3.4 Numerical integration

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "3.4 Numerical integration"

Transcription

1 3.4. Numericl integrtion Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [, b], i.e. I( f )= w(x) f (x) dx. The weight function my be the identity function w(x) =1, so tht the integrl represents the re under the function f in the intervl. In other pplictions the weight function could lso be the probbility density function of continuous rndom vrible x with support [, b], sothti( f ), represents the expected vlue of f ( x). In the following we discuss so-clled numericl qudrture methods. In order to pproximte the bove integrl, qudrture methods choose discrete set of qudrture nodes x i nd pproprite weights w i in wy tht n I( f ) w i f (x i ). The integrl is therefore pproximted by the sum of function vlues t the nodes x 0 < < x n b nd the respective qudrture weights w i. Of course, the pproximtion improves with rising number n. Different qudrture methods only differ with respect to the chosen nodes x i nd weights w i. In the following we will concentrte on Newton-Cotes method nd Gussin qudrture method. The former pproximtes the integrnd f between nodes using low-order polynomils nd sum the integrls of the polynomils to estimte the integrl of f. The ltter methods choose the nodes nd weights in order to mtch specific moments(such s expected vlue, vrince etc.) of the pproximted function Summed Newton-Cotes methods In this section we set w(x) =1. A weighted integrl cn then simply be computed by setting g(x) =w(x) f (x) nd pproximting g(x) dx. Summed Newton-Cotes formuls prtition the intervl [, b] into n subintervls of equl length h by computing the qudrture nodes x i := + ih, i = 0, 1,..., n, mit h := b n. A summed Newton-Cotes formul of degree k then interpoltes the function f on every subintervl [x i, x i+1 ] with polynomil of degree k. Wefinlly integrte the interpolting polynomil on every sub-intervl nd sum up ll the resulting sub-res. Figure 3.10 shows this pproch for k = 0ndk = 1.

2 64 Chpter 3 Numericl Solution Methods f(x) f(x) f(x) f(x) = x 1 x x 3 x 4 x 5 b = x 6 x = x 1 x x 3 x 4 x 5 b = x 6 x Figure 3.10: Summed Newton-Cotes formuls for k = 0 nd k = 1 Summed rectngle rule If k = 0, the interpolting functions re polynomils of degree 0, i.e. constnt functions. We cn write ny re below this constnt functions on the intervl [x i, x i+1 ] s I (0) [x i,x i+1 ] ( f )=(x i+1 x i ) f (x i )=hf(x i ), which is the surfce of rectngle. The degree 0 formul is therefore clled the summed rectngle rule, the explicit form of which is given by I (0) n 1 ( f )= hf(x i ). The weights of the summed rectngle rule consequently re w i = h for i = 0,...,n 1 nd w n = 0. Summed trpezoid rule If k = 1, the function f in the i-th subintervl [x i, x i+1 ] is pproximted by the line segment pssing through the two points (x i, f (x i )) nd (x i+1, f (x i+1 )). The re under this line segment is the surfce of trpezoid, i.e. I (1) [x i,x i+1 ] ( f )=hf(x i)+ 1 h[ f (x i+1) f (x i )] = h [ f (x i)+ f (x i+1 )]. It is immeditely cler tht the summed trpezoid rule improves the pproximtion of I( f ) compred to the summed rectngle rule discussed bove. Summing up the res of the trpezoids cross sub-intervls yields the pproximtion { } I (1) = h n 1 f ()+ f (x i )+ f (b) i=1 of the whole integrl I( f ). The weights of the rule therefore re w 0 = w n = h nd w i = h for i = 1,...,n 1. (3.5)

3 3.4. Numericl integrtion 65 Summed rectngle nd trpezoid rule re simple nd robust. Hence, computtion does not need much effort. In ddition, the ccurcy of the pproximtion of I( f ) increses with rising n. It is lso cler tht the trpezoid rule will exctly compute the integrl of ny first-order polynomil, i.e. line. Progrm 3.11 shows how to pply the summed trpezoid rule to the function cos(x). We Progrm 3.11: Summed trpezoid rule with cos(x) progrm NewtonCotes! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: h, x(0:n), w(0:n), f(0:n) integer :: i! clculte qudrture nodes h = (b-)/dble(n) x = (/( + dble(i)*h,,n)/)! get weights w(0) = h/d0 w(n) = h/d0 w(1:n-1) = h! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)-sin(0d0) end progrm thereby first declre ll the vribles needed. Specil ttention should be devoted to the declrtion of x, w nd f. These rrys will store the qudrture nodes x i, the weights w i nd the function vlues t the nodes f (x i ), respectively. We then first compute h nd the nodes x i = + ih nd the weights w i s in (3.5). With the respective function vlues, the pproximtion to the integrl 0 cos(x) dx = sin() sin(0) is the given by sum(w*f, 1). Note tht the pproximtion of the integrl is quite bd. If we increse the number of qudrture nodes n increses the ccurcy, however, we need bout 600 nodes in order to perfectly mtch numericl nd nlyticl result on 6 digits.

4 66 Chpter 3 Numericl Solution Methods Summed Simpson rule Finlly if k =, the integrnd f in the i-th subintervl [x i, x i+1 ] is pproximted by second-order polynomil function c 0 + c 1 x + c x.nowthreegrph points re required to specify the polynomil prmeters c i in the subintervls. Given these prmeters, one cn compute the integrl of the qudrtic function in the subintervl [x i, x i+1 ] s [ ( ) ] I () [x i,x i+1 ] ( f )=h xi + x f (x i )+4f i+1 + f (x i+1 ). 3 Summing up the different res on the sub-intervls yields { I () ( f )= h n 1 n 1 ( ) } xi + x f ()+ 6 f (x i )+4 f i+1 + f (b). i=1 The summed Simpson rule is lmost s simple to implement s the trpezoid rule. If the integrnd is smooth, Simpson s rule yields pproximtion error tht with rising n flls twice s fst s tht of the trpezoid rule. For this reson Simpson s rule is usully preferred to the trpezoid nd rectngle rule. Note, however, tht the trpezoid rule will often be more ccurte thn Simpson s rule if the integrnd exhibits discontinuities in its first derivtive, which cn occur in economic pplictions exhibiting corner solutions. Of course, summed Newton-Cotes rules lso exist for higher order piecewise polynomil pproximtions, but they re more difficult to work with nd thus rrely used Gussin Qudrture In opposite to the Newton-Cotes methods, Gussin qudrture tkes explicitly into ccount weight functions w(x). Here obviously, qudrture nodes x i nd weights w i re computed differently. Guss-Legendre qudrture Suppose gin, for the moment, tht w(x) =1. The Guss-Legendre qudrture nodes x i [, b] nd weights w i re computed in wy tht they stisfy the n + momentmtching conditions x k w(x) dx = x k n dx = w i xi k for k = 0, 1,..., n 1. (3.6) With the bove conditions holding, we cn write every integrl over polynomil with degree m n 1s p(x) dx = c 0 p(x) =c 0 + c 1 x c m x m 1 dx +c 1 xdx } {{} = n w ix i }{{} = n w i c m x m dx } {{} = n w ixi m

5 3.4. Numericl integrtion 67 n = w i [c 0 + c 1 x i c m x m n i ] = w i p(x i ). Consequently, qudrture nodes nd weights tht stisfy the moment-mtching conditions in (3.6) re ble to integrte ny polynomil p of degree m n 1exctly. Clculting the nodes nd weights of the Guss-Legendre qudrture formul is not so esy. One method is to set up the non-liner eqution system defined in (3.6). For n = 3 e.g. we hve w 0 x 0 + w 1 x 1 + w x = 1 ) (b = 1 dx, w 0 x 0 + w 1 x 1 + w x = 1 ( b ) = xdx,. w 0 x0 5 + w 1x1 5 + w x 5 = 1 ( b 6 6) = x 5 dx. 6 This nonliner eqution system cn be solved by rootfinding lgorithm like fzero. However, there is more efficient, but lso less intuitive wy to clculte qudrture nodes nd weights of the Guss-Legendre qudrture implemented in the subroutine legendre in the module gussin_int. Progrm3.1 demonstrtes how to use it. The Progrm 3.1: Guss-Legendre qudrture with cos(x) progrm GussLegendre! module use use gussin_int! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: x(0:n), w(0:n), f(0:n)! clculte nodes nd weights cll legendre(0d0, d0, x, w)! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)-sin(0d0) end progrm subroutine tkes two rel*8 rguments defining the left nd right intervl borders nd

6 68 Chpter 3 Numericl Solution Methods b. In ddition, we hve to pss two rrys of equl length to the routine. The first of these will be filled with the nodes x i, wheres the second will be given the weights w i. After hving clculted the function vlues f (x i ), we cn gin clculte the numericl pproximtion of the integrl like in Progrm Here, with 10 qudrture nodes, we lredy mtch the nlyticl integrl vlue by 6 digits. Note tht we needed bout 600 nodes with the trpezoid rule. Guss-Hermite qudrture If we set the weight function to w(x) = 1 π exp( x ),weresultintheguss-hermite qudrture method. Note tht the weight function now is equl to the density function of the stndrd norml distribution. Now, with m n + 1 nd normlly distributed rndom vrible x, wehve E( x) = 1 exp( x )x m n dx = π w i xi m = I GH (x m ). Consequently, with Guss-Hermite qudrture, we re ble to perfectly mtch the first n 1 moments of normlly distributed rndom vrible x. An pproximtion procedure for normlly distributed rndom vribles is included in the module normlprob. The procedure norml_discrete(x, w, mu, sig) is exctly bsed on the Guss-Hermite qudrture method. The subroutine receives four input rguments, the lst of which re expected vlue nd vrince of the norml distribution tht should be pproximted. The routine then stores pproprite nodes x i nd weights w i in the rrys x nd w (tht should be of sme length) which cn be used to compute moments of normlly distributed rndom vrible x with expecttion mu nd vrince sig. For pplying this subroutine we consider the following exmple: Exmple Consider n griculturl commodity mrket, where plnting decisions re bsed on the price expected t hrvest A = E(p), (3.7) with A denoting crege supply nd E(p) defining expected price. After the crege is plnted, normlly distributed rndom yield y N(1, 0.1) is relized, giving rise to the quntity q s = Ay which is sold t the mrket clering price p = 3 q s. In order to solve this system we substitute nd therefore q s = [ E(p)] y p = 3 [ E(p)] y.

7 3.4. Numericl integrtion 69 Tking expecttions on both sides leds to E(p) =3 [ E(p)] E(y) nd therefore E(p) = 1. Consequently, equilibrium crege is A = 1. Finlly, the equilibrium price distribution hs vrince of Vr(p) =4 [ E(p)] Vr(y) =4Vr(y) =0.4. Suppose now tht the government introduces price support progrm which gurntees ech producer minimum price of 1. If the mrket price flls below this level, the government pys the producer the difference per unit produced. Consequently, the producer now receives n effective price of mx(p,1) nd the expected price in (3.7) thenis clculted vi E(p) =E [mx (3 Ay,1)]. (3.8) The equilibrium crege supply finlly is the supply A tht fulfills (3.7) with the bove price expecttion. Agin, this problem cnnot be solved nlyticlly. In order to compute the solution of the bove crege problem, one hs to use two modules. normlprob on the one hnd provides the method to discretize the normlly distributed rndom vrible y. The solution to (3.7) cn on the other hnd be clculted by using the method fzero from the rootfinding module. Progrm 3.13 shows how to do this. Due to spce restrictions, we do not show the whole progrm. For running the progrm, we lso need module globls in which we store the expecttion mu nd vrince sig of the normlly distributed rndom vrible y s well s the minimum price minp gurnteed by the government. In ddition, the module stores the qudrture nodes y nd weights w obtined by discretizing the norml distribution for y. In the progrm, we first discretize the distribution for y by mens of the subroutine norml_discrete. This routine receives the expected vlue nd vrince of the norml distribution nd stores the respective nodes y nd weights w in two rrys of sme size. Next we set strting guess for crege supply. We then let fzero find the root of the function mrket tht clcultes the mrket equilibrium condition in (3.7). This function only gets A s n input. From A we cn clculte the expected price E(p) by mens of (3.8) nd finlly the mrket clering condition. Hving found the root of mrket, i.e. the equilibrium crege supply, we cn clculte the expected vlue nd vrince of the price p. Inorder to showthe effects of minimum price, wefirst set the minimum price gurntee minp to lrge negtive vlue, sy 100. This lrge negtive minimum price will never be binding, hence, the outcome is exctly the sme s in the model without price gurntee, see the bove exmple description. If we now set the minimum price t 1, equilibrium crege supply increses by bout 9.7percent. This is becuse the expected effective price

8 70 Chpter 3 Numericl Solution Methods Progrm 3.13: Agriculturl problem progrm griculture...! discretize y cll norml_discrete(y, w, mu, sig)! initilize vribles A = 1d0! get optimum cll fzero(a, mrket, check)! get expecttion nd vrince of price Ep = sum(w*mx(3d0-d0*a*y, minp)) Vrp = sum(w*(mx(3d0-d0*a*y, minp) - Ep)**)... end progrm function mrket(a) use globls rel*8, intent(in) :: A rel*8 :: mrket rel*8 :: Ep! clculte expected price Ep = sum(w*mx(3d0-d0*a*y, minp))! get equilibrium eqution mrket = A - (0.5d0+0.5d0*Ep) end function for the producer rises from the minimum price gurntee. In ddition, the uncertinty for the producers is reduced, since the vrince of the price distribution decreses from 0.4 to bout 0.115, which is gin due to the fct tht the possible price reliztion now hve lower limit of 1.

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

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

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9 III Lecture on Numericl Integrtion File fclib/dttb/lecture-notes/numerical-inter03.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 information

Continuous Random Variables

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

More information

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

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

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

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

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 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 information

Math 360: A primitive integral and elementary functions

Math 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 information

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2-Dimensionl Gussin Qudrture 20 4

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres 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 information

31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes

31.2. 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 information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 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 information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

Numerical Integration

Numerical 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 information

20 MATHEMATICS POLYNOMIALS

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

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

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

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial 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 information

OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS

OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS T. Rmchndrn R.Priml

More information

S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:

S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality: FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy

More information

7. Numerical evaluation of definite integrals

7. Numerical evaluation of definite integrals 7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh - Adptilit Hn Hldíková Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re

More information

Math 113 Exam 2 Practice

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

More information

New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs

New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute

More information

Harmonic Mean Derivative - Based Closed Newton Cotes Quadrature

Harmonic Mean Derivative - Based Closed Newton Cotes Quadrature IOSR Journl of Mthemtics (IOSR-JM) e-issn: - p-issn: 9-X. Volume Issue Ver. IV (My. - Jun. 0) PP - www.iosrjournls.org Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd

More information

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula. Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using

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

CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES

CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED

More information

Introduction to Numerical Analysis

Introduction to Numerical Analysis Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes

38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which

More information

Problem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume

Problem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping

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

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

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

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

Convex Sets and Functions

Convex Sets and Functions B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Math 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx

Math 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx Mth 5 2006-2007) Yum-Tong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

LECTURE NOTE #12 PROF. ALAN YUILLE

LECTURE NOTE #12 PROF. ALAN YUILLE LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, K-mens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of

More information

D01BBF NAG Fortran Library Routine Document

D01BBF NAG Fortran Library Routine Document D01 Qudrture D01BBF NAG Fortrn Librry Routine Document Note. Before using this routine, plese red the Users Note for your implementtion to check the interprettion of bold itlicised terms nd other implementtion-dependent

More information

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

More information

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

Week 10: Riemann integral and its properties

Week 10: Riemann integral and its properties Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

More information

Numerical Methods 2007

Numerical Methods 2007 Physics Mster Course: Numericl Methods 2007 Hns Mssen Rdboud Universiteit Nijmegen Onderwijsinstituut Wiskunde, Ntuur- en Sterrenkunde Toernooiveld 1 6525 ED Nijmegen September 2007 1 Introduction In these

More information

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35 7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

More information

Chapter 4. Additional Variational Concepts

Chapter 4. Additional Variational Concepts Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.

More information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Line and Surface Integrals: An Intuitive Understanding

Line and Surface Integrals: An Intuitive Understanding Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

More information

Week 7 Riemann Stieltjes Integration: Lectures 19-21

Week 7 Riemann Stieltjes Integration: Lectures 19-21 Week 7 Riemnn Stieltjes Integrtion: Lectures 19-21 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0

More information

Efficient Computation of a Class of Singular Oscillatory Integrals by Steepest Descent Method

Efficient Computation of a Class of Singular Oscillatory Integrals by Steepest Descent Method Applied Mthemticl Sciences, Vol. 8, 214, no. 31, 1535-1542 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/1.12988/ms.214.43166 Efficient Computtion of Clss of Singulr Oscilltory Integrls by Steepest Descent

More information

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

More information

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations 18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following

More information

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco Numericl Methods I Olof Widlund Trnscribed by In Tobsco Abstrct. This is prt one of two semester course on numericl methods. The course ws offered in Fll 011 t the Cournt Institute for Mthemticl Sciences,

More information

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q. Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

More information

The Definite Integral

The Definite Integral CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls

More information

Ordinary Differential Equations- Boundary Value Problem

Ordinary Differential Equations- Boundary Value Problem Ordinry Differentil Equtions- Boundry Vlue Problem Shooting method Runge Kutt method Computer-bsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

More information

ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance

ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between

More information

Solutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7

Solutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7 Solutions to Problems in Merzbcher, Quntum Mechnics, Third Edition Homer Reid April 5, 200 Chpter 7 Before strting on these problems I found it useful to review how the WKB pproimtion works in the first

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

The Dirac distribution

The Dirac distribution A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

More information

A basic logarithmic inequality, and the logarithmic mean

A basic logarithmic inequality, and the logarithmic mean Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of

More information

l 2 p2 n 4n 2, the total surface area of the

l 2 p2 n 4n 2, the total surface area of the Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes. Paul Carnig. January ODE s vs PDE s 1 PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

More information

1 Online Learning and Regret Minimization

1 Online Learning and Regret Minimization 2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.

More information

Math 324 Course Notes: Brief description

Math 324 Course Notes: Brief description Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

More information

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information

C1M14. Integrals as Area Accumulators

C1M14. Integrals as Area Accumulators CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms

More information

Anonymous Math 361: Homework 5. x i = 1 (1 u i )

Anonymous Math 361: Homework 5. x i = 1 (1 u i ) Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define

More information

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

Summary of Elementary Calculus

Summary of Elementary Calculus Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s number-line nd its elements re often referred

More information

Problem Set 3

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

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

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

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Chapter 4 Models for Stationary Time Series

Chapter 4 Models for Stationary Time Series Chpter 4 Models for Sttionry Time Series This chpter discusses the bsic concepts of brod clss of prmetric time series models the utoregressive-moving verge models (ARMA. These models hve ssumed gret importnce

More information

Non-Linear & Logistic Regression

Non-Linear & Logistic Regression Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

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

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information