Review of Gaussian Quadrature method

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Review of Gaussian Quadrature method"

Transcription

1 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 over the rel line. This mens to evlute I = f x) dx Geometriclly, this integrl represents the re under fx) from to Solution We cn lwys pproximte the re y dividing it in equl width strips nd then sum the res of ll the strips. In generl, there will lwys e n error in the estimte of the re using this method. The error will ecome smller the more strips we use which implies smller strip width). Hence we cn write N ) f x) dx = x f x i ) + E i=1 Where E is the error etween the ctul re nd the pproximted re using the ove method of numericl integrtion. N ove is the numer of strips or cn lso e refereed to s the numer of integrtion points. Insted of keep referring to the width of the strip ll the time, we will cll this quntity the weight w i tht we will multiply the vlue of the function with to otin the re. Hence the ove ecomes N ) f x) dx = w i f x i ) + E i=1 Using implied summtion on indices the ove ecomes f x) dx = w i f x i ) + E 1

2 y=fx) f x x Are under curve f x dx Figure 1: Integrting function f x f x 1 f x f x 4 y=fx) f x 1 4 x x 4 f x dx i 1 x f x i Figure : Numericl integrtion

3 In the ove we divided the rnge of the integrtion the distnce etween the upper nd lower limits of integrtion) into equl intervls. We cn decide to evlute fx i ) t the middle of the strip or t the strt of the strip or t the end of the strip. In the digrm ove we evluted the fx) t the left end of the strip. Our gol is to evlute this integrl such s the error E is minimum nd using the smllest numer of integrtion points. In sense this cn e considered n optimiztion prolem with constrints: minimize the error of integrtion using the smllest possile numer of points. To e le to do this minimiztion, we need to consider wht re the vriles involved. We see tht there re two degrees of freedom to this prolem. One is the width of the strip or w i. We do not hve to use fixed vlue of width, we cn use different width for different strips if the resulting integrl gives etter pproximtion. The second degree of freedom is the point t which to evlute fx i ) ssocited with ech strip. In the exmple ove we choose to evlute fx i ) t left end point of the strip. We cn choose to select different x i point if this will result in etter pproximtion. This is the min ide of Guss Qudrture numericl integrtion. It is to e le to choose specific vlues for these two degrees of freedom, the w i nd the x i. It turns out tht if the function fx) is polynomil, then there is n optiml solution. There is n optiml {w i, x i } for ech polynomil of order n. We cn determine these degrees of freedom such tht the error E is zero, nd with the lest possile numer of integrtion points. We re le to tulte these two degrees of freedom for ech polynomil of specific order. In other words, if the function fx) is polynomil of order n then we know efore the computtion strts wht these degrees of freedom should e. We know the loctions of x i nd we know weight w i tht we need to multiply fx i ) with to otin the re with minumum error. You might sk how cn this method of integrtion know the loctions of the integrtion points x i eforehnd without eing given the integrtion rnge of the function to integrte? It turns out tht we will mp f x) into new known nd specific rnge of integrtion from to +1) for the method tht we will now discuss..1 Guss Qudrture From now on we will ssume the function fx) to e integrted is polynomil in x of some order n. Guss qudrture is optiml when the function is polynomil The min strting point is to represent the function f x) s comintion of linerly independent sis. Insted of using strips of equl width, we ssume the width cn vry from one strip to the next. Let us cll the width of the strip w i. Insted of tking the height of the i th strip to e the vlue of the function t the left edge of the strip, let us lso e flexile on the loction of the x ssocited with strip w i nd cll the height of the strip w i s fx i ) where x i is to e determined. Hence the ove integrtion ecomes

4 N ) f x) dx = w i f x i ) + E i=1 N w i f x i ) i=1 So our gol is to determine w i nd x i such s the error E is minimized in the ove eqution. We would relly like to find w i nd x i such tht the error is zero with the smllest vlue for N. It seems s if this is very hrd prolem. We hve N unknown quntities to determine. N different widths, nd N ssocited x points to evlute the height of ech specific strip t. And we only hve s n input fx) nd the limits of integrtion, nd we need to determine these N quntities such tht the error in integrtion is zero. In other words, the prolem is to find w i, x i such tht I = f x) dx = w 1 f x 1 ) + w f x ) + w N f x N ) 1) One wy to mke some progress is to expnd fx) s series. We cn pproximte fx) s convergent power series for exmple. If fx) hppens to e polynomil insted, we cn represent it exctly using finite sequence of Legendre polynomils. It is in this second cse where this method mkes the most sense to use due to the dvntges we mke from the second representtion. We show oth methods elow. Expnding f x) s convergent power series over the rnge, gives Sustituting ) into 1) gives f x) = x + x + m x m + ) I = But x + x + m x m + ) dx = w 1 f x 1 ) + w f x ) + w N f x N ) ) x + x + m x m + ) dx = = 0 ) + 1 ) Sustituting the ove into ) results in 0 dx + + ) 1 x dx + x dx m m+1 m+1) m m x m dx + 4

5 ) ) m+1 m+1) 0 ) m + m + 1 = w 1 f x 1 ) + w f x ) + + w N f x N ) 4) But from ) we see tht f x 1 ) = x 1 + x 1 + m x m 1 + f x ) = x + x + m x m + Sustituting the ove into 4) gives f x N ) = x N + x N + m x m N + ) ) m+1 m+1) 0 ) m + = m + 1 w x 1 + x 1 + m x m 1 + ) +w x + x + m x m + ) Rerrnging gives +w N x N + x N + m x m N + ) ) ) m m ) 0 ) m + = m 0 w 1 + w + + w N ) + 1 w 1 x 1 + w x + + w N x N ) + w1 x 1 + w x + + w N x ) N + m w 1 x m 1 + w x m + + w N x m N) Equting coefficients on oth sides results in w 1 + w + + w N = 5) ) w 1 x 1 + w x + + w N x N = w 1 x 1 + w x + + w N x ) N = w 1 x m 1 + w x m + + w N x m N = m m ) m 5

6 Since we hve N unknown quntities to solve for N weights w i nd N points x i ) we need N equtions. In other words, we need to hve m = N. The ove set of simultneous N equtions cn now e solved for the unknown w i, x i nd this will give us the numericl integrtion we wnted. The ove ssumed tht the function fx) cn e represented exctly y the power series expnsion with m terms. We now consider the representtion of fx) s series of Legendre polynomils. We do this since when fx) itself is polynomil. We cn represent f x) exctly y finite numer of Legendre polynomils. But since Legendre polynomils P n x) re defined over [, 1] we strt y trnsforming f x) to this new rnge nd then we cn expnd the mpped f x) which we will cll g ζ)) in terms of the Legendre polynomils. y=fx) f x Liner trnsformtion g x An esy wy to find this mpping is to lign the rnges over ech others nd tke the rtio etween s the scle fctor. This digrm shows this for generl cse where we mp f x) defined over [, ] to new rnge defined over [c 1, c ] f x dx x g dx d c c 1 c 1 c d We see from the digrm tht ζ = c 1 + dζ But dζ dx is the sme rtio s Hence c c 1 dx dζ = c c 1 6) 6

7 The ove is clled the Jcoin of the trnsformtion. Now, From the digrm we see tht And dx = x Hence 6) ecomes dζ = ζ c 1 Hence we otin tht And x ζ c 1 = c c 1 ζ = x c c 1 ) + c 1 x = c c 1 ζ c 1 ) + For the specific cse when c 1 = nd c = +1 the ove expressions ecome Hence ζ = x ) 1 x ) = = x x = ζ + 1) + ) ζ + + ) = Before we proceed further, It will e interesting to see the effect of this trnsformtion on the shpe of some functions. Below I plotted some functions under this trnsformtion. The left plots re the originl functions plotted over some rnge, in this cse [4, 10] nd the right side plots show the new shpe the function g ζ)) over the new rnge [, 1] 7

8 We must rememer tht in the following nlysis, we re integrting now the function g ζ) over [, 1] nd not f x) over [, ]. Hence to otin the required integrl we need to trnsform ck the finl result. We will show how to do this elow. We cn pproximte ny function f x) s series expnsion in terms of weighted sums of set of sis functions. We do this for exmple when we use Fourier series expnsion. Hence we write f x) = i Φ i x) 7) i 8

9 We cn give n intuitive justifiction to the ove formultion s follows. If we think in terms of vectors. A vector V in n n-dimensionl spce is written in terms of its components s follows V = 1 e 1 + e + N e N N = i e i i Where e i is the sis vectors in tht spce. If we now consider generl infinite dimensionl vector spce where ech point in tht spce is function, then we see tht we cn lso represent tht function s weighted series of sis functions just s we did for norml vector. There re mny sets of sis functions we cn choose to represent the function f x) with. The min requirements for the sis functions is tht they re complete This mens they spn the whole spce) nd there is defined n inner product on them. For our purposes here, we re interested in the clss of function f x) tht re polynomils in x. The sis we will select to use re the Legendre sis s explined ove. To do this, we trnsform f x) to g ζ) s shown ove nd then now our integrl ecomes f x) dx = ) ) f x ζ)) dζ This is ecuse we found tht dx = ) dζ from ove. If we cll f x ζ)) s g ζ) the integrl ecomes f x) dx = ) g ζ) dζ Since ) is the Jcoin of the trnsformtion, we write the integrl s f x) dx J g ζ) dζ Since the Jcoin is constnt in this cse, we will only worry out g ζ) dζ nd we just need to rememer to scle the result t the end y J. This is the integrl we will now numericlly integrte. Eqution 7) is now written s g ζ) = i P i ζ) i Where P i ζ) is the Legendre polynomil of order i nd g ζ) is polynomil in ζ or some order m. 9

10 Now we crry the sme nlysis we did when we expressed f x) s power series. The difference now is tht the limit of integrtion is symmetricl nd the sis re now the Legendre polynomils insted of the polynomils x n s the cse ws in the power series expnsion. So now eqution 1) ecomes I = g ζ) dζ = w 1 g ζ 1 ) + w g ζ ) + + w N g ζ N ) 8) And eqution ) ecomes g ζ) = 0 P 0 ζ) + 1 P 1 ζ) + P ζ) + m P m ζ) + 9) Sustituting 9) into 8) we get the equivlent of eqution ) I = 0 P 0 ζ) + 1 P 1 ζ) + P ζ) + m P m ζ) + ) dζ 10) = w 1 g ζ 1 ) + w g ζ ) + + w N g ζ N ) 1) 0 P P 1 + P + m P m + ) dζ = 0 P 0 dζ + 1 P 1 dζ + P dζ + + m P m dζ + = 0 ) = 0 The ove occurs since the integrl of ny Legendre polynomil of order greter thn zero will e zero over [, 1] Sustituting the ove into 10) we otin But from 9) we see tht I = 0 = w 1 g ζ 1 ) + w g ζ ) + + w N g ζ N ) 11) g ζ 1 ) = 0 P 0 ζ 1 ) + 1 P 1 ζ 1 ) + P ζ 1 ) + m P m ζ 1 ) + g ζ ) = 0 P 0 ζ ) + 1 P 1 ζ ) + P ζ ) + m P m ζ ) + g ζ N ) = 0 P 0 ζ N ) + 1 P 1 ζ N ) + P ζ N ) + m P m ζ N ) + Sustituting the ove in 11) gives 0 = w 1 0 P 0 ζ 1 ) + 1 P 1 ζ 1 ) + P ζ 1 ) + m P m ζ 1 ) + ) + + w 0 P 0 ζ ) + 1 P 1 ζ ) + P ζ ) + m P m ζ ) + ) + + w N 0 P 0 ζ N ) + 1 P 1 ζ N ) + P ζ N ) + m P m ζ N ) + ) 10

11 Rerrnging results in 0 = 0 w 1 P 0 ζ 1 ) + w P 0 ζ ) + + w N P 0 ζ N )) + 1 w 1 P 1 ζ 1 ) + w P 1 ζ ) + + w N P 1 ζ N )) + + m w 1 P m ζ 1 ) + w P m ζ ) + + w N P m ζ N )) Equting coefficients gives = w 1 P 0 ζ 1 ) + w P 0 ζ ) + + w N P 0 ζ N ) 0 = w 1 P 1 ζ 1 ) + w P 1 ζ ) + + w N P 1 ζ N ) 0 = w 1 P ζ 1 ) + w P ζ ) + + w N P ζ N ) 0 = w 1 P m ζ 1 ) + w P m ζ ) + + w N P m ζ N ) If we select the points ζ i to e the roots of P i we cn write the ove s = w 1 P 0 ζ 1 ) + w P 0 ζ ) + + w N P 0 ζ N ) 0 = w 1 P 1 ζ 1 ) + w P 1 ζ ) + + w N P 1 ζ N ) 0 = w 1 P ζ 1 ) + w P ζ ) + + w N P ζ N ) 0 = w 1 P m ζ 1 ) + w P m ζ ) + + w N P m ζ N ) References 1. Mthemtic Structurl Mechnics help pge. MIT course 16.0 lecture notes. MIT open course wesite.. Theory of elsticity y S. P. Timoshenko nd J. N. Goodier. chpter 10 11

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Polynomials and Division Theory

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

More information

DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.

DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1. 398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

September 13 Homework Solutions

September 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 information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

Orthogonal Polynomials

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

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous 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 information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Linear Inequalities. Work Sheet 1

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

More information

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17 EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

More information

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature

Lecture 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 information

Bases for Vector Spaces

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

More information

Numerical Analysis: Trapezoidal and Simpson s Rule

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

More information

Section 4: Integration ECO4112F 2011

Section 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 information

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

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

More information

1. Gauss-Jacobi 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. Gauss-Jacobi 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. Guss-Jcobi 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 information

Best Approximation. Chapter The General Case

Best 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 information

Numerical integration

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

More information

Chapter 6 Techniques of Integration

Chapter 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 information

p-adic Egyptian Fractions

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

More information

Lecture 2e Orthogonal Complement (pages )

Lecture 2e Orthogonal Complement (pages ) Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process

More information

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 17

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 17 CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking

More information

The Evaluation Theorem

The Evaluation Theorem These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not

More information

10 Vector Integral Calculus

10 Vector Integral Calculus Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Tony Mrtino My 7, 20 Theorem 9 negtiveorder Theorem 11 clss2 Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin

More information

Section 6.1 Definite Integral

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

More information

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

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

More information

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

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

More information

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 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 information

Chapter 9 Definite Integrals

Chapter 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 information

Interpreting Integrals and the Fundamental Theorem

Interpreting 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 information

Theoretical foundations of Gaussian quadrature

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

More information

1B40 Practical Skills

1B40 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 information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

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

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

More information

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

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

More information

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

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

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, 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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014 Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

More information

Lecture 2: January 27

Lecture 2: January 27 CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full

More information

Review of Calculus, cont d

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

More information

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

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

More information

The Regulated and Riemann Integrals

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

More information

Lecture 19: Continuous Least Squares Approximation

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

More information

Lecture 19: Continuous Least Squares Approximation

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

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

Bridging the gap: GCSE AS Level

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

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties 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 information

5.1 How do we Measure Distance Traveled given Velocity? Student Notes

5.1 How do we Measure Distance Traveled given Velocity? Student Notes . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

More information

1 The Lagrange interpolation formula

1 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 information

Summary: Method of Separation of Variables

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

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

Numerical Methods I Orthogonal Polynomials

Numerical Methods I Orthogonal Polynomials Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

1 Probability Density Functions

1 Probability Density Functions Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

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

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

More information

1 Linear Least Squares

1 Linear Least Squares Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

More information

Surface maps into free groups

Surface maps into free groups Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The

More information

NUMERICAL INTEGRATION

NUMERICAL 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 information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

Abstract inner product spaces

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

More information

Quadratic Forms. Quadratic Forms

Quadratic 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 information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

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

More information

Mathematics Number: Logarithms

Mathematics 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 information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f 1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

More information

308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:

308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices: 8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion

More information

Minimal DFA. minimal DFA for L starting from any other

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

More information

Lecture 4 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell

Lecture 4 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell Lecture 4 Notes, Electromgnetic Theory I Dr. Christopher S. Bird University of Msschusetts Lowell 1. Orthogonl Functions nd Expnsions - In the intervl (, ) of the vrile x, set of rel or complex functions

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

Interpolation. Gaussian Quadrature. September 25, 2011

Interpolation. Gaussian Quadrature. September 25, 2011 Gussin Qudrture September 25, 2011 Approximtion of integrls Approximtion of integrls by qudrture Mny definite integrls cnnot be computed in closed form, nd must be pproximted numericlly. Bsic building

More information

Construction of Gauss Quadrature Rules

Construction of Gauss Quadrature Rules Jim Lmbers MAT 772 Fll Semester 2010-11 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht Newton-Cotes qudrture

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

Chapter 14. Matrix Representations of Linear Transformations

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

More information

Designing Information Devices and Systems I Discussion 8B

Designing Information Devices and Systems I Discussion 8B Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 Completed 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 Completed 1 CCBC Dundalk Bo Brown Mth Clculus Chpter, Section Completed CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse

More information

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions

More information

Torsion in Groups of Integral Triangles

Torsion in Groups of Integral Triangles Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

More information

Improper Integrals, and Differential Equations

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

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Lecture 14: Quadrature

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

More information

Math 8 Winter 2015 Applications of Integration

Math 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 information

CMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature

CMDA 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 information

Evaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals.

Evaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals. Evluting Definite Integrls There re few properties tht you should rememer in order to ssist you in evluting definite integrls. f x dx= ; where k is ny rel constnt k f x dx= k f x dx ± = ± f x g x dx f

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Chapter 6 Continuous Random Variables and Distributions

Chapter 6 Continuous Random Variables and Distributions Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The 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 information

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors: Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

More information

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

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

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information