M344  ADVANCED ENGINEERING MATHEMATICS


 Emily Fowler
 1 years ago
 Views:
Transcription
1 M3  ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If the het eqution is written in terms of 3 spce vriles,, nd z, it hs the form u t = (u + u + u zz ) = u, where is red del squred, nd is clled the Lplcin opertor. When the temperture u reches sted stte, tht is, it stops chnging with time, then u t = nd we hve the sted stte het eqution u = u + u + u zz =. This is Lplce s eqution in three spce vriles. The eqution we re going to solve is Lplce s eqution in two spce vriles, which cn e written s u = u + u =. (1) It cn e seen tht eqution (1) is n elliptic prtil differentil eqution, ccording to our definition, noting tht A = C = 1 nd B = impl B AC = <. As n emple of phsicl sitution where this eqution rises, consider the temperture u(, ) in rectngulr metl plte which is insulted on the top nd ottom, so tht het cnnot flow in the zdirection. If the tempertures on ll four edges of the rectngle re specified, then s t, the temperture in the interior of the rectngulr plte will pproch the solution of eqution (1). u = g() u = h() u + u = u = f() u = k() Sted stte temperture in n insulted rectngulr plte 1
2 Solution of Lplce s Eqution Seprtion of Vriles The method of seprtion of vriles tht we pplied to the het eqution nd the wve eqution cn lso e used to solve eqution (1), if it is ssumed tht three sides of the rectngle re held t temperture. The temperture on the fourth side cn e specified n ritrr piecewise continuous function. Since Lplce s eqution is liner nd homogeneous, we cn find four different series solutions, ech one stisfing nonzero condition on different side, nd dd them together to get solution which stisfies ritrr conditions round the entire oundr of the rectngle. Note tht there is no initil condition on u, since we re looking for the sted stte temperture inside the rectngle R = {, }. We will ssume first tht the oundr conditions re s shown in the figure elow. u(, ) = R u + u = u(, ) = u(, ) = u(, ) = f() If u(, ) = X()Y () is sustituted into eqution (1), nd the result is divided XY, X Y XY + XY XY = X = Y X Y = λ. The two ordinr differentil equtions in nd re X () + λx() =, Y () λy () =. The oundr conditions u(, ) = X()Y () = nd u(, ) = X()Y () = for ll in the intervl [, ] impl tht X() = X() =. These oundr conditions give us the sme SturmLiouville prolem for X + λx = tht we hve solved twice efore. The eigenvlues will e λ n = n π nd the corresponding eigenfunctions re X n () = c n sin( nπ). The eqution for Y n, with λ n = n π, ecomes Y n n π Y n =, which hs solution Y n () = n cosh( nπ ) + n sinh( nπ ).
3 The series solution, which will e clled u 1 (, ), cn e written s u 1 (, ) = sin( nπ )[A n cosh( nπ ) + B n sinh( nπ )], nd the coefficients A n nd B n must e chosen to stisf the remining two conditions u 1 (, ) =, u 1 (, ) = f(), for. The condition u 1 (, ) = f() = A n sin( nπ ) implies tht the A n re the coefficients in the Fourier Sine Series for f(); therefore, A n = nπ f() sin( )d. The other condition implies tht u 1 (, ) = sin( nπ )[A n cosh( nπ ) + B n sinh( nπ )], nd therefore, for ech n = 1,,, we must hve A n cosh( nπ)+b n sinh( nπ) =. This mens tht nd the solution u 1 cn e written s u 1 (, ) = cosh( nπ B n = A ) n sinh( nπ) = A n coth( nπ ), sin( nπ )A n1[cosh( nπ ) coth(nπ ) sinh(nπ )], A n1 = f() sin( nπ )d. The other three cses correspond to oundr conditions specified s shown elow: g() u (, ) The corresponding solutions re: u 3 (, ) h() 3 u (, ) k()
4 u (, ) = u 3 (, ) = u (, ) = sin( nπ )[B n sinh( nπ )], B n = nπ g() sin( )d sinh( nπ) sin( nπ )A n3[cosh( nπ ) coth(nπ ) sinh(nπ )], A n3 = h() sin( nπ )d sin( nπ )[B n sinh( nπ )], B n = nπ k() sin( )d sinh( nπ) In the eercises ou will e sked to derive the solution u (, ). In this cse the SturmLiouville eqution is the eqution in, rther thn the eqution in. Emple 1 Consider the rectngle R = { 15, 1} with tempertures long the oundr given u(, ) = f() =.7(15 ), u(, 1) = g(), u(, ) = h() = sin( π ), nd u(15, ) = k() = (1 ). 5 Using MAPLE progrm to compute the coefficients, the series for u(, ) = u 1 (, ) + u 3 (, ) + u (, ), with terms in ech series ws plotted in Figure 1() s threedimensionl surfce ove the rectngle R. A contour plot showing where the temperture hs the vlues 1,, 5, 1, 15, nd 3 is lso shown, in Figure 1(). In this emple, the oundr functions were chosen so tht u = t ll four corners. This gurnteed continuous solution everwhere inside the rectngle. temperture in R 1 Contour plot of temperture Figure 1: () Figure 1: ()
5 Numericl Solution of Lplce s Eqution If the rectngle R is prtitioned long the nd es, letting = /N nd = /M for integers N nd M, the centrl difference formul for u nd u cn e used to write the following difference pproimtion to eqution (1): u( +, ) u(, ) + u(, ) u(, + ) u(, ) + u(, ) ( ) + ( ) =. If nd cn e chosen to e equl, then letting = = h, the eqution cn e multiplied h on oth sides, resulting in u( + h, ) u(, ) + u( h, ) + u(, + h) u(, ) + u(, h) =. This cn e solved for u(, ) in the form u(, ) = u( + h, ) + u( h, ) + u(, + h) + u(, h). () Note tht this ss tht the temperture t ech grid point in the interior of R is the verge of the tempertures t the four nerest grid points. If ll of the oundr vlues re given, this produces sstem of liner equtions for the unknown tempertures in the grid. The numer of equtions in the liner sstem is equl to the numer of interior grid points. Emple We will numericll pproimte the tempertures tht were clculted the series solution in Emple 1. With = 15 nd = 1, we cn let = = h = 5, nd use the grid shown elow. There re onl two unknown tempertures to e computed, lelled T 1 nd T. The should e pproimtions to u(5, 5) nd u(1, 5), respectivel. The oundr tempertures were otined from the formuls for f(), g(), h(), nd k() in Emple 1. T 1 T
6 Using eqution (), the two liner equtions for T 1 nd T re: Written in the form T 1 = + T , T = T 1 T 1 T = 35 T 1 + T = 6 the equtions cn e solved to give T 1 = nd T = These compre to the vlues T 1 = u(5, 5) 1.6 nd T = u(1, 5) 16.3 otined from the series solution in Emple 1. It is cler tht much etter numericl pproimtion would result if the step size h is decresed. If we tke h =.5, which is one hlf of the originl h, the numer of unknown tempertures inside the rectngle increses to 15 (Check it!). Similrl, the numer of liner equtions in the sstem increses to 15. Computer methods for solving lrge sstems of liner equtions hve een round for long time, nd the re ver es to ppl. This is topic tht is covered in oth Liner Alger nd Numericl Anlsis courses. Prctice Prolems: 1. * Derive the series solution for u(, ) where u stisfies Lplce s eqution inside the rectngle R, nd the oundr conditions re: u(, ) = u(, ) =, ; u(, ) =, u(, ) k(),. Your nswer should look like u (, ) on pge.. * Use the numericl method for solving Lplce s eqution to find pproimtions to T 1, T, nd T 3 in the Lshped region shown here. The oundr tempertures re ll given in the digrm. 6
7 38 T T T 3 5 Note: The method of seprtion of vriles does not work for nonrectngulr region of this tpe. 3. Etr Credit: Set up nd solve the sstem of liner equtions to find the numericl solution to Emple, using h =.5. The corresponding 15 interior tempertures given the series in Emple 1 (with terms) re: = = = Compre our numericl solution to these vlues. Is the greement etter thn it ws with h = 5? Eplin. Hint: The mtri ou need cn e set up s 3 3 lock mtri, where the locks consist of 3 different 5 5 mtrices. One of these is the zero mtri, nother is I, where I is the 5 5 identit mtri, nd the third hs specil form. To see how to set up lock mtri in MAPLE, eecute the instruction?lockmtri.. * If nd re hlved gin, so tht h = 1.5, how mn unknown tempertures must e computed in the numericl solution in Emple? Determine wht the mtri of coefficients will look like for this sstem. Give complete description of it s lock mtri. 7
MA 201: Partial Differential Equations Lecture  12
Two dimensionl Lplce Eqution MA 201: Prtil Differentil Equtions Lecture  12 The Lplce Eqution (the cnonicl elliptic eqution) Two dimensionl Lplce Eqution Two dimensionl Lplce Eqution 2 u = u xx + u yy
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationModule 9: The Method of Green s Functions
Module 9: The Method of Green s Functions The method of Green s functions is n importnt technique for solving oundry vlue nd, initil nd oundry vlue prolems for prtil differentil equtions. In this module,
More informationPhysics 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 information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More information(PDE) u t k(u xx + u yy ) = 0 (x, y) in Ω, t > 0, (BC) u(x, y, t) = 0 (x, y) on Γ, t > 0, (IC) u(x, y, 0) = f(x, y) (x, y) in Ω.
Seprtion of Vriles for Higher Dimensionl Het Eqution 1. Het Eqution nd Eigenfunctions of the Lplcin: An 2D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More 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 informationChapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
More informationLecture 24: Laplace s Equation
Introductory lecture notes on Prtil Differentil Equtions  c Anthony Peirce. Not to e copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 24: Lplce s Eqution
More informationWhat Is Calculus? 42 CHAPTER 1 Limits and Their Properties
60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of
More informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr SturmLiouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More information1 2D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More information50 AMC Lectures Problem Book 2 (36) Substitution Method
0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem
More informationSection  2 MORE PROPERTIES
LOCUS Section  MORE PROPERTES n section , we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationDifferential Equations 2 Homework 5 Solutions to the Assigned Exercises
Differentil Equtions Homework Solutions to the Assigned Exercises, # 3 Consider the dmped string prolem u tt + 3u t = u xx, < x , u, t = u, t =, t >, ux, = fx, u t x, = gx. In the exm you were supposed
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationWave Equation on a Two Dimensional Rectangle
Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c
More informationu(x, y, t) = T(t)Φ(x, y) 0. (THE EQUATIONS FOR PRODUCT SOLUTIONS) Plugging u = T(t)Φ(x, y) in (PDE)(BC) we see: There is a constant λ such that
Seprtion of Vriles for Higher Dimensionl Wve Eqution 1. Virting Memrne: 2D Wve Eqution nd Eigenfunctions of the Lplcin Ojective: Let Ω e plnr region with oundry curve Γ. Consider the wve eqution in Ω
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationMath 017. Materials With Exercises
Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson
More information1 E3102: a study guide and review, Version 1.0
1 E3102: study guide nd review, Version 1.0 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in very
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More informationChapter 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
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More information5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationIntroduction to Algebra  Part 2
Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger  Prt Sttement of Prerequisite
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information5.7 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 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 informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationSturmLiouville Theory
LECTURE 1 SturmLiouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More informationITERATIVE SOLUTION REFINEMENT
Numericl nlysis f ngineers Germn Jdnin University ITRTIV SOLUTION RFINMNT Numericl solution of systems of liner lgeric equtions using direct methods such s Mtri Inverse, Guss limintion, GussJdn limintion,
More informationImproper Integrals with Infinite Limits of Integration
6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationMA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1
MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]
More information10 Elliptic equations
1 Elliptic equtions Sections 7.1, 7.2, 7.3, 7.7.1, An Introduction to Prtil Differentil Equtions, Pinchover nd Ruinstein We consider the twodimensionl Lplce eqution on the domin D, More generl eqution
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationBob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions
More 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 information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
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 informationCalculus  Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus  Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More informationThe Trapezoidal Rule
SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proilit Distriutions 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationMT Integral equations
MT58  Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
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 informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationu 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 timeindependent problems Introductory emples Onedimensionl het eqution Consider the onedimensionl het eqution with boundry conditions nd initil condition We lredy know
More informationAT100  Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a
Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7
More informationEquations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationSECTION 94 Translation of Axes
94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More information1. Twelve less than five times a number is thirty three. What is the number
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationChapter 7: Applications of Integrals
Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As
More information5.1 How do we Measure Distance Traveled given Velocity? Student Notes
. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the xxis & yxis
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More information