*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully
|
|
- Clementine Jennings
- 6 years ago
- Views:
Transcription
1 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. Sectio A assesses the Uits Numerical Aalysis ad Sectio B assesses the Uit Mathematics for Applied Mathematics. Full credit will be give oly where the solutio cotais appropriate workig.. Numerical Aalysis Formulae ca be foud o pages two ad three of this Questio Paper. LI 0/70 6/70 *X0/70*
2 NUMERICAL ANALYSIS FORMULAE Taylor polyomials For a fuctio f, defied ad times differetiable for values of x close to a, the Taylor polyomial of degree is f () a f () a f( a) + f ( a)( x a) + ( x a) ( x a)!! ( ) f () a f () a ad fa ( + h) fa ( ) + f ( ah ) + h h!! ( ) Newto forward differece iterpolatio formula p p p fp = f0 + f0 + f0 + f Lagrage iterpolatio formula p ( x) = L ( x) y i i i= 0 ( x x0)( x x)... ( x xi )( x xi+ )... ( x x) where Li ( x) = ( x x )( x x )... ( x x )( x x )... ( x x ) i 0 i i i i i+ i Newto-Raphso formula For a equatio f(x) = 0, with x 0 give, + = fx ( ) x x f ( x ) Composite trapezium rule h fxdx ( ) = { f + f + ( f+ f f )} + E b 0 a a with h = ad fk = f( a+ kh) where E is (approximately) bouded by a (i) hm with f ( x ) M for a x b a or (ii) D with f D for a x b [X0/70] Page two
3 Simpso s composite rule h fxdx ( ) = { f + f + ( f+ f f ) + ( f + f f )} + E b 0 a with h = a ad fk = f( a+ kh) where E is (approximately) bouded by (i) a hm (iv) with f ( x ) M for a x b 80 or (ii) a D with f 80 D for a x b Richardso s formula Trapezium rule: Simpso s rule: I I + ( I I ) I I + ( I I ) Euler s method dy For a equatio = fxy (, ) with ( x0, y0) give, dx y = y + hf( x, y ) + Predictor-Corrector method: Euler-Trapezium Rule y = y + hf( x, y ) + y+ = y + h ( f ( x, y ) + f ( x+, y+ )) [Tur over [X0/70] Page three
4 Sectio A (Numerical Aalysis ad ) Aswer all the questios. A. The followig data is available for a fuctio f: x f(x) Use the Lagrage iterpolatio formula to obtai a quadratic approximatio to f(x), simplifyig your aswer. Marks A. The polyomial p is the Taylor polyomial of degree three for the fuctio f ear x = π/. For the fuctio f(x) = cos x, express p(π/ + h) i the form c 0 + c h + c h + c h. Estimate the value of cos 6 usig the secod degree approximatio, statig your aswer rouded to four decimal places. Write dow the pricipal trucatio error term for this secod degree approximatio ad calculate its value. Hece state the secod degree approximatio to a appropriate degree of accuracy. A. Derive the Newto forward differece formula of degree two to fit a polyomial through the poits (x 0, f(x 0 )), (x, f(x )), (x, f(x )). A. The followig data (accurate to the degree implied) ad differece table are available for a fuctio f. i x i f(x i ) f f f f (a) Calculate the maximum roudig error i f 0. (b) (c) (d) Idetify the value of f i this table. State a feature of the table which suggests that a third degree polyomial would be a good approximatio for this fuctio. Use the Newto forward differece formula of degree three to estimate f(. 6), workig to four decimal places. [X0/70] Page four
5 A. Use the Jacobi iterative procedure with x = x = x = 0 as a first approximatio to solve the followig equatios correct to two decimal places. 8x + 0 x + 0 8x = 0 6x + 9x 0 6x = 7 0 x + 0x = 79 Marks A6. Use sythetic divisio to obtai (without roudig) the quotiet Q(x) ad remaider R whe the polyomial f(x) = x + x x +. is divided by g(x) = x Give that all coefficiets are exact except for the costat terms i f(x) ad g(x), which are rouded to the degree of accuracy implied, determie to two decimal places the maximum possible value of R. (You may assume that f(x) is icreasig o the iterval [0. 7, 0. 9].) dy x A7. The differetial equatio = e ( x y ) with y() = is to be solved umerically. dx Use the predictor-corrector method with Euler s method as predictor ad the trapezium rule as corrector to obtai a solutio of this equatio at x =.. Use a step size h = 0. ad oe applicatio of the corrector o each step. Perform the calculatios usig four decimal place accuracy. 6 A8. Gaussia elimiatio with partial pivotig is used to obtai the iverse of the matrix A ( ) 6 7 = 6 0. Durig the process, the followig augmeted matrix is obtaied. ( ) Complete the determiatio of the iverse of A, givig all elemets i the iverse matrix to two decimal places. Explai what is meat by a ill-coditioed matrix. Is A ill-coditioed? Give a reaso for your aswer. Why is the use of partial pivotig importat i dealig with a ill-coditioed matrix? 6 [Tur over [X0/70] Page five
6 A9. The equatio f(x) = 0 has a root close to x = x 0. Use the Taylor series expasio of f(x) about x = x 0 to derive the Newto-Raphso method of solutio of f(x) = 0. It is give that the equatio x x + = 0, has three distict real roots. Use the Newto-Raphso method to determie the root ear x = correct to two decimal places. Show that the iterative scheme may be suitable to determie the root ear x = x x + = Usig x = 0. as a startig value ad recordig successive iterates to three decimal places, use Simple Iteratio to determie this root to two decimal places. The third root lies i the iterval [.,. 0]. Use three applicatios of the bisectio method to determie a more accurate estimate of the iterval i which this root lies. Marks A0. The fuctio f is defied by f(x) = x l( + x), x 0. (a) Use Simpso s rule with two strips ad the composite Simpso s rule with four strips to obtai two estimates I ad I respectively for the itegral I = f( x) dx. Perform the calculatios usig four decimal places. (iv) ( x + x+ 6) (b) It is give that for this fuctio f ( x) = ad that f (v) (x) ( x + ) has o zero o the iterval [, ]. Use this iformatio to obtai a estimate of the maximum trucatio error i I. Hece state the value of I to a suitable accuracy. (c) Establish Richardso s formula to improve the accuracy of Simpso s rule by iterval halvig. Use Richardso extrapolatio to obtai a improved estimate for I based o the values of I ad I obtaied i part (a) of the questio. [END OF SECTION A] [X0/70] Page six
7 Sectio B (Mathematics for Applied Mathematics) Aswer all the questios. Marks B. Calculate A where A = ( 0 ). Hece solve the system of equatios x + y = x + y + z = x + y + z =. B. Give that y = l( + si x), d y where 0 < x < π, show that = dx + si. x B. Defie S = r,. Write dow formulae for S ad S +. r= Obtai a formula for (). B. Solve the differetial equatio dy cos x y, dx = give that y > 0 ad that y = whe x = 0. B. Use the substitutio + x = u to obtai x + x dx. B6. (a) (b) (c) 0 x Evaluate xe dx. x Use part ( a) to evaluate x e dx. 0 x Hece obtai (x + x) e dx. 0 [X0/70] [END OF SECTION B] [END OF QUESTION PAPER] Page seve
8 [BLANK PAGE]
Section 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 information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
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 informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationFind quadratic function which pass through the following points (0,1),(1,1),(2, 3)... 11
Adrew Powuk - http://www.powuk.com- Math 49 (Numerical Aalysis) Iterpolatio... 4. Polyomial iterpolatio (system of equatio)... 4.. Lier iterpolatio... 5... Fid a lie which pass through (,) (,)... 8...
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
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 informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
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 informationNUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK
NUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK For this piece of coursework studets must use the methods for umerical itegratio they meet i the Numerical Methods module
More informationUnit 4: Polynomial and Rational Functions
48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad
More informationLecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods
Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace
More informationMath 113 (Calculus 2) Section 12 Exam 4
Name: Row: Math Calculus ) Sectio Exam 4 8 0 November 00 Istructios:. Work o scratch paper will ot be graded.. For questio ad questios 0 through 5, show all your work i the space provided. Full credit
More informationME NUMERICAL METHODS Fall 2007
ME - 310 NUMERICAL METHODS Fall 2007 Group 02 Istructor: Prof. Dr. Eres Söylemez (Rm C205, email:eres@metu.edu.tr ) Class Hours ad Room: Moday 13:40-15:30 Rm: B101 Wedesday 12:40-13:30 Rm: B103 Course
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use
More informationPC5215 Numerical Recipes with Applications - Review Problems
PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 9 Coimisiú a Scrúduithe Stáit State Examiatios Commissio LEAVING CERTIFICATE EXAMINATION, 006 MATHEMATICS HIGHER LEVEL PAPER 1 ( 00 marks ) THURSDAY, 8 JUNE MORNING, 9:0 to 1:00 Attempt SIX QUESTIONS
More informationCARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION EXAMINATION ADDITIONAL MATHEMATICS. Paper 02 - General Proficiency
TEST CODE 01254020 FORM TP 2015037 MAY/JUNE 2015 CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION CERTIFICATE@ EXAMINATION ADDITIONAL MATHEMATICS Paper 02 - Geeral Proficiecy 2 hours 40 miutes
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 information(5x 7) is. 63(5x 7) 42(5x 7) 50(5x 7) BUSINESS MATHEMATICS (Three hours and a quarter)
BUSINESS MATHEMATICS (Three hours ad a quarter) (The first 5 miutes of the examiatio are for readig the paper oly. Cadidate must NOT start writig durig this time). ------------------------------------------------------------------------------------------------------------------------
More informationTaylor Series (BC Only)
Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor
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 informationCALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve
More information(A) 0 (B) (C) (D) (E) 2.703
Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released
More informationHOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
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 informationMATH 129 FINAL EXAM REVIEW PACKET (Revised Spring 2008)
MATH 9 FINAL EXAM REVIEW PACKET (Revised Sprig 8) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial exam, but they will
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 informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationMath 116 Practice for Exam 3
Math 6 Practice for Exam Geerated October 0, 207 Name: SOLUTIONS Istructor: Sectio Number:. This exam has 7 questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationMath 116 Second Midterm November 13, 2017
Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationLøsningsførslag i 4M
Norges tekisk aturviteskapelige uiversitet Istitutt for matematiske fag Side 1 av 6 Løsigsførslag i 4M Oppgave 1 a) A sketch of the graph of the give f o the iterval [ 3, 3) is as follows: The Fourier
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 8 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayaavaam Road 5758 QUESTION BANK (DESCRIPTIVE Subject with Code : (6HS6 Course & Brach: B.Tech AG Year & Sem: II-B.Tech& I-Sem
More informationSection 13.3 Area and the Definite Integral
Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2017
Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationEdexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary
Cetre No. Cadidate No. Surame Sigature Paper Referece(s) 6667/0 Edexcel GCE Further Pure Mathematics FP Advaced/Advaced Subsidiary Moday 28 Jauary 203 Morig Time: hour 30 miutes Materials required for
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
More informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationAP Calculus BC 2005 Scoring Guidelines
AP Calculus BC 5 Scorig Guidelies The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad
More informationMath 128A: Homework 1 Solutions
Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
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 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 informationCOMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More informationPhysicsAndMathsTutor.com
PhysicsAdMathsTutor.com physicsadmathstutor.com Jue 005 4. f(x) = 3e x 1 l x, x > 0. (a) Differetiate to fid f (x). (3) The curve with equatio y = f(x) has a turig poit at P. The x-coordiate of P is α.
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 informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
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 informationFall 2018 Exam 3 HAND IN PART 0 10 PIN: 17 INSTRUCTIONS
MARK BOX problem poits HAND IN PART 0 10 1 10 2 5 NAME: Solutios 3 10 PIN: 17 4 16 65=13x5 % 100 INSTRUCTIONS This exam comes i two parts. (1) HAND IN PART. Had i oly this part. (2) STATEMENT OF MULTIPLE
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success
More informationENGI 9420 Engineering Analysis Assignment 3 Solutions
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
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
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 informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
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 informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationMathematics 3 Curs /Q1 - First exam. 30/10/14 Group M1 Lecturer: Yolanda Vidal
Mathematics Curs 2014-2015/Q1 - First exam. 0/10/14 Group M1 Lecturer: Yolada Vidal Name: Calculator: 1. [ poits] You are desigig a spherical tak to hold water for a small village i a developig coutry.
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationMTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1
MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces
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 informationMH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME
MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationME 501A Seminar in Engineering Analysis Page 1
Accurac, Stabilit ad Sstems of Equatios November 0, 07 Numerical Solutios of Ordiar Differetial Equatios Accurac, Stabilit ad Sstems of Equatios Larr Caretto Mecaical Egieerig 0AB Semiar i Egieerig Aalsis
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 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 informationMath 220B Final Exam Solutions March 18, 2002
Math 0B Fial Exam Solutios March 18, 00 1. (1 poits) (a) (6 poits) Fid the Gree s fuctio for the tilted half-plae {(x 1, x ) R : x 1 + x > 0}. For x (x 1, x ), y (y 1, y ), express your Gree s fuctio G(x,
More informationIYGB. Special Extension Paper E. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
YGB Special Extesio Paper E Time: 3 hours 30 miutes Cadidates may NOT use ay calculator. formatio for Cadidates This practice paper follows the Advaced Level Mathematics Core ad the Advaced Level Further
More informationNATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 1 EXAMINATION ADVANCED CALCULUS II. November 2003 Time allowed :
NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER EXAMINATION 003-004 MA08 ADVANCED CALCULUS II November 003 Time allowed : hours INSTRUCTIONS TO CANDIDATES This examiatio paper cosists of TWO
More informationMath 116 Final Exam December 12, 2014
Math 6 Fial Exam December 2, 24 Name: EXAM SOLUTIONS Istructor: Sectio:. Do ot ope this exam util you are told to do so. 2. This exam has 4 pages icludig this cover. There are 2 problems. Note that the
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
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 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 information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationPRACTICE FINAL/STUDY GUIDE SOLUTIONS
Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)
More informationMath 122 Test 3 - Review 1
I. Sequeces ad Series Math Test 3 - Review A) Sequeces Fid the limit of the followig sequeces:. a = +. a = l 3. a = π 4 4. a = ta( ) 5. a = + 6. a = + 3 B) Geometric ad Telescopig Series For the followig
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 informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More information