ENGI 9420 Engineering Analysis Assignment 3 Solutions
|
|
- Jacob Barnett
- 5 years ago
- Views:
Transcription
1 ENGI 9 Egieerig Aalysis Assigmet Solutios Fall [Series solutio of ODEs, matri algebra; umerical methods; Chapters, ad ]. Fid a power series solutio about =, as far as the term i 7, to the ordiary differetial equatio d y π + y = sec, < d [This ODE is also i Assigmet Questio.] You may quote the Maclauri series epasio for sec : sec = ( < π ) 7 86 This ODE is aalytic for all i ( π + π ),. Therefore = is ot a sigular poit. Attempt a simple power series solutio: = = ( ) y= a y = a = = y = + + a = a Substitute the series for y, y" ad sec ito the ODE: ( a+ ( + )( + ) + a ) = = Matchig coefficiets: : a + a = a = a a a a a : + = = 6 : + = = = ( ) = + a a a a a a : = 5 = = 6 = a a a a a a
2 ENGI 9 Assigmet Solutios Page of 8. (cotiued) : 7 6a 7 + a5 = a7 = a5 = a = 5 a 6 : 8 7a a6 = 7 : 6 5a + a = a = a = a = 5 a 6 6 ( 5 ) ( 56 ) a = a = a = + a The leadig coefficiets a ad a are arbitrary costats, which are also the iitial values y ad The geeral solutio is y respectively. Rewrite them as A ad B respectively. 6 = ( ) B( ) ( ) y A 7 Upo recogizig the presece of factorial fuctios, we ca write the geeral solutio as y = 6 A + +!! 6! B + +!! 5! 7! ! 6! 8! Oe ca recover the complemetary fuctio quicly, upo recogizig the Maclauri series for the sie ad cosie fuctios: y = Acos + Bsi ! 6! 8! The particular solutio turs out to be the Maclauri series epasio for si + ( cos ) l cos A brief Maple worsheet provides this Maclauri series epasio.
3 ENGI 9 Assigmet Solutios Page of 8. Use the method of Frobeius to fid the geeral solutio of the ordiary differetial equatio d y dy + ( + ) + ( ) y = d d as a series about =, as far as the fourth o-zero term. d y dy + ( + ) + ( ) y = P d d F Q R Q R F = ( + ), = ad = are all aalytic. P P P Therefore = is a regular sigular poit of this ODE. + r + r = = y a ( r)( r + r ) = y= a y = a + r = + + Substitute the series ito the ODE: + r + + r + a( + r)( + r ) + a + r = = + r + + r+ + r a( r) a a = = = = Adjust the ide of summatio, such that the epoet of is +r i all cases. The ODE becomes + r a ( + r)( + r ) + ( + r) = + a ( ( + r ) + r + ) = = + r + r ( r )( r ) a ( ( r) ) a = = r ( r )( r+ ) a + + r ( + r )( + r+ ) a + ( ( + r) ) a = = This equatio must be true for all values of. Therefore the coefficiet of every power of must be zero. But a caot be idetically zero. r r+ = r = or
4 ENGI 9 Assigmet Solutios Page of 8. (cotiued) Case r = : This value of r is a positive iteger, so that the series becomes a ordiary power series. ( + )( + + ) a + ( ( + ) ) a = > ( + ) a = a ( + ) Employig this recurrece relatio to fid a to a, ( + ) 6 a = a = a + 5 ( + ) ( + ) ( + ) a = a = a = + a a = a = a = a a = a = a = + a y = a = a = Case r = /: ( )( + ) a + ( ( ) ) a = > a ( ) = a ( ) > ( ) a = a ( ) = a = > This series is therefore fiite, with oly oe term: y = a / The fuctio y ( ) is clearly idepedet of the fuctio Therefore the geeral solutio is y. A 5 = + ( ) y B (where A ad B are arbitrary costats).
5 ENGI 9 Assigmet Solutios Page 5 of 8. Fid the etire series solutio (usig the method of Frobeius, adapted to a first order ODE) about = of the ordiary differetial equatio dy y + = d P=, Q=, F = Q F = ad = are both aalytic P P Therefore = is a regular sigular poit of this ODE. + r + r = = y= a y = a + r Substitute the series ito the ODE: r y + + y = a + r+ = = r a r The leadig term ( = ) is ( + ) 5 If r = the a + = a = The ODE the becomes 5 ( 5 + a + ) = a = > 5 = + this is a particular solutio. y = a = 5 = If r = the a ( r+ ) = r = ( a caot be zero) = 5 a = a = ecept for =, where a is arbitrary, (relabel it as A ) ad for = 5, where 5a 5 = a5 = 5 y = a = A = 5 which icludes the case r =. Therefore the geeral solutio is 5 A = + y [Note that this ODE is liear. It is easy to verify that the solutio above is correct.]
6 ENGI 9 Assigmet Solutios Page 6 of 8. I Chapter of the lecture otes, dimesioal aalysis is used to derive the fuctioal form of the Plac legth L i terms of the uiversal costats G, h ad c. P I the study of the steady flow of icompressible fluid through a pipe, it is ow that the volume of liquid issuig per secod from a pipe, Q, depeds o the coefficiet of viscosity η (measured i the uits g m - s - ), the radius a of the pipe (measured i metres) ad the pressure gradiet p/l set up alog the pipe (measured i the uits N m - m - or, equivaletly, g m - s - ). Use a similar dimesioal aalysis to derive the fuctioal form for Q i terms of η, a ad p/l. z y p Q = η a, where is a costat of proportioality that caot be determied l by dimesioal aalysis aloe. Q is a volume per uit time (m s - ). Coductig the dimesioal aalysis, y z y z z z LT ML T [ L] ML T + M L + = = T y z M R + R L R+ R T R ( ) R R R + R =, y =, z = Therefore Q a p = η l which is Poiseuille s formula. By methods beyod the scope of this course, it ca be show that = π. 8 Plai tet output from the liear system reductio program is available at this li.
7 ENGI 9 Assigmet Solutios Page 7 of 8 5. Fid the itersectio of the three plaes + y + z = + 5y + z + = + y + z = Where the three plaes all meet, all three equatios must be true simultaeously. Form ad reduce the augmeted matri that is equivalet to this triple of simultaeous liear equatios. A b R R = 5 R R + Upo reachig this triagular form, oe ca see that ra A = ra [A b] = so that there is a uique solutio. R R 9 R R R + R Geometrically, the three plaes itersect each other i a sigle poit. From this reduced echelo form, we ca read the uique solutio (, y, z ) = ( 9,, ) 6. By reducig the appropriate liear system to echelo form, show that the itersectio of the three plaes + y + z + = + y + z + = + y + z + = y ( ) z is the lie = =. The liear system for (, y, z) correspodig to the simultaeous equatios of the three plaes is
8 ENGI 9 Assigmet Solutios Page 8 of 8 6 (cotiued) y z Plae Plae Plae R R R R At this poit, oe ca see that (row ) = (row ), so that row will cacel dow to a row of all zeroes ad ra A = ra [A b] = <. There are therefore ifiitely may solutios. More precisely, oe ca detect a oe-parameter family of solutios. Completig the row reductio, R R R+ R This is the reduced row echelo form for [A b]. There are leadig oe s for ad y oly. z is therefore a free parameter (call it t ). Readig the solutios from this echelo form, z = t; y + t = y = t ; t = = t+ (, y, z) = ( t+, t, t) = (,, ) + (,, ) t This oe vector equatio is also a set of three simultaeous scalar equatios for, y, z i terms of t. Mae t the subject of all three equatios, the t ( ) y z = = = which is the equatio of a lie. Geometrically, a oe parameter family of solutios should geerate a oe-dimesioal object (the lie).
9 ENGI 9 Assigmet Solutios Page 9 of 8 7. Fid the value of the determiat ad, if it eists, the iverse matri, for the matri A = Form the augmeted matri [ ] A I = Trasform the matri to reduced echelo form: R R R + R R R The left matri is ow i triagular form. The oly row operatio so far that has chaged the value of the determiat is the row iterchage. The value of the determiat of the origial matri A is therefore ( ½) =. No-zero determiat A - eists. Resumig the row reductio: R
10 ENGI 9 Assigmet Solutios Page of 8 7. (cotiued) R R R+ R R R R R + R = I A R R The oly row operatios that chaged the value of the determiat were oe row iterchage ad two cacellig row multiplicatios (the first by a factor of ½ ad the secod by a factor of ). All other row operatios were of the type (from oe row subtract or add a multiple of aother row), which do ot affect the value of the determiat. Therefore the determiat of the origial matri A is the determiat of the echelo form I. But det I =. Therefore det A =. Also, the fact that the reduced echelo form of A is the idetity matri allows us to coclude that A is ivertible ad that the iverse is the right-had matri i the reduced echelo form of the augmeted matri. Therefore det A ad A = = [Oe may chec the aswer, usig A A = A A = I.]
11 ENGI 9 Assigmet Solutios Page of 8 7. (cotiued) Additioal Note: It is possible to use a cofactor epasio to evaluate the determiat ad iverse of the matri A = However, this is a much slower method tha row reductio (Gaussia elimiatio) to echelo form. The details of this alterative method are o a separate web page. 8. I Assigmet, Questio 5, fid, correct to the earest millimetre, the iitial head H of water eeded i the coical ta i order for the ta to drai completely i eactly oe miute, usig (a) the method of bisectio (or a graphical zoom-i, i which case provide setches or screeshots) ad eplai your choice of iitial values. The time T required to drai the coical ta of head H completely was foud i assigmet to be ( ) H T = H + H + g A ta of head cm was foud to drai i approimately 6 s. The fuctioal form for T (H) ivolves a higher power tha the square. It is therefore certai that a ta of head 6 cm will require more tha 6 secods to drai. I the bisectio method, tae as startig values H = ad H = 6. [et page]
12 ENGI 9 Assigmet Solutios Page of 8 8 (a) (cotiued) H T At this poit oe ca deduce that, correct to the earest millimetre, a draiage time of eactly oe miute will occur whe H =. cm A alterative to tabular bisectio is a sequece of graphical zooms, as illustrated here.
13 ENGI 9 Assigmet Solutios Page of 8 8 (a) (cotiued) This process of graphical zooms ca be cotiued, i order to fid the solutio to greater precisio: from which H =. cm, correct to four decimal places. T (.) = 6. s (b) Newto s method ad eplai your choice of iitial value. As i part (a), we ca deduce that the required value of H is somewhere i (, 6). The o-liear fuctio suggests that the value is closer to tha it is to 6. Therefore select H = as a iitial guess. + = f f ( ) ( ), where f = ( ) g ( 5 5 ) f = + + g f ( ) f ' ( ) f / f ' Therefore, correct to oe decimal place, H =. cm [A Ecel spreadsheet file for this solutio is available here. Oe may eperimet with differet startig values.]
14 ENGI 9 Assigmet Solutios Page of 8 9. Why should Newto s Method ot be used to fid a root of e = ta, (ecept whe the iitial guess is very ear the true value)? Demostrate the problem by usig Newto s Method to try to fid the first positive root, with iitial guesses of.99 ad.. The fuctio ta has a ifiite umber of ifiite discotiuities. It is quite possible for the taget lie from a value of to cross oe or more such discotiuities before reachig its ais itercept. Newto s Method ca thus become very ustable, jumpig from ear oe root to ear some other root. Graph of y = e ta Also ote the proimity to the first positive root of two turig poits (where f ' () = ). Usig Newto s Method ayway, with the startig values.99 ad.: f ( ) + =, where f = e ta f = e sec f ( )
15 ENGI 9 Assigmet Solutios Page 5 of 8 9 (cotiued) f( ) f '( ) f / f ' Note that the method has coverged o the wrog root! f( ) f '( ) f / f ' ad =.6... is the first positive root. Note how two startig values so close together have coverged to very differet roots. This is a illustratio of how ustable Newto s Method is i this case. [A Ecel spreadsheet file for this solutio is available here. Oe may eperimet with differet startig values.]
16 ENGI 9 Assigmet Solutios Page 6 of 8. Use the stadard fourth order Ruge-Kutta method, with a step size of h =., to fid the value at =.6 of the solutio of the iitial value problem dy ( y y), y. d = = Show your calculatios for the costats,,, ad for y i the first iteratio. You may submit output from a spreadsheet program (such as Ecel) for the other steps. y For this ODE, the RK algorithm becomes = f, y = y y = y y ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (, ) ( ) ( )(( ) ) = f + h, y + h = + h y + h y + h = f + h, y + h = + h y + h y + h = f + h y + h = + h y + h y + h h y+ = y + ( ) 6 with startig values = ad y =.. I the first iteratio, =... = = =.6 = =.6 = =.8. y(.) = y = + ( ) = Secod iteratio: = (.) (.9996) ((.9996) ) =.8 ( ) ( ) = (. +.) ( )( ) (.) = y = = =.7 = = = =.98978
17 ENGI 9 Assigmet Solutios Page 7 of 8 (cotiued) y Fial iteratio: = (.) (.98978) ((.98978) ) =. ( ) ( ) = (. +.) ( )( ) (.6) = y = = =.986 = = = =.9866 Therefore, correct to four decimal places, y(.6) =.99 A directio field plot from Maple cofirms the behaviour of the solutio to this iitial value problem: [A Ecel spreadsheet file for this solutio is available here.] [A Maple worsheet, from which the plot above was derived, is available here.]
18 ENGI 9 Assigmet Solutios Page 8 of 8 (cotiued) Note that the ODE is Beroulli: dy + y = y d P R ( ) h / h = P d = d = e = e y h / / e R d = e d = e h h + / / w = = e e R d + C = e e + C y The geeral solutio is y = / + Ae + The iitial coditio y () =. leads to A =. The eact value of y (.6) is The RK value is correct to seve decimal places! Retur to the ide of assigmets
ENGI 9420 Lecture Notes 3 - Numerical Methods Page 3.01
ENGI 940 Lecture Notes 3 - Numerical Methods Page 3.0 3. Numerical Methods The majority of equatios of iterest i actual practice do ot admit ay aalytic solutio. Eve equatios as simple as = e ad I = e d
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationMATHEMATICS 9740 (HIGHER 2)
VICTORIA JUNIOR COLLEGE PROMOTIONAL EXAMINATION MATHEMATICS 970 (HIGHER ) Frida 6 Sept 0 8am -am hours Additioal materials: Aswer Paper List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write our ame
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 4 Solutions [Numerical Methods]
ENGI 3 Advaced Calculus or Egieerig Facult o Egieerig ad Applied Sciece Problem Set Solutios [Numerical Methods]. Use Simpso s rule with our itervals to estimate I si d a, b, h a si si.889 si 3 si.889
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationTEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2
TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS 9740 9 Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write your
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationMathematics Extension 1
016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More information( ) ( ) ( ) ( ) ( + ) ( )
LSM Nov. 00 Cotet List Mathematics (AH). Algebra... kow ad use the otatio!, C r ad r.. kow the results = r r + + = r r r..3 kow Pascal's triagle. Pascal's triagle should be eteded up to = 7...4 kow ad
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationSOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,
4. Series Solutios of Differetial Equatios:Special Fuctios 4.. Illustrative examples.. 5. Obtai the geeral solutio of each of the followig differetial equatios i terms of Maclauri series: d y (a dx = xy,
More informationCalculus 2 Test File Spring Test #1
Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationSection A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics
X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationx x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula
NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More informationA) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks
More informationCS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationVICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015
VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationA widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α
Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows
More information7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.
Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud
More informationMathematics: Paper 1
GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationMon Feb matrix inverses. Announcements: Warm-up Exercise:
Math 225-4 Week 6 otes We will ot ecessarily fiish the material from a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More informationCalculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)
Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics
More informationL 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!
aylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. At ay poit i the eighbourhood of =0, the fuctio ca be represeted as a power series of the followig form: X 0 f(a) f() ƒ() f()= ( ) f( ) (
More informationA.1 Algebra Review: Polynomials/Rationals. Definitions:
MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole
More informationNBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?
NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More information3 Show in each case that there is a root of the given equation in the given interval. a x 3 = 12 4
C Worksheet A Show i each case that there is a root of the equatio f() = 0 i the give iterval a f() = + 7 (, ) f() = 5 cos (05, ) c f() = e + + 5 ( 6, 5) d f() = 4 5 + (, ) e f() = l (4 ) + (04, 05) f
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationChapter 2: Numerical Methods
Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationNorthwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet
Northwest High School s Algebra /Hoors Algebra Summer Review Packet This packet is optioal! It will NOT be collected for a grade et school year! This packet has bee desiged to help you review various mathematical
More informationMathematics Extension 2
004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationMathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang
Vectors (9/0 pers) Mathematics 3 Outcome / Revise positio vector, PQ = q p, commuicative, associative, zero vector, multiplicatio by a scalar k, compoets, magitude, uit vector, (i, j, ad k) as well as
More informationU8L1: Sec Equations of Lines in R 2
MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informatione to approximate (using 4
Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationENGI Series Page 5-01
ENGI 344 5 Series Page 5-01 5. Series Cotets: 5.01 Sequeces; geeral term, limits, covergece 5.0 Series; summatio otatio, covergece, divergece test 5.03 Series; telescopig series, geometric series, p-series
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationTaylor expansion: Show that the TE of f(x)= sin(x) around. sin(x) = x - + 3! 5! L 7 & 8: MHD/ZAH
Taylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. A ay poit i the eighbourhood of 0, the fuctio ƒ() ca be represeted by a power series of the followig form: X 0 f(a) f() f() ( ) f( ) ( )
More informationFUNCTIONS (11 UNIVERSITY)
FINAL EXAM REVIEW FOR MCR U FUNCTIONS ( UNIVERSITY) Overall Remiders: To study for your eam your should redo all your past tests ad quizzes Write out all the formulas i the course to help you remember
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 9 Set partitios ad permutatios It could be said that the mai objects of iterest i combiatorics are subsets, partitios ad permutatios of a fiite set. We have spet some time coutig
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationRoot Finding COS 323
Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationHonors Calculus Homework 13 Solutions, due 12/8/5
Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y
More informationSeries: Infinite Sums
Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree
More informationSubject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points
Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More information*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully
X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More information( ) ( ) ( ) notation: [ ]
Liear Algebra Vectors ad Matrices Fudametal Operatios with Vectors Vector: a directed lie segmets that has both magitude ad directio =,,,..., =,,,..., = where 1, 2,, are the otatio: [ ] 1 2 3 1 2 3 compoets
More informationFourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationAfter the completion of this section the student should recall
Chapter III Liear Algebra September 6, 7 6 CHAPTER III LINEAR ALGEBRA Objectives: After the completio of this sectio the studet should recall - the cocept of vector spaces - the operatios with vectors
More informationPhysics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing
Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationNational Quali cations SPECIMEN ONLY
AH Natioal Quali catios SPECIMEN ONLY SQ/AH/0 Mathematics Date Not applicable Duratio hours Total s 00 Attempt ALL questios. You may use a calculator. Full credit will be give oly to solutios which cotai
More informationNATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2
NATIONAL JUNIOR COLLEGE SENIOR HIGH PROMOTIONAL EXAMINATIONS Higher MATHEMATICS 9758 9 September 06 hours Additioal Materials: Aswer Paper List of Formulae (MF6) Cover Sheet READ THESE INSTRUCTIONS FIRST
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationWe will conclude the chapter with the study a few methods and techniques which are useful
Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationLINEAR ALGEBRA. Paul Dawkins
LINEAR ALGEBRA Paul Dawkis Table of Cotets Preface... ii Outlie... iii Systems of Equatios ad Matrices... Itroductio... Systems of Equatios... Solvig Systems of Equatios... 5 Matrices... 7 Matrix Arithmetic
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationMID-YEAR EXAMINATION 2018 H2 MATHEMATICS 9758/01. Paper 1 JUNE 2018
MID-YEAR EXAMINATION 08 H MATHEMATICS 9758/0 Paper JUNE 08 Additioal Materials: Writig Paper, MF6 Duratio: hours DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO READ THESE INSTRUCTIONS FIRST Write
More information