Topic 1 Notes Jeremy Orloff


 Aubrey Payne
 1 years ago
 Views:
Transcription
1 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 to derive the differentil eqution modeling phsicl or geometric sitution. 4. Be ble to solve seprble differentil eqution, including finding lost solutions. 5. Be ble to solve n initil vlue problem (IVP) b solving the differentil eqution nd using the initil condition to find the constnt of integrtion. 1.2 Differentil equtions nd solutions A differentil eqution (DE) is n eqution with derivtives! Emple 1.1. (DE s modeling phsicl processes, i.e., rte equtions) 1. Newton s lw of cooling: dt = k(t A), where T is the temperture of bod in n dt environment with mbient temperture A. 2. Grvit ner the erth s surfce: m d2 = mg, where is the height of mss m dt2 bove the surfce of the erth. 3. Hooke s lw: m d2 dt 2 = k, where is the displcement from equilibrium of spring with spring constnt k. Other emples: Below we will give some emples of differentil equtions modeling some geometric situtions. A solution to differentil eqution is n function tht stisfies the DE. Let s focus on wht this mens b contrsting it with solving n lgebric eqution. The unknown in n lgebric eqution, such s = 0 is the number. The eqution is solved b finding numericl vlue for tht stisfies the eqution. You cn check b substitution tht = 1 is solution to the eqution shown. The unknown in the differentil eqution d 2 d d d + = 0 1
2 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 2 is the function ().The eqution is solved b finding function () tht stisfies the eqution One solution to the eqution shown is () = e. You cn check this b substituting () = e into the eqution. Agin, note tht the solution is function. More often we will s tht the solution is fmil of functions, e.g. = Ce t. The prmeter C is like the constnt of integrtion in Ever vlue of C gives different function which solves the DE. 1.3 The most importnt differentil eqution in Here, in the ver first clss, we stte nd give solutions to our most importnt differentil equtions. In this cse we will check the solutions b substitution. As we proceed in the course we will lern methods tht help us discover solutions to equtions. The most importnt DE we will stud is d dt =, (1) where is constnt (in units of 1/time). In words the eqution ss tht the rte of chnge of is proportionl to. Becuse of its importnce we will write down some other ws ou might see it: = ; d dt = (t); = 0;. = 0. In the lst eqution we used the phsicist dot nottion to indicte the derivtive is with respect to time. You should recognize tht ll of these re the sme eqution. The solution to this eqution is (t) = Ce t, where C is n constnt Checking the solution b substitution The bove solution is esil checked b substitution. Becuse this eqution is so importnt we show the detils. Substituting (t) = Ce t into Eqution 1 we hve: Left side of (1): Right side of (1): = Ce t = Ce t Since fter substitution the left side equls the right, we hve shown tht (t) = Ce t is indeed solution of Eqution 1.
3 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS The phsicl model of the most importnt DE As phsicl model this eqution ss tht the quntit chnges t rte proportionl to. Becuse of the form the solution tkes we s tht Eqution 1 models eponentil growth or dec. In this course we will lern mn techniques for solving differentil equtions. We will test lmost ll of them on Eqution 1. You should of course understnd how to use these techniques to solve (1). However: whenever ou see this eqution ou should remind ourself tht it models eponentil growth or dec nd ou should know the solution without computtion. 1.4 The second most importnt differentil eqution Our second most importnt DE is m + k = 0, (2) where m nd k re constnts. You cn esil check tht, with ω = k/m, the function (t) = C 1 cos(ωt) + C 2 sin(ωt) is solution. Eqution 2 models simple hrmonic oscilltor. More prosicll, it models mss m oscillting t the end of spring with spring constnt k. 1.5 Solving differentil equtions b the method of optimism In our first nd second most importnt equtions bove we simpl told ou the solution. Once ou hve possible solution it is es to check it b substitution into the differentil eqution. We will cll this method, where ou guess solution nd check it b plugging our guess into the eqution, the method of optimism. In ll seriousness, this will be n importnt method for us. Of course, its utilit depends on lerning how to mke good guesses! 1.6 Generl form of differentil eqution We cn lws rerrnge differentil eqution so tht the right hnd side is 0. For emple, = cn be written s = 0. With this in mind the most generl form for differentil eqution is F (t,,,..., (n) ) = 0, where F is function. For emple, ( ) 2 + e sin(t) (4) = 0. The order of differentil eqution is the order of the highest derivtive tht occurs. So, the emple just bove shows DE of order 4.
4 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS Constructing differentil eqution to model phsicl sitution We use rte equtions, i.e. differentil equtions, to model sstems tht undergo chnge. The following rgument using t should be somewht fmilir from clculus. Emple 1.2. Suppose popultion P (t) hs constnt birth nd deth rtes: β = 2%/er, δ = 1%/er Build differentil eqution tht models this sitution. nswer: In the intervl [t, t + t], the chnge in P is given b P = number of births  number of deths. Over smll time intervl t the popultion is roughl constnt so: Births in the time intervl P (t) β t Deths in the time intervl P (t) δ t Combining these we hve P (β δ)p (t). t Finll, letting t go to 0 we hve derived the differentil eqution dp dt = (β δ)p = kp. Notice tht if β > δ then the popultion is incresing. Of course, this DE is our most importnt DE (1): the eqution of eponentil growth or dec. We know the solution is P = P 0 e kt. Note: If β nd δ re more complicted nd depend on t, s β = P + 2t nd δ = P/t. The derivtion of the DE is the sme, but becuse β nd δ re no longer constnts this is not sitution of eponentil growth nd the solution will be more complicted (nd probbl hrder to find). Emple 1.3. Bcteri growth. Suppose popultion of bcteri is modeled b the eponentil growth eqution P = kp. Suppose tht the popultion doubles ever 3 hours. Find the growth constnt k. nswer: The eqution P = kp hs solution P (t) = Ce kt. From the initil condition we hve tht P (0) = C. Since the popultion doubles ever 3 hours we hve P (3) = Ce 3k = 2C. Solving for k we get k = 1 3 ln 2 (in units of 1/hours.) 1.8 Initil vlue problems An initil vlue problem (IVP) is just differentil eqution where one vlue of the solution function is specified. We illustrte with some simple emples. Emple 1.4. Initil vlue problem. Solve the IVP ẏ = 3, (0) = 7.
5 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 5 nswer: We recognize this s n eponentil growth eqution, so (t) = Ce 3t. Using the initil condition we hve (0) = 7 = C. Therefore (t) = 7e 3t. Emple 1.5. Initil vlue problem. Solve the IVP = 2, (2) = 7. nswer: Note, the use of indictes tht the independent vrible in this problem is. This is rell n problem: integrting we get = 3 /3+C. Using the initil condition we find C = 7 8/ Seprble Equtions Now it s time to lern our first technique for solving differentil equtions. A first order DE is clled seprble if the vribles cn be seprted from ech other. We illustrte with series of emples. Emple 1.6. Eponentil growth. Use seprtion of vribles to solve the eponentil growth eqution = k. nswer: We rewrite the eqution s d = 4. Net we seprte the vribles b getting dt ll the s on one side nd the t s on the other. d = 4 dt. Now we integrte both sides: d = 4 dt ln = 4t + C. Now we solve for b eponentiting both sides: = e C e 4t or = ±e C e 4t. Since ±e C is just constnt we renme it simpl K. We now hve the solution we knew we d get: = Ke 4t. Emple 1.7. Here is stndrd emple where the solution goes to infinit in finite time (i.e. the solutions blow up ). One of the fun fetures of differentil equtions is how ver simple equtions cn hve ver surprising behvior. Solve the initil vlue problem d dt = 2 ; (0) = 1. nswer: We cn seprte the vribles b moving ll the s to one side nd the t s to the other d 2 = dt Integrting both sides we get: 1 = t + C
6 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 6 Think: The constnt of integrtion is importnt, but we onl need it on one side. Solving for we get the solution: = 1 t + C. Finll, we use initil condition (0) = 1 to find tht C = 1. (t) = 1 1 t. So the solution is: We grph this below. Note tht the grph of the solution hs verticl smptote t t = 1. = 1 1 t t Grph of solution Technicl definition of solution Looking t the previous emple we see the domin of consists of two intervls: (, 1) nd (1, ). For technicl resons we will require tht the domin consist of ectl one intervl. So the bove grph rell shows two solutions: Solution 1: (t) = 1/(1 t), where is in the intervl (, 1) Solution 2: (t) = 1/(1 t), where is in the intervl (1, ) In the emple problem, since our IVP hd (0) = 1 the solution must hve t = 0 in its domin. Therefore, solution 1 is the solution to the emple s IVP Lost solutions We hve to cover one more detil of seprble equtions. Sometimes solutions get lost nd hve to be recovered. This is smll detil, but ou wnt to p ttention since it s worth 1 es point on ems nd psets. Emple 1.8. In the emple = 2 we found the solution = 1. But it is es to + C check b substitution tht (t) = 0 is lso solution. Since this solution cn not be written s = 1/( + C) we cll it lost solution. The simple eplntion is tht it got lost when we divided b 2. After ll if = 0 it ws not legitimte to divide b 2. Generl ide of lost solutions for seprble DE s
7 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 7 Suppose we hve the differentil eqution = f()g() If g( 0 ) = 0 then ou cn check b substitution the () = 0 is solution to the DE. It m get lost in when we seprte vribles becuse dividing b b g() would then men dividing b 0. Emple 1.9. Find ll the (possible) lost solutions of = ( 2)( 3). nswer: In this cse g() = ( 2)( 3). The lost solutions re found b finding ll the roots of g(). Tht is, the lost solutions re () = 2 nd () = Implicit solutions Sometimes solving for s function of is too hrd, so we don t! Emple Implicit solutions. Solve = nswer: This is seprble nd fter seprtting vribles nd integrting we hve = C. This is too hrd to solve for s function of so we leve our nswer in this implicit form More emples Emple Solve d d =. nswer: Seprting vribles: d = d. Therefore d = d, which implies ln = C. Finll fter eponentition nd replcing ec b K we hve = Ke 2 /2. Think: There is lost solution tht ws found b some slopp lgebr. Cn ou spot the solution nd the slopp lgebr? Emple Solve d d = 3 2. nswer: Seprting vribles nd integrting gives: 1 = 4 4 There is lso lost solution: () = 0. Emple Solve + p() = 0. 4 = 4 + 4C. + C. Solving for we hve nswer: We first rewrite this so tht it s clerl seprble: d = p() d. After the usul seprtion nd integrtion we hve log( ) = p() d + C Therefore, () = e C e p() d nd () = 0 is lost solution.
8 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS Geometric Applictions of DEs Since the slope of curve is given b its derivtives we cn often use differentil equtions to describe curves. Emple An hev object is drgged through the snd b rope. Suppose the object strts t (0, ) with the puller t the origin, so the rope hs length. The puller moves long the is so tht the rope is lws tut nd tngent to the curve followed b the object. This curve is clled trctri. Find n eqution for it. nswer: The digrm below shows tht d d = 2 2 (, ) 2 2 The trctri Thus 2 2 d = d. Integrting (detils below) we get ( + ) ln = + C. ( + ) The initil position (, ) = (0, ) implies C = 0. Therefore = ln To finish the problem we show tht the integrl is wht we climed it ws: Let I = 2 2 d. Now use the trig. substitution: = sin u: cos u cos 2 u I = cos u du = sin u sin u du 1 sin 2 u = du = csc u sin u du sin u = ln(csc u + cot u) cos u Bck substituting we get I = ( ) + ln 2 2, which is wht we climed bove.
9 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 9 Emple Suppose = () is curve in the first qudrnt nd tht the prt of the curve s tngent line tht lies in the first qudrnt is bisected b the point of tngenc. Find nd solve the DE for this curve. nswer: The figure shows the piece of the tngent bisected b the point (, ) on the curve. Thus, the slope of tngent = d d =. This differentil eqution is seprble nd is esil solved: = C/. 2 (, )
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 informationCalculus 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 informationChapter 6: Transcendental functions: Table of Contents: 6.3 The natural exponential function. 6.2 Inverse functions and their derivatives
Chpter 6: Trnscendentl functions: In this chpter we will lern differentition nd integrtion formuls for few new functions, which include the nturl nd generl eponentil nd the nturl nd generl logrithmic function
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationWe 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 informationMath 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 information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationPDE 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 informationIntegration by Substitution. Pattern Recognition
SECTION Integrtion b Substitution 9 Section Integrtion b Substitution Use pttern recognition to find n indefinite integrl Use chnge of vribles to find n indefinite integrl Use the Generl Power Rule for
More informationC1M14. 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 informationThe 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 informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationUsing integration tables
Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl
More informationUniversity 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 informationArithmetic & Algebra. NCTM National Conference, 2017
NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible
More informationContinuous 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 relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationA little harder example. A block sits at rest on a flat surface. The block is held down by its weight. What is the interaction pair for the weight?
Neton s Ls of Motion (ges 999) 1. An object s velocit vector v remins constnt if nd onl if the net force cting on the object is zero.. hen nonzero net force cts on n object, the object s velocit chnges.
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationFunctions of bounded variation
Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics Clevel thesis Dte: 20060130 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054700 10
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More information11.1 Exponential Functions
. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function
More informationThe Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2
The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published
More informationPhysics 207 Lecture 7
Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions)
More informationMapping 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 information5: 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 informationFact: All polynomial functions are continuous and differentiable everywhere.
Dierentibility AP Clculus Denis Shublek ilernmth.net Dierentibility t Point Deinition: ( ) is dierentible t point We write: = i nd only i lim eists. '( ) lim = or '( ) lim h = ( ) ( ) h 0 h Emple: The
More informationAnonymous 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 informationIII. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet
III. AB Review The mteril in this section is review of AB concepts. www.mstermthmentor.com  181  Illegl to post on Internet R1: Bsic Differentition The derivtive of function is formul for the slope of
More information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More information7  Continuous random variables
71 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 informationExponents and Polynomials
C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse
More informationMath 115 ( ) YumTong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx
Mth 5 20062007) YumTong 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 informationNot for reproduction
AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type
More informationThe Thermodynamics of Aqueous Electrolyte Solutions
18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong
More information4.1. Probability Density Functions
STT 1 4.14. 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 informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationF is n ntiderivtive èor èindeæniteè integrlè off if F 0 èxè =fèxè. Nottion: F èxè = ; it mens F 0 èxè=fèxè ëthe integrl of f of x dee x" Bsic list: xn
Mth 70 Topics for third exm Chpter 3: Applictions of Derivtives x7: Liner pproximtion nd diæerentils Ide: The tngent line to grph of function mkes good pproximtion to the function, ner the point of tngency.
More informationCHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics
More information7.6 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 informationThe Solution of Volterra Integral Equation of the Second Kind by Using the Elzaki Transform
Applied Mthemticl Sciences, Vol. 8, 214, no. 11, 52553 HIKARI Ltd, www.mhikri.com http://dx.doi.org/1.12988/ms.214.312715 The Solution of Volterr Integrl Eqution of the Second Kind by Using the Elzki
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, Kmens, 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 informationPhysics Honors. Final Exam Review Free Response Problems
Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting
More informationContinuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim
Recll the following properties of limits. Theorem. Suppose tht lim f() =L nd lim g() =M. Then lim [f() ± g()] = L + M, lim [f()g()] = LM, if M = 0, lim f() g() = L M. Furthermore, if f() g() for ll, then
More informationSeries Solutions of ODEs. Special Functions
CHAPTER 5 Series Solutions of ODEs. Specil Functions In the previous chpters, we hve seen tht liner ODEs with constnt coefficients cn be solved by lgebric methods, nd tht their solutions re elementry functions
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationDepartment of Chemical Engineering ChE101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII
Tutoril VII: Liner Regression Lst updted 5/8/06 b G.G. Botte Deprtment of Chemicl Engineering ChE0: Approches to Chemicl Engineering Problem Solving MATLAB Tutoril VII Liner Regression Using Lest Squre
More informationl 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 nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationChapter 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 informationWaveguide Guide: A and V. Ross L. Spencer
Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it
More informationThermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report
Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block
More information7 Green s Functions and Nonhomogeneous Problems
7 Green s Functions nd Nonhomogeneous Problems The young theoreticl physicists of genertion or two erlier subscribed to the belief tht: If you hven t done something importnt by ge 3, you never will. Obviously,
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationNonLinear & Logistic Regression
NonLiner & 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 informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI 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 informationCalculus MATH 172Fall 2017 Lecture Notes
Clculus MATH 172Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems
More informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 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 informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationa 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
More informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
More informationLecture 8 Wrapup Part1, Matlab
Lecture 8 Wrpup Prt1, Mtlb Homework Polic Plese stple our homework (ou will lose 10% credit if not stpled or secured) Submit ll problems in order. This mens to plce ever item relting to problem 3 (our
More information6.5 Plate Problems in Rectangular Coordinates
6.5 lte rolems in Rectngulr Coordintes In this section numer of importnt plte prolems ill e emined ug Crte coordintes. 6.5. Uniform ressure producing Bending in One irection Consider first the cse of plte
More informationMATH 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 relvlued function on I), nd let L 1 (I) denote the completion
More informationConvex 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 information4. CHEMICAL KINETICS
4. CHEMICAL KINETICS Synopsis: The study of rtes of chemicl rections mechnisms nd fctors ffecting rtes of rections is clled chemicl kinetics. Spontneous chemicl rection mens, the rection which occurs on
More informationSULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAHSEKOLAH MENENGAH ZON A KUCHING
SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAHSEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationChapter 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
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationThe Fundamental Theorem of Algebra
The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive
More informationE1: CALCULUS  lecture notes
E1: CALCULUS  lecture notes Ştefn Blint Ev Kslik, Simon Epure, Simin Mriş, Aureli Tomoiogă Contents I Introduction 9 1 The notions set, element of set, membership of n element in set re bsic notions of
More informationTransfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions
5/4/6 PM : Trnfer Function Chpter 5 Trnfer Function Defined G() = Y()/U() preent normlized model of proce, i.e., cn be ued with n input. Y() nd U() re both written in devition vrible form. The form of
More informationNUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL
NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL V. Bkhnov, O. Kryvook, B. Dormn Ukrinin Hydrometeorologicl Reserch Institute, Avenue of Science 37,
More informationOverview. Before beginning this module, you should be able to: After completing this module, you should be able to:
Module.: Differentil Equtions for First Order Electricl Circuits evision: My 26, 2007 Produced in coopertion with www.digilentinc.com Overview This module provides brief review of time domin nlysis of
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationData Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3dVritionl Method
More information(2) x't(t) = X> (í)mm0)píj'sgnfoíffyít))], l<i<n, 3 = 1
proceedings of the mericn mthemticl society Volume 104, Number 4, December 1988 SYSTEMS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ASYMPTOTICALLY CONSTANT SOLUTIONS WILLIAM F. TRENCH AND TAKASI KUSANO (Communicted
More informationLecture 8. Band theory con.nued
Lecture 8 Bnd theory con.nued Recp: Solved Schrodinger qu.on for free electrons, for electrons bound in poten.l box, nd bound by proton. Discrete energy levels rouse. The Schrodinger qu.on pplied to periodic
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016
CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More informationINVESTIGATION OF MATHEMATICAL MODEL OF COMMUNICATION NETWORK WITH UNSTEADY FLOW OF REQUESTS
Trnsport nd Telecommuniction Vol No 4 9 Trnsport nd Telecommuniction 9 Volume No 4 8 34 Trnsport nd Telecommuniction Institute Lomonosov Rig LV9 Ltvi INVESTIGATION OF MATHEMATICAL MODEL OF COMMUNICATION
More informationFrom the Numerical. to the Theoretical in. Calculus
From the Numericl to the Theoreticl in Clculus Teching Contemporry Mthemtics NCSSM Ferury 67, 003 Doug Kuhlmnn Phillips Acdemy Andover, MA 01810 dkuhlmnn@ndover.edu How nd Why Numericl Integrtion Should
More informationDiophantine Steiner Triples and PythagoreanType Triangles
Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 15341178 Diophntine Steiner Triples nd PythgorenType Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer
More informationFlow in porous media
Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ
More informationChapter 2 Fundamental Concepts
Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationThe Deltanabla Calculus of Variations for Composition Functionals on Time Scales
Interntionl Journl of Difference Equtions ISSN 973669, Volume 8, Number, pp. 7 47 3) http://cmpus.mst.edu/ijde The Deltnbl Clculus of Vritions for Composition Functionls on Time Scles Monik Dryl nd Delfim
More informationMATH 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 informationa n+2 a n+1 M n a 2 a 1. (2)
Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More information1.3 The log laws. Inverse functions. We have defined logx as the index needed to write x as a power of 10. For example:
1.3 The log lws Inverse functions. We hve defined log s the inde needed to write s power of. For emple: since 1.8 = 66.07 we hve log66.07 = 1.8 Note tht wht this sys is tht the functions f() = nd g() =
More informationPart III. Calculus of variations
Prt III. Clculus of vritions Lecture notes for MA342H P. Krgeorgis pete@mths.tcd.ie 1/43 Introduction There re severl pplictions tht involve expressions of the form Jy) = Lx,yx),y x))dx. For instnce, Jy)
More informationLecture 9: LTL and Büchi Automata
Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled
More informationMETHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS
Journl of Young Scientist Volume III 5 ISSN 448; ISSN CDROM 449; ISSN Online 445; ISSNL 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist
More informationf (x) dx = f(b) f(a) f (x i ) x i i=1
1 Cse Study: Flood Wtch c 00 Donld Kreider nd Dwight L Animtion: Flood Wtch To get you going on the Cse Study! In this section, we hve lerned tht if we re given the continuous derivtive f of function on
More information