Pi evaluation. Monte Carlo integration

Size: px
Start display at page:

Download "Pi evaluation. Monte Carlo integration"

Transcription

1 Pi evlution y 1 1 x Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (311) Monte Crlo integrtion we wnt to evlute the following integrl: F = f (x) dx remember tht the expecttion vlue of the function f (x) for x distributed ccording to PDF p(x) f = f (x) p(x) dx with: p(x) dx = 1 choosing x to be uniformly distributed in the intervl [, b], one hs: p(x) = 1 b f = 1 b f (x) p(x) dx = b f (x) dx MC integrtion F = f (x) dx = (b ) f = (b ) i=1 f (x i) x i is rndom vrible uniformly distributed in the intervl [, b] error estimtion σ F = (b )σ f σ 2 f = f 2 f 2 σ 2 f = σ2 f σ F = (b ) σ f 1 1 = (b ) ( f (xi ) ) 2 i=1 1 2 f (x i ) i=1 Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (312)

2 MC integrtion (cont.) Let s compute the integrls of the functions: dx 2 = cos(x) dx = (x + b) dx = Throwing 100 rndom vrible uniformly distributed we obtin the following results: dx 2 = ± cos(x) dx = ± (x + b) dx = ± Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (313) MC integrtion (cont.) double xmin=tmth::pi()*0.2; double xmx=tmth::pi()*0.5; int = 1000; lgorithm TF1 *f1 = new TF1( f1, TMth::Abs(cos(x)),xmin,xmx); TF1 *f2 = new TF1( f2, 0.5,xmin,xmx); TF1 *f3 = new TF1( f3, -0.4/TMth::Pi()*x+0.8,xmin,xmx);... for (int i=0; i<; i++) { double x = xmin + (xmx-xmin)*grndom->uniform(); double func1 = f1->evl(x); double func2 = f2->evl(x); double func3 = f3->evl(x); F1 += func1; F2 += func2; F3 += func3; f1s += func1*func1; f2s += func2*func2; f3s += func3*func3; } double f1m = F1/; //men double f2m = F2/; double f3m = F3/; lgorithm // integrls double I1 = f1m*(xmx-xmin); double I2 = f2m*(xmx-xmin); double I3 = f3m*(xmx-xmin); // vrinces double Vr1 = f1s/ - f1m*f1m; double Vr2 = f2s/ - f2m*f2m; double Vr3 = f3s/ - f3m*f3m; // errors double E1 = (xmx-xmin)/sqrt()*sqrt(vr1); double E2 = (xmx-xmin)/sqrt()*sqrt(vr2); double E3 = (xmx-xmin)/sqrt()*sqrt(vr3); Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (314)

3 MC integrtion (cont.) Let s check the integrl vlue s function of the number of rndom vribles generted F = cos(x) dx F = dx 2 F = (x + b) dx Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (315) Empty Slide Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (316)

4 Reduction vrince techniques The cos(x) function vries much more in the intervl of integrtion tht the others Its integrl vlue evlution presents the lrgest vrince. Why? Becuse we re smpling uniformly nd the regions close to zero where the function is more importnt re smpled with the sme importnce s others where the function is smller! In the frmework of the importnce smpling technique n dditionl pdf p(x) cn be used to rend the integrnd smooth! Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (317) Importnce smpling Rend smooth our integrnd by pplying pdf p(x) F = f (x) dx = f (x) p(x) p(x) dx If the pdf is normlized in the integrl intervl [, b] p(x) dx = 1 nd x is vrible distributed ccording to p(x), then f b = p f (x) p(x) p(x) dx Let s mke vrible chnge f (x) p(x) p(x) dx } {{ } p(y)dy p(x)dx = p(y)dy if y is distributed uniformly in [0, 1] then 1 0 p(y)dy = 1 p(y) = 1 The trnsformtion between x nd Y cn be obtined by: x p(x )dx = y 0 dy y = x p(x )dx Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (318)

5 Importnce smpling (cont.) From the trnsformtion of vribles we hve reltion between x nd y y = x p(x )dx x(y) Generting rndom vrible y uniformly between [0, 1] nd pplying the trnsformtion reltion x(y) we get rndom vribles x distributed ccording to p(x) F = f (x) dx = f (x) p(x) p(x) dx = 1 0 f [ x(y) ] f p [ x(y) ] dy = = 1 p y i=1 f [ x(y i ) ] p [ x(y i ) ] Exercise: mke the following integrl cos(x) dx expected = MC = / (100 devites generted) Wht bout using importnce smpling with pdf: p(x) e x? Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (319) Importnce smpling (cont.) The function PDF shpe mtters? Let s study the vrition of the integrl error with the prmeter of the exponentil Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (320)

6 Importnce smpling (cont.) Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (321) Simultion Simultion is very importnt for understnding rel situtions or for modelling the behviour of system it is lrgely used on prticle nd stroprticle physics for designing the instruments used for prticles detection the vrious rel conditions the system hs cn be introduced esily in simulted process Suppose you hd to design detector system for detecting photons coming from Compton scttering on mteril? I ssume my gmm source emits bem very colimted long n xis (x for instnce) nd in between I hve block of mteril where Compton is going to hppen... Wht we need to know? Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (322)

7 Buffon s needle problem Buffon s needle problem is question first posed in the 18th century by Georges-Louis Leclerc where simultion cn help us lot! A needle of length l is thrown rndomly onto grid of prllel lines, seprted by distnce d, with d > l Wht is the probbility tht needle intersects line? Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (323) Buffon s needle problem The probbility of crossing the line it s the rtio between the needle "perpendiculr distnce", l sin θ 2 nd the lines seprtion distnce, d P(θ) = l sin θ 2 d All θ re likely, so we need to clculte n verge probbility for θ rnge: 2 [0, π] Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (324)

8 Generting rndom vribles The common problem is to hve vrible x distributed ccording given distribution function p(x) we re going to mke chnge of vrible such tht the number of events (rndoms) generted is independent of the used vrible d = p(x)dx = p(y)dy Suppose tht y is rndom vrible distributed in [0, 1] nd x strts t x 0 p(y) = 1 y 0 dy = x x p(x )dx y = p(x )dx x 0 x 0 For generting rndom vrible x in intervl [x 0, x 1 ] we just mke sure tht: x1 x 0 p(x)dx = 1 nd we invert the reltion bove (boxed) giving us x(y). Provided rndom y in [0, 1] we generte rndom x in [x 0, x 1 ]. Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (325) uniform [0,1] uniform [,b] Let s trnsform vrible y uniformly distributed in [0, 1] into vrible x uniformly distributed in [, b] ormliztion of p(x): p(x) = 1 b Trnsformtion: y = x 1 b dx = x b x = (b )y x = + (b ) y Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (326)

9 uniform [0,1] exponentil [0, ] Let s trnsform vrible y uniformly distributed in [0, 1] into vrible x distributed ccording to p(x) e x in [0, ] ormliztion of p(x): k + e x dx = 1 0 k [ e x] 0 = 1 k = 1 p(x) = e x Trnsformtion: y = x 0 e x dx = [ e x ] x = 1 0 e x x = ln(1 y) Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (327) rndoms gen: cceptnce-rejection For pplying the method before sid we need to integrte the pdf nd invert it In cse it is not possible we cn use the more generic but less efficient method of cceptnce-rejection For generting x vrible in the intervl [, b] distributed ccording to pdf p(x) wo do the following: generte uniform rndom x R between [, b] compute p(x R ) nd the rtio p(x R) p mx generte second rndom u R from U(0, 1) if u R p(x R) p mx ccept the vrible x R Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (328)

10 MC integrtion: cc-rej method introduced by von eumnn for integrting the function we define n envelope with n re A = (x mx x min ) f mx generte two rndom vribles x R in in rnge [x min, x mx ] f R in in rnge [0, f mx ] count the number of events R tht f R f (x R ) the integrl I = (x mx x min ) f mx R the integrl error σ I = (x mx x min ) f mx R ( 1 R ) Using 100 rndoms: cos(x) dx = ± Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (329) rndom gen: f (x) = 3x 2 The cceptnce-rejection method is inefficient if the function vries quickly The frction of rndoms ccepted (efficiency) is given by: ε = f (x) dx (b ) f mx The efficiency cn be improved using n uxilir function S (x) tht hs shpe close to the one we wnt to smple ε S = f (x) dx q(x) dx where q(x) = C S (x) f (x) for x in [, b] efficiencies: ε = R ε S = R 0.91 Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (330)

11 cc-rej with ux function For generting rndom vrible x distributed ccording to f (x) in the intervl [, b] 1. Find uxilir function S (x) with shpe close to the function f (x) we wnt to smple integrble, invertible from S (x) we define pdf p(x) nd we pply the trnsformtion y U[0, 1] x U(p(x)) invert trnsformtion eqution: x(y) using x(y) nd generting y uniformly between [0, 1] we obtin the rndom vrible x R distributed ccording to p(x) 2. Define function q(x) = C S (x) such tht q(x) f (x) in the intervl [, b] 3. Generte rndom vrible x R ccording to p(x) nd compute f (x R) q(x R ) 4. Generte rndom u R U[0, 1] nd ccept x R if: u R f (x R) q(x R ) Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (331) rndom gen: f (x) = 3x 2 We wnt to generte vrible ccording to function f (x) = 3x 2 1. Define the uxilir function S (x) to improve cceptnce-rejection efficiency 2. define q(x) which in this cse is = S (x) nd generte x R ccording to p(x) 3.,4. generte u R u[0, 1] nd ccept x R if u R f (x R) q(x R ) S (x) = 30x 63 generte rndom in [3,7] intervl ccording to uxilir function get pdf(x): S (x) p(x) with 7 3 p(x)dx = 1 k 7 1 S (x) dx = 1 k = p(x) = (30x 63) mke trnsformtion: y[0, 1] x ccording to p(x) y = x 3 p(x ) dx = x (30x 63) dx Derive x(y) from: 15x 2 63x y = 0 Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (332)

12 C++ clsses clss Func1D { public: Func1D(TF1 *fp=ull); // other constructors? ~Func1D(); void Drw(); double Evlute(); (...) protected: TF1 *p; //integrnd function }; clss IntegrtorMC: public Func1D { public: Integrtor(double fx0, double fx1, TF1 *fp=ull) : x0(fx0), x1(fx1), Func1D(fp) {;} ~Integrtor(); // set function void SetIntegrndFunction(TF1*); // simple integrtion void IntegrlMC(double xmin, double xmx, int, double& result, double& error); // importnce smpling void IntegrlMCIS(double xmin, double xmx, int, double& result, double& error, TF1* pdf); // cceptnce-rejection void IntegrlMCIS(double xmin, double xmx, int, double& result, double& error, TF1* pdf); // other methods (...) protected: double x0, x1; // integrnd limits }; Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (341) Computtionl Physics Universe prticle ccelertor Fernndo Bro, Phys Deprtment IST (Lisbon) Computtionl Physics (Phys Dep IST, Lisbon) Fernndo Bro (342)

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

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

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

Monte Carlo method in solving numerical integration and differential equation

Monte Carlo method in solving numerical integration and differential equation Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The

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

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

Chapter 5 : Continuous Random Variables

Chapter 5 : Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

More information

CS 109 Lecture 11 April 20th, 2016

CS 109 Lecture 11 April 20th, 2016 CS 09 Lecture April 0th, 06 Four Prototypicl Trjectories Review The Norml Distribution is Norml Rndom Vrible: ~ Nµ, σ Probbility Density Function PDF: f x e σ π E[ ] µ Vr σ x µ / σ Also clled Gussin Note:

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

Expectation and Variance

Expectation and Variance Expecttion nd Vrince : sum of two die rolls P(= P(= = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(=2) = 1/36 P(=3) = 1/18 P(=4) = 1/12 P(=5) = 1/9 P(=7) = 1/6 P(=13) =? 2 1/36 3 1/18 4 1/12 5 1/9 6 5/36 7 1/6

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

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

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

Integral equations, eigenvalue, function interpolation

Integral equations, eigenvalue, function interpolation Integrl equtions, eigenvlue, function interpoltion Mrcin Chrząszcz mchrzsz@cernch Monte Crlo methods, 26 My, 2016 1 / Mrcin Chrząszcz (Universität Zürich) Integrl equtions, eigenvlue, function interpoltion

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

INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

More information

Main topics for the Second Midterm

Main topics for the Second Midterm Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

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

Method: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume

Method: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping

More information

Numerical Integration

Numerical Integration Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

Math 135, Spring 2012: HW 7

Math 135, Spring 2012: HW 7 Mth 3, Spring : HW 7 Problem (p. 34 #). SOLUTION. Let N the number of risins per cookie. If N is Poisson rndom vrible with prmeter λ, then nd for this to be t lest.99, we need P (N ) P (N ) ep( λ) λ ln(.)

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

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Lecture 21: Order statistics

Lecture 21: Order statistics Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for

More information

TP 10:Importance Sampling-The Metropolis Algorithm-The Ising Model-The Jackknife Method

TP 10:Importance Sampling-The Metropolis Algorithm-The Ising Model-The Jackknife Method TP 0:Importnce Smpling-The Metropoli Algorithm-The Iing Model-The Jckknife Method June, 200 The Cnonicl Enemble We conider phyicl ytem which re in therml contct with n environment. The environment i uully

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

More information

Tutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.

Tutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits. Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce

More information

1.9 C 2 inner variations

1.9 C 2 inner variations 46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for

More information

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8 Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More 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

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h AM Mthemticl Anlysis Oct. Feb. Dte: October Exercises Lecture Exercise.. If h, prove the following identities hold for ll x: sin(x + h) sin x h cos(x + h) cos x h = sin γ γ = sin γ γ cos(x + γ) (.) sin(x

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

Tests for the Ratio of Two Poisson Rates

Tests for the Ratio of Two Poisson Rates Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

More information

Math 116 Calculus II

Math 116 Calculus II Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function? Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

More information

X Z Y Table 1: Possibles values for Y = XZ. 1, p

X Z Y Table 1: Possibles values for Y = XZ. 1, p ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent

More information

JDEP 384H: Numerical Methods in Business

JDEP 384H: Numerical Methods in Business BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Instructor: Thoms Shores Deprtment of Mthemtics Lecture 20, Februry 29, 2007 110 Kufmnn Center Instructor:

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,

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

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

cos 3 (x) sin(x) dx 3y + 4 dy Math 1206 Calculus Sec. 5.6: Substitution and Area Between Curves

cos 3 (x) sin(x) dx 3y + 4 dy Math 1206 Calculus Sec. 5.6: Substitution and Area Between Curves Mth 126 Clculus Sec. 5.6: Substitution nd Are Between Curves I. U-Substitution for Definite Integrls A. Th m 6-Substitution in Definite Integrls: If g (x) is continuous on [,b] nd f is continuous on the

More information

1 Techniques of Integration

1 Techniques of Integration November 8, 8 MAT86 Week Justin Ko Techniques of Integrtion. Integrtion By Substitution (Chnge of Vribles) We cn think of integrtion by substitution s the counterprt of the chin rule for differentition.

More information

Week 10: Line Integrals

Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Quantum Physics I (8.04) Spring 2016 Assignment 8

Quantum Physics I (8.04) Spring 2016 Assignment 8 Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

Consequently, the temperature must be the same at each point in the cross section at x. Let:

Consequently, the temperature must be the same at each point in the cross section at x. Let: HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the

More information

Lecture 1: Introduction to integration theory and bounded variation

Lecture 1: Introduction to integration theory and bounded variation Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

More information

Final Exam - Review MATH Spring 2017

Final Exam - Review MATH Spring 2017 Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

More information

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher). Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

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

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

p(t) dt + i 1 re it ireit dt =

p(t) dt + i 1 re it ireit dt = Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning

Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (

More information

Big idea in Calculus: approximation

Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics Semester 1, 2002/2003 MA1505 Math I Suggested Solutions to T. 3

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics Semester 1, 2002/2003 MA1505 Math I Suggested Solutions to T. 3 NATIONAL UNIVERSITY OF SINGAPORE Deprtment of Mthemtics Semester, /3 MA55 Mth I Suggested Solutions to T. 3. Using the substitution method, or otherwise, find the following integrls. Solution. ) b) x sin(x

More information

Math Bootcamp 2012 Calculus Refresher

Math Bootcamp 2012 Calculus Refresher Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More 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

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

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

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

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

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

5.5 The Substitution Rule

5.5 The Substitution Rule 5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

More information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Section 17.2 Line Integrals

Section 17.2 Line Integrals Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More 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

( ) 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

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

f(a+h) f(a) x a h 0. This is the rate at which

f(a+h) f(a) x a h 0. This is the rate at which M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

More information

CBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0

CBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0 CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section

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

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information