid mathematical cocepts ad skills covered i Algebra I, II ad Geometry.
More informationMTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1
MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces
More informationFor use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel)
For use oly i Badmito School November 0 C Note C Notes (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i Badmito School November 0 C Note Copyright www.pgmaths.co.uk
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationSolving Third Order Boundary Value Problem Using. Fourth Order Block Method
Applied Matematical Scieces, Vol. 7,, o. 5, 69-65 HIKARI Ltd, www.m-ikari.com Solvig Tird Order Boudar Value Problem Usig Fourt Order Block Metod Amad Sa Abdulla, *Zaaria Abdul Maid, ad Norazak Seu, Istitute
More informationNumerical Methods for Ordinary Differential Equations
Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute
Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,
More informationSubstitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationMaximum and Minimum Values
Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationTEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by
Advaced Placemet Specialty Coferece TEACHING THE IDEAS BEHIND POWER SERIES Preseted by LIN McMULLIN Sequeces ad Series i Precalculus Power Series Itervals of Covergece & Covergece Tests Error Bouds Geometric
More informationCHAPTER 5: FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION PLOT PERIODIC GRAPH
CHAPTER : FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION POT PERIODIC GRAPH PROPERTIES OF EVEN AND ODD FUNCTION Fuctio is said to be a eve uctio i: Fuctio is said to be a odd uctio i: Fuctio is said
More informationNewton s Method. y f x 1 x x 1 f x 1. Letting y 0 and solving for x produces. x x 1 f x 1. x 1. x 2 x 1 f x 1. f x 1. x 3 x 2 f x 2 f x 2.
460_008.qd //04 :7 PM Page 9 SECTION.8 Newto s Method 9 (a) a a Sectio.8 (, ( )) (, ( )) Taget lie c Taget lie c b (b) The -itercept o the taget lie approimates the zero o. Figure.60 b Newto s Method Approimate
More informationCurve Sketching Handout #5 Topic Interpretation Rational Functions
Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials
More informationSTAT 516 Answers Homework 6 April 2, 2008 Solutions by Mark Daniel Ward PROBLEMS
STAT 56 Aswers Homework 6 April 2, 28 Solutios by Mark Daiel Ward PROBLEMS Chapter 6 Problems 2a. The mass p(, correspods to either o the irst two balls beig white, so p(, 8 7 4/39. The mass p(, correspods
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationLESSON 2: SIMPLIFYING RADICALS
High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More information9.3 Taylor s Theorem: Error Analysis for Series. Tacoma Narrows Bridge: November 7, 1940
9. Taylor s Theorem: Error Aalysis or Series Tacoma Narrows Bridge: November 7, 940 Last time i BC So the Taylor Series or l x cetered at x is give by ) l x ( ) ) + ) ) + ) ) 4 Use the irst two terms o
More information(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.
The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More information*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully
X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationMA Lesson 26 Notes Graphs of Rational Functions (Asymptotes) Limits at infinity
MA 1910 Lesso 6 Notes Graphs of Ratioal Fuctios (Asymptotes) Limits at ifiity Defiitio of a Ratioal Fuctio: If P() ad Q() are both polyomial fuctios, Q() 0, the the fuctio f below is called a Ratioal Fuctio.
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationCalculus II exam 1 6/18/07 All problems are worth 10 points unless otherwise noted. Show all analytic work.
9.-0 Calculus II exam 6/8/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. (5 poits) Prove that the area eclosed i the circle. f( x) = x +, 0 x. Use the approximate the area
More information