Math 1B, lecture 4: Error bounds for numerical methods


 Wilfrid Rodgers
 5 years ago
 Views:
Transcription
1 Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the integrl f(x)dx by dividing the intervl [, b] into n slices of equl width, nd pproximte the re under the curve in ech slice by ssuming tht the curve hs certin shpe (flt, liner, or qudrtic), nd pproximting the re by wht the re would be in this cse. Using ny of these numericl methods, it is necessry to understnd the error tht my result from these ssumptions. This lecture presents method for computing bounds on the error of ech numericl method. The objective of this topic is not tht you should remember these error bounds fter the course; in prctice, ny time you need to the vlue of n integrl, you will use mchine tht implements these methods (or even more sophisticted methods which we do not discuss). Rther, the objective is to fmilirize you with how these methods work, so tht you will be wre of the bsic qulittive spects of the methods tht mchines use. Furthermore, we wish to emphsize tht n pproximtion is useless without knowledge of its possible error, so tht you will lwys remember to demnd error bounds whenever you use pproximte methods. As secondry objective, we wish to use these methods nd their error bounds to illustrte how polynomils re, in some sense, the fundmentl functions in clculus; ech of these numericl methods hs the feture tht it is exct for polynomils of certin degree, nd its error cn be found by considering polynomils of higher degree. The bsic philosophy tht function cn be understood by mens of the polynomils it resembles is centrl feture of clculus, which will be mde especilly vivid by the development of Tylor series in couple weeks. The reding for tody is Gottlieb 6.. The homework is to do problems, 3, 4, 5 from 6., s well s Topic Outline. Summry of results We first recll the definitions of the five numericl methods discussed lst time. integrl: We re evluting the f(x)dx. In order to do this, divide the intervl [, b] on which the integrl is being computed into n pieces (where n cn be chosen to be nything), thus dividing the vlue of the integrl into n slices, with endpoints x 0, x,..., x n, s shown.
2 f(x)dx = = ( re under the curve in the k th slice ) k= xk k= x k f(x)dx Here we mke the following definitions. x 0 = x n = b x = (b )/n x k = + k x Thus [x 0, x ] is the intervl corresponding to the first slice, [x, x ] corresponds to the seconds slice, nd in generl [x k, x k ] corresponds to the k th slice. Using this nottion, the five numericl methods tht we consider cn be summrized s follows. Left pproximtion L n = Right pproximtion R n = Midpoint pproximtion M n = f(x k ) x k= f(x k ) x k= ( ) xk + x k f x k= Trpezoid pproximtion T n = (L n + R n ) = k= k= [f(x k ) + f(x k )] x Simpson s rule S n = 3 M n + 3 T n [ = f(x k + 4f 6 ( xk + x k ) ] + f(x k ) x Note tht both the trpezoid pproximtion nd Simpson s rule cn be defined either in terms of erlier pproximtions or directly in terms of vlues of the function f. Note lso tht ech pproximtion is sum of estimtes for individul slices, ech of which consists of the width of the slice times n estimte for the verge vlue of f on tht slice. The estimte for this verge vlue depends on the ssumed shpe of the function on the slice. The error bounds which we present will include constnt fctor determined by the size of some derivtive of f(x). We shll use the following nottion. The symbol M is clled Gothic M, or Germn M. It is frequently used by mthemticins in situtions where the ordinry letter M would be mbiguous, s it certinly would be here.
3 M = mx { f (x) : x in [, b]} M = mx { f (x) : x in [, b]} { } M 4 = mx f (4) (x) : x in [, b] Using this nottion, the error bounds tht we shll use re expressed by the following theorem (which will not be proved in clss). Theorem.. If the integrl f(x)dx is pproximted using the methods bove, then the following bounds hold. L n f(x)dx ( M (b ) /n = ) M ( x) n R n f(x)dx ( M (b ) /n = ) M ( x) n M n f(x)dx 4 M (b ) 3 /n (= ) 4 M ( x) 3 n T n f(x)dx M (b ) 3 /n (= ) M ( x) 3 n ( S n f(x)dx 80 M 4(b ) 5 /(n) 4 = ) 80 M 4( x/) 5 n The expressions in prentheses re included simply to show how much ech of the n slices contributes to the totl error of the pproximtion. The proof of the first two bounds is not difficult; we will sketch the rgument in section 4. The ltter three bounds re somewht more technicl, nd require the use of integrtion by prts. The prticulr forms of these bounds re not importnt (nd you will lmost certinly forget them fter the exm). However, the following fetures re importnt. The error terms hve constnt fctors coming from the mximum vlue of some derivtive of f(x). The better the pproximtion, the higher the derivtive which governs the error. The error bound shrinks s n grows. The better the pproximtion, the fster the error bound will shrink. Theorem., together with the qulittive descriptions discusses in the lst lecture, re summrized in the following tble. Remember tht the conditions under the columns overestimtes nd underestimtes must hold t ll points in the intervl [, b] in order to conclude whether the method gives n overestimte or underestimte. Method Overestimtes Underestimtes Error bound Left L n f (x) < 0 f (x) > 0 M (b ) /n Right R n f (x) > 0 f (x) < 0 M (b ) /n Midpoint M n f (x) < 0 f (x) > 0 4 M (b ) 3 /n Trpezoid T n f (x) > 0 f (x) < 0 M (b ) 3 /n Simpson S n f (4) (x) > 0 f (4) (x) < 0 80 M 4(b ) 5 /(n) 4 3
4 3 Exmples Exmple 3.. How ccurte will Simpson s rule be for n slices be in computing 0 x3 dx? The error bound is 80 M 4(b ) 5 /(n) 4. For this prticulr function, the fourth derivtive is 0, nd therefore M 4 = 0. Therefore the error bound is 0, no mtter which n is used. So Simpson s rule should be exctly correct for this integrl. This my seem too good to be true, so here is computtion, for n = (i.e. computtion of S ). By definition of Simpson s rule, S 4 = 6 ( ) = ( ) 6 = 4. Indeed, the stndrd clcultion shows tht 0 x3 dx = 4. Exmple 3.. Consider the integrl 0 e x dx. Determine sufficient number of slices for the midpoint pproximtion to be ccurte within The error bound for M n is 4 M (b ) 3 /n. Since the length of the intervl is (b ) =, this is equl to 3 M /n in this cse. We need to select n n lrge enough tht this bound is no more tht 0.00, i.e M. So we simply need to find M. such tht 3 M /n Equivlently, n M, or n In fct, it is not necessry to find M exctly; n upper bound on M will do. To find such n upper bound, begin by computing derivtives of the function (detils of the computtion re omitted). f(x) = e x f (x) = ( x)e x f (x) = (4x )e x It would be hssle to ctully explicitly find the mximum of this second derivtive, so simply get n upper bound. For this purpose, observe tht for ll x, x 0, hence e x e 0 =. Therefore f (x) = (4x )e x 4x. Now, the function 4x is n incresing function on [0, ], with vlues rnging from to 4. Thus n upper bound on its size is 4. Tken together, this implies tht f (x) 4 on [0, ], nd thus M 4. Now, since this is n upper bound on M, if we cn ensure tht n hve tht n 3 M, nd in turn tht the error of M n is less thn Now 69 slices will be sufficient to ensure error of less thn 0.00 for the midpoint pproximtion. 3 4, then we will certinly Thus Exmple 3.3. Consider the integrl dx x, whose exct vlue is equl to ln. Numericl methods give one wy to clculte this vlue to rbitrry ccurcy (better methods will come from the notion of Tylor series). How mny slices re needed in order to compute this integrl with error t most one millionth (tht is, 0 6 ), using the five numericl methods discussed? To begin, we need the mximum vlues M, M nd M 4, for which we need the derivtives of the function. 4
5 f(x) = /x f (x) = /x f (x) = /x 3 f (x) = 6/x 4 f (4) (x) = 4/x 5 Ech of these functions decrese in mgnitude s x increses, so ll of them hve mximum bsolute vlue t x =. From this we obtin the following vlues. M = M = M 4 = 4 From this, s well s the fct tht the length of the intervl is (b ) =, the following bounds for the error of ech pproximtion cn be found. L n ln M (b ) /n = n R n ln M (b ) /n = n M n ln 4 M (b ) 3 /n = n T n ln M (b ) 3 /n = 6n S n ln 80 M 4(b ) 5 /(n) 4 = 0n 4 Now, observe tht n 0 6 n 500, 000, thus 500, 000 slices will ensure tht L n nd R n re both within 0 6 of the true vlue. For the midpoint pproximtion, n 0 6 n 0 6 / n 0 6 / So 89 slices will ensure tht M n is within 0 6 of the true vlue. For the trpezoid pproximtion, 6n 0 6 n 0 6 /6 n 0 6 / So 409 slices will ensure tht T n is within 0 6 of the true vlue. For Simpson s rule, 0n 0 6 n /0 n / So 0 slices will ensure tht 4 S n is within 0 6 of the true vlue. As this result indictes, the error bounds gurntee much fster convergence for Simpson s rule thn the less sophisticted pproximtions. In fct, explicit computtion revels tht, in order to chieve error of less thn one millionth, only 7 slices re needed for Simpson (the bounds ensure tht 0 will work), 77 slices re needed for midpoint pproximtion (the bounds ensure tht 89 will work), nd 50 slices re needed for trpezoid pproximtion (the bounds ensure tht 409 will work). This illustrtes the fct tht the error bounds, while obviously not exct, give good rough ide for how mny slices re needed to chieve desired ccurcy of computtion. 5
6 4 Error of L n nd R n In order to understnd the error bound for left nd right pproximtion, we begin by considering the error of these pproximtions in the specil cse where f(x) is liner function, with slope m. Tht is, f (x) = m for ll x, where m is constnt. Then we cn see the error of the left pproximtion L n in the following picture. The left pproximtions L n undershoots the re of ech slice by precisely the re of one of the shdes tringles. Observe tht the bse of ech tringle is equl to the width of the slice, nmely x, nd the height must be equl to m x (where m is the slope of the function, i.e. the constnt vlue of its first derivtive). Therefore the re of ech shded tringle is ( x)(m x) = m( x). Since there re n such tringles, the totl error is precisely n m( x) = mn ( b n ) = m (b ) n. Becuse m is the vlue of f (x) t every point, m is equl to the vlue M, nd so in fct the error is precisely equl to the bound M (b ) n. Now suppose tht f(x) is ny function, not necessrily liner. Then lthough we cn no longer view the error s the res of right tringles, essentilly the sme technique will pply: since the function never grows t rte fster thn M, the error of pproximtion in ech slice is still bounded by the re of tringle with bse x nd height M x. A complete proof of the error bound cn be given by using the men vlue theorem; we omit the detils. The result is, s stted in section, b f(x)dx L n M (b ) /n. 5 Error of M n nd T n In this section, we give only rough sketch of the principles behind the error bounds for M n nd T n. To give complete detils would require somewht more technicl nottion thn seems pproprite t the moment. Consider the following imge, depicting the error of both the midpoint pproximtion nd trpezoid pproximtion on single slice. 6
7 Wht cn mke these errors s lrge s possible? First of ll, if f is liner in this slice, then both errors will be 0. So the error is governed by the second derivtive. The lrgest possible error will result when the function begins, t the lest end of the slice, by incresing (respectively, decresing) very quickly, turns round in the middle, nd then decreses (respectively, increses) rpidly to the other endpoint. Such behvior involves drmtic chnge in the first derivtive, i.e. lrge second derivtive. Upon formlizing this rough ide nd doing some (not entirely trivil) computtion, the following result cn be obtined. error in single slice for M n 4 M ( x) 3 error in single slice for T n M ( x) 3 Replcing x by (b )/n s usul, nd multiplying by n (since there re n slices, ech contributing some error) gives the error bounds from section. The get n ide for where the constnts 4 nd come from, you my evlute the error in the cse of qudrtic function, where these error bounds re exct. From this it follows (mong other things), the clssicl fct (known to the Greeks by much more elementry methods) tht the re of prbolic segment is equl to 3 of the bse times the height. 7
Review of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 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 informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
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 informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationUnit #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 informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationLecture 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 relvlues nd smooth The pproximtion of n integrl by numericl
More informationNUMERICAL 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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMORE 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 informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationThe 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 informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More information63. 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 information7.2 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 informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationLecture 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 informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
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 informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationAbstract 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 informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationNumerical 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 informationx = b a N. (131) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is
Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More information6.5 Numerical Approximations of Definite Integrals
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationAdvanced 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 informationNumerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1
AMSC/CMSC 46/466 T. von Petersdorff 1 umericl Integrtion 1 Introduction We wnt to pproximte the integrl I := f xdx where we re given, b nd the function f s subroutine. We evlute f t points x 1,...,x n
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
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 information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationMath& 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 informationReview 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 informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
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 informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 ThreePoint
More informationSuppose 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 informationSection 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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
More informationSUMMER 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 informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More information38 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 xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationAn Overview of Integration
An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is
More informationZ 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 informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More information4.4 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 informationTheoretical 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 informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationNumerical 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 information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
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 informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationLecture 23: Interpolatory Quadrature
Lecture 3: Interpoltory Qudrture. Qudrture. The computtion of continuous lest squres pproximtions to f C[, b] required evlutions of the inner product f, φ j = fxφ jx dx, where φ j is polynomil bsis function
More informationMidpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
More informationImproper 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 informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationLECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and
LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
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 informationRiemann 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 informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
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 informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More information!0 f(x)dx + lim!0 f(x)dx. The latter is sometimes also referred to as improper integrals of the. k=1 k p converges for p>1 and diverges otherwise.
Chpter 7 Improper integrls 7. Introduction The gol of this chpter is to meningfully extend our theory of integrls to improper integrls. There re two types of soclled improper integrls: the first involves
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More informationOrthogonal 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 informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More information