# Teaching Mathematics Concepts via Computer Algebra Systems

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Caot help solvig this problem Remark :The mai poit here is to fid the sum of 5 ( ), ad the fid ( ) lim. ( ) The fuctio is ot cotiuous whe the value of the limit is ot equal to the value of the fuctio at.the limit ivolves L Hopial s rule but we caot do it before havig the sum of 5 ( ). Distiguish Tricky Problem : - IV. DEFINITE INTEGRAL 4 Page

3 Remark :The fuctio f ( ) must be cotiuous o [a,b] to fid f ( ) d ad its value is the area uder the curve f ( ) o the iterval [a,b]. The fuctio b a f ( ) is ot cotiuous at [,]. By igorig the cotiuity coditio, the result is d l( ) l( ) l( ) l( ) which is udefied value. Distiguish Tricky Problem : d Remark :Completig this without techology would give aswer. Improper Itegral Distiguish Tricky Problem 4:The improper itegral e d. ad this is a icorrect 5 Page

4 Remark 4:Let us graph e by the Matlab commad, So, e e,, the it is easy to complete without techology [ ] e d e d e ( ). Imagiary Itegratio Distiguish Tricky Problem 5: l d Remark 5:To graph of f ( ) l, by 6 Page

5 Based o the output warig provided by Matlab, the solutio refers to igorig the imagiary parts of f ( ) whe ad. Itegratio Test Distiguish Tricky Problem 6:To check the coverget of the series l( ) l(l( )) by the itegratio test, is etremely difficult without techology. It is recommeded to use CAS to make sese of the eample. Remark 6: (I) The result is as = If meas which meas the series is diverget. But the series is coverget whe the itegratio value is a fiite value. (II) The cocept ivolves the defiite itegral which could be discussed ad eplored by CAS. For istace, the area uder a curve i a iterval i its domai. ) R V. DOUBLE INTEGRAL AND POLAR COORDINATES f (, y da is the volume of the solid regio betwee a bouded rectagular or o- rectagular regio R i the plae R ad the surface Z f (, y ). It is the etesio to the defiite itegral but for a cotiuous fuctio of two variables f (, y ). Distiguish Tricky Problem 7: y e y ddy ad e y dyd. 7 Page

6 Remark 7: Some of CAS tried to compute the itegratio of two variable fuctio i the Cartesia coordiates, the importace of trasformig ito the polar coordiate ad the area elemet i it r ad da = rdrd comes to accout. The boud of the itegratio for are foud from the relatio betwee the Cartesia ad the polar coordiates. Noe of the CAS techologies provided a correct solutio. For istace, trasformig ito polar coordiates by CAS, by usig Mathematica the output is: ddy ( e ) The itegratio is possible usig e r rdrd, that ca be doe by o-cas techology. However, ( e ) Teas Istrumet provided the same result as Reversig the itegratio order 6 Distiguish Tricky Problem 8: 4 y cos( ) ddy 8 Page

7 Remark 8: SomeCAS techology caot evaluate some double itegratiowithout reversig it ad ca help after doig so. Net the importace of reversig the itegratio is show: VI. DIFFERENTIAL EQUATION Distiguish Tricky Problem 9:The followig first order differetial equatio: dy y ( y ), y ( ) d Caot solve differetial equatios Remark 9: (I) the two solutio are y, ad y satisfyig the differetial ( e ) ( ) equatio ad its iitial coditio. (II) Without the iitial coditio Matlab gives: Where, C = or C =- by usig the iitial coditio y ( ), therefore the solutio is coicide with VII. CONCLUSION. CAS is ot oly used for solvig time-cosumig mathematics problems but also is used to eplore mathematics cocepts by solvig distiguish tricky problems.. Sometimes CAS will ot help studets i solvig some mathematics problems ad the studets must be aware whe to use ad ot use CAS.. Use of CAS features to focus o mathematics cocepts ad how to adopt the solutio techique avoidig the theoretical calculatio steps. 9 Page

8 4. Mathematical backgroud ad mathematical software skills are ecessary i solvig mathematical problems whe usig CAS techology. 5. Mathematics course descriptios should iclude three parts. First, teachig of mathematics should lik to the studets major. Secod, CAS should be a essetial compoet i mathematical problem solvig. Third, some mathematical problems should relate to the studet s major. 6. A uderstadig of how these tools are adopted ad applied i professioal eviromets is valuable, both i guidig improvemets to these tools ad i idetifyig ew tools which ca aid mathematicias. REFERENCES []. Rashwa, Osama A. (4), Teachig Egieerig Mathematics Dilemma.Joural of Mathematical Scieces ;5-56 []. Johso, J. D. (). Prospective middle grades teachers of mathematics usig had-held CAS techology to create rich mathematical task. Joural of Mathematical Scieces ad Mathematics Educatio. 5; Page

### 62. 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

### C. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions

C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe

### Infinite Sequences and Series

Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

### 10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.

0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece

### Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

### TEACHER 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

### AP 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

### Carleton 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

### A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

### Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Section 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

### NUMERICAL METHODS FOR SOLVING EQUATIONS

Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

### (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

### Fourier 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

### THE 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

### Summary: 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

### Solutions to quizzes Math Spring 2007

to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x

### Algebra 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.

### AP 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

### JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or

### Lesson 10: Limits and Continuity

www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

### Ma 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

### Math 106 Fall 2014 Exam 3.2 December 10, 2014

Math 06 Fall 04 Exam 3 December 0, 04 Determie if the series is coverget or diverget by makig a compariso (DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write

### Fooling Newton s Method

Foolig Newto s Method You might thik that if the Newto sequece of a fuctio coverges to a umber, that the umber must be a zero of the fuctio. Let s look at the Newto iteratio ad see what might go wrog:

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### 7.1 Finding Rational Solutions of Polynomial Equations

Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial

### The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

### INFINITE 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

### MATH 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,

### P.3 Polynomials and Special products

Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

### Seunghee 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

### Using Spreadsheets as a Computational Tool in Teaching Mechanical. Engineering

Proceedigs of the th WSEAS Iteratioal Coferece o COMPUTERS, Vouliagmei, Athes, Greece, July 3-5, 6 (pp35-3) Usig Spreadsheets as a Computatioal Tool i Teachig Mechaical Egieerig AHMADI-BROOGHANI, ZAHRA

### MAT1026 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

### Subject: 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

### The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

### 9.3 Power Series: Taylor & Maclaurin Series

9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0

### AP Calculus BC 2007 Scoring Guidelines Form B

AP Calculus BC 7 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

### Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions

Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group

### Lecture 4 Conformal Mapping and Green s Theorem. 1. Let s try to solve the following problem by separation of variables

Lecture 4 Coformal Mappig ad Gree s Theorem Today s topics. Solvig electrostatic problems cotiued. Why separatio of variables does t always work 3. Coformal mappig 4. Gree s theorem The failure of separatio

### A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios

### A) 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

### CALCULUS II. Sequences and Series. Paul Dawkins

CALCULUS II Sequeces ad Series Paul Dawkis Table of Cotets Preface... ii Sequeces ad Series... 3 Itroductio... 3 Sequeces... 5 More o Sequeces...5 Series The Basics... Series Covergece/Divergece...7 Series

### MAS111 Convergence and Continuity

MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

### Tennessee Department of Education

Teessee Departmet of Educatio Task: Comparig Shapes Geometry O a piece of graph paper with a coordiate plae, draw three o-colliear poits ad label them A, B, C. (Do ot use the origi as oe of your poits.)

### Notes 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

### Representing Functions as Power Series. 3 n ...

Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.

### Diploma Programme. Mathematics HL guide. First examinations 2014

Diploma Programme First eamiatios 014 33 Topic 6 Core: Calculus The aim of this topic is to itroduce studets to the basic cocepts ad techiques of differetial ad itegral calculus ad their applicatio. 6.1

### Math 106 Fall 2014 Exam 3.1 December 10, 2014

Math 06 Fall 0 Exam 3 December 0, 0 Determie if the series is coverget or diverget by makig a compariso DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write Coverget

### Analytic 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

### MEI Casio Tasks for Further Pure

Task Complex Numbers: Roots of Quadratic Equatios. Add a ew Equatio scree: paf 2. Chage the Complex output to a+bi: LpNNNNwd 3. Select Polyomial ad set the Degree to 2: wq 4. Set a=, b=5 ad c=6: l5l6l

### Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### Løsningsførslag 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

### ( ) ( ) ( ) ( ) ( + ) ( )

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

### Complex Numbers Primer

Before I get started o this let me first make it clear that this documet is ot iteded to teach you everythig there is to kow about complex umbers. That is a subject that ca (ad does) take a whole course

### The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

### Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

### ON THE EIGENFUNCTION EXPANSION METHOD FOR THE CALCULATION OF GREEN S FUNCTIONS

July 3. Vol. 4, No. Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF. All rights reserved www.eaas-joural.org ISSN35-869 ON THE EIGENFUNCTION EXPANSION METHOD FOR THE CALCULATION OF GREEN S

### ON THE EIGENFUNCTION EXPANSION METHOD FOR THE CALCULATION OF GREEN S FUNCTIONS

July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 ON THE EIGENFUNCTION EXPANSION METHOD FOR THE CALCULATION OF GREEN S FUNCTIONS

### Lecture 3 The Lebesgue Integral

Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

### CALCULUS II Sequences and Series. Paul Dawkins

CALCULUS II Sequeces ad Series Paul Dawkis Table of Cotets Preface... ii Sequeces ad Series... Itroductio... Sequeces... 5 More o Sequeces... 5 Series The Basics... Series Covergece/Divergece... 7 Series

### UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questios. For the quadratic fuctio show below, the coordiates of its verte are () 0, (), 7 (3) 6, (4) 3, 6. A quadratic

### Solutions to Tutorial 3 (Week 4)

The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/

### Poisson s remarkable calculation - a method or a trick?

Poisso s remarkable calculatio - a method or a trick? Deis Bell 1 Departmet of Mathematics, Uiversity of North Florida 1 UNF Drive, Jacksoville, FL 34, U. S. A. email: dbell@uf.edu The Gaussia fuctio e

### 2 n = n=1 a n is convergent and we let. i=1

Lecture 3 : Series So far our defiitio of a sum of umbers applies oly to addig a fiite set of umbers. We ca exted this to a defiitio of a sum of a ifiite set of umbers i much the same way as we exteded

### An Analysis of a Certain Linear First Order. Partial Differential Equation + f ( x, by Means of Topology

Iteratioal Mathematical Forum 2 2007 o. 66 3241-3267 A Aalysis of a Certai Liear First Order Partial Differetial Equatio + f ( x y) = 0 z x by Meas of Topology z y T. Oepomo Sciece Egieerig ad Mathematics

### Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically

ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab

### Analysis of Experimental Data

Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both

### POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS

Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity

### On an Application of Bayesian Estimation

O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma

### Section 11.8: Power Series

Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

### AP Calculus. Notes and Homework for Chapter 9

AP Calculus Notes ad Homework for Chapter 9 (Mr. Surowski) I do t feel that the textbook does a particularly good job at itroducig the material o ifiite series, power series (Maclauri ad Taylor series),

### CALCULATING FIBONACCI VECTORS

THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet

### Singular Continuous Measures by Michael Pejic 5/14/10

Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

### Confidence Intervals for the Population Proportion p

Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:

### Nernst Equation. Nernst Equation. Electric Work and Gibb's Free Energy. Skills to develop. Electric Work. Gibb's Free Energy

Nerst Equatio Skills to develop Eplai ad distiguish the cell potetial ad stadard cell potetial. Calculate cell potetials from kow coditios (Nerst Equatio). Calculate the equilibrium costat from cell potetials.

### Probability, Expectation Value and Uncertainty

Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

### Math 116 Practice for Exam 3

Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur

### A numerical Technique Finite Volume Method for Solving Diffusion 2D Problem

The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed

### Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

### Sequences of Definite Integrals, Factorials and Double Factorials

47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

### Numerical 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

### Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

### (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

### Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

### IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

### A Block Cipher Using Linear Congruences

Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &

### SUMMARY OF SEQUENCES AND SERIES

SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger

### De la Vallée Poussin Summability, the Combinatorial Sum 2n 1

J o u r a l of Mathematics ad Applicatios JMA No 40, pp 5-20 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper

### sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

### Practice Test Problems for Test IV, with Solutions

Practice Test Problems for Test IV, with Solutios Dr. Holmes May, 2008 The exam will cover sectios 8.2 (revisited) to 8.8. The Taylor remaider formula from 8.9 will ot be o this test. The fact that sums,

### Most 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

### Differentiable Convex Functions

Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for

### Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;

### Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations

Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif

### Math 210A Homework 1

Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called

### Recurrence Relations

Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The