Chapter Direct Method of Interpolation More Examples Chemical Engineering
|
|
|
- Gabriel Martin
- 5 years ago
- Views:
Transcription
1 hter 5. Direct Method of Interoltion More Exmles hemicl Engineering Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg Determine the vlue of the secific het t T 6 using the direct method of interoltion nd first order olynomil. 5..
2 5.. hter 5. Figure Secific het of wter vs. temerture. Solution For first order olynomil interoltion (lso clled liner interoltion), we choose the secific het given by T T y x, y f x x, y x Figure Liner interoltion.
3 Direct Method of Interoltion More Exmles: hemicl Engineering 5.. Since we wnt to find the secific het t T 6, nd we re using first order olynomil, we need to choose the two dt oints tht re closest to T 6 tht lso brcket T 6 to evlute it. The two oints re T 5 nd T 8. Then T 5, T T 8, T 499 gives Writing the equtions in mtrix form, we hve Solving the bove two equtions gives Hence T T T, 5 T 8 At T 6, J kg Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg
4 5..4 hter 5. Determine the vlue of the secific het t T 6 using the direct method of interoltion nd second order olynomil. Find the bsolute reltive roximte error for the second order olynomil roximtion. Solution For second order olynomil interoltion (lso clled qudrtic interoltion), we choose the secific het given by T T T y x, y x, y f x x, y x Figure Qudrtic interoltion. Since we wnt to find the secific het t T 6, nd we re using second order olynomil, we need to choose the three dt oints tht re closest to T 6 tht lso brcket T 6 to evlute it. The three oints re T 4, T 5, nd T 8. Then T 4, T 479 T 5, T T 8, T 499 gives Writing the three equtions in mtrix form, we hve
5 Direct Method of Interoltion More Exmles: hemicl Engineering 5..5 Solving the bove three equtions gives Hence T T T, 4 T 8 At T 6, J 49. kg The bsolute reltive roximte error obtined between the results from the first nd second order olynomil is % Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg Determine the vlue of the secific het t T 6 using the direct method of interoltion nd third order olynomil. Find the bsolute reltive roximte error for the third order olynomil roximtion. Solution For third order olynomil interoltion (lso clled cubic interoltion), we choose the secific het given by T T T T
6 5..6 hter 5. y x, y x, y f x x, y x, y x Figure 4 ubic interoltion. Since we wnt to find the secific het t T 6, nd we re using third order olynomil, we need to choose the four dt oints closest to T 6 tht lso brcket T 6 to evlute it. The four oints re T 4, T 5, T 8 nd T. (hoosing the four oints s T, T 4, T 5 nd T 8 is eqully vlid.) Then T 4, T 479 T 5, T T 8, T 499 T, T 47 gives Writing the four equtions in mtrix form, we hve Solving the bove four equtions gives
7 Direct Method of Interoltion More Exmles: hemicl Engineering 5..7 Hence T T T T T.67T T J 49. kg The bsolute reltive roximte error nd third order olynomil is % T, 4 T obtined between the results from the second INTERPOLATION Toic Direct Method of Interoltion Summry Exmles of direct method of interoltion. Mjor hemicl Engineering Authors Autr Kw Dte November, 9 Web Site htt://numericlmethods.eng.usf.edu
Chapter Direct Method of Interpolation More Examples Civil Engineering
Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this
Chapter Direct Method of Interpolation More Examples Electrical Engineering
Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o
Chapter Direct Method of Interpolation More Examples Mechanical Engineering
Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by
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
a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
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
Chapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
Best Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
Chapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
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
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
AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
A Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
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
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
Chapter 3 Solving Nonlinear Equations
Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,
NUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
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
OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved.
Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B
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
Midpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
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
4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.
4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)
CBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
Quadratic Residues. Chapter Quadratic residues
Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue
Mon 2:04:28 PM. Introduction
Mon 2:04:28 PM Slide Nr. 0 of 18 Slide Introduction Air in the tmohere normlly contin ome wter vor (or moiture) nd i referred to tmoheric ir. By contrt, ir tht contin no wter vor i clled dry ir. It i often
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
HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
Best Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
Predict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
Lecture 12: Numerical Quadrature
Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review
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
Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?
University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the
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
Solution to HW 4, Ma 1c Prac 2016
Solution to HW 4 M c Prc 6 Remrk: every function ppering in this homework set is sufficiently nice t lest C following the jrgon from the textbook we cn pply ll kinds of theorems from the textbook without
Undergraduate Research
Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm,
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,
Math 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
ad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4)
10 Most proofs re left s reding exercises. Definition 10.1. Z = Z {0}. Definition 10.2. Let be the binry reltion defined on Z Z by, b c, d iff d = cb. Theorem 10.3. is n equivlence reltion on Z Z. Proof.
TABLE OF CONTENTS 3 CHAPTER 1
TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER
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
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
Instructor: Marios M. Fyrillas HOMEWORK ASSIGNMENT ON INTERPOLATION
AMAT 34 Numericl Methods Instructor: Mrios M. Fyrills Emil: m.fyrills@fit.c.cy Office Tel.: 34559/6 Et. 3 HOMEWORK ASSIGNMENT ON INTERPOATION QUESTION Using interpoltion by colloction-polynomil fit method
The Gear Hub Profiling for Machining Surfaces with Discreetly Expressed Surfaces
he Ger Hub Profiling for Mchining Surfces with iscreetly Exressed Surfces NILE NE, VIRGIL ER, INUŢ PP, GBRIEL UR Mnufcturing Science nd Engineering ertment unăre de Jos University of Glţi omnescă street,
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](/thumbs/72/66633133.jpg "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
Chapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
Jim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
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
DOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES
DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OE-DIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m
Lesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
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
DERIVATIVES NOTES HARRIS MATH CAMP Introduction
f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we
Student Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
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
The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues
Section 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
Math RE - Calculus II Area Page 1 of 12
Mth --RE - Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=
Chapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
u t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
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...
k and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
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
ECE 451 Automated Microwave Measurements. TRL Calibration
utomted Microwve Mesurements Clibrtion Jose E. Schutt-ine Electricl & Computer Engineering University of Illinois jose@emlb.uiuc.edu Copyright by Jose E. Schutt ine, ll ights eserved Coxil Microstrip rnsition
Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
Numerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
Example Sheet 6. Infinite and Improper Integrals
Sivkumr Exmple Sheet 6 Infinite nd Improper Integrls MATH 5H Mteril presented here is extrcted from Stewrt s text s well s from R. G. Brtle s The elements of rel nlysis. Infinite Integrls: These integrls
Numerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
The Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule
PHYS 75: Clssicl echnics Sll Oscilltions: Exple A Liner Tritoic olecule A Liner Tritoic olecule x b b x x3 x Experientlly, one ight be interested in the rdition resulted fro the intrinsic oscilltion odes
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.
Duke Math Meet
Duke Mth Meet 01-14 Power Round Qudrtic Residues nd Prime Numers For integers nd, we write to indicte tht evenly divides, nd to indicte tht does not divide For exmle, 4 nd 4 Let e rime numer An integer
Math 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1
Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks
On the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model.
Popultion Dynmics Definition Model A model is defined s physicl representtion of ny nturl phenomen Exmple: 1. A miniture building model. 2. A children cycle prk depicting the trffic signls 3. Disply of
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
MTH 4-16a Trigonometry
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
Chapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
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
First Semester Review Calculus BC
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.
Orthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
CHAPTER 4a. ROOTS OF EQUATIONS
CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment
Abstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)
1. Extend QR downwards to meet the x-axis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
Joint distribution. Joint distribution. Marginal distributions. Joint distribution
Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i
Lecture 6: Isometry. Table of contents
Mth 348 Fll 017 Lecture 6: Isometry Disclimer. As we hve textook, this lecture note is for guidnce nd sulement only. It should not e relied on when rering for exms. In this lecture we nish the reliminry
Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
APPLIED THERMODYNAMICS TUTORIAL 6 AIR-VAPOUR MIXTURES
APPLIED THERMODYNAMICS TUTORIAL 6 AIR-APOUR MIXTURES In thi tutoril you will do the following. Revie the UNIERSAL GAS LAW Lern DALTONS LAW OF PARTIAL PRESSURES Aly thee lw to mixture of wter vour nd ir.
Chapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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
ODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
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
Homework Problem Set 1 Solutions
Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:
Precalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
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](/thumbs/80/80533932.jpg "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: