DOCSLIB.ORG

Basic Mathematical Tools Outline Summation Operation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Mathematical Tools Outline Summation Operation Outline I. Summation Operation and Descriptive Statistics Basic Mathematical Tools II. Properties of Linear Functions III. Proportions and Percentages Read Wooldridge, Appendix A IV. Special Functions V. Differential Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 2 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Summation Operation • Summation operator () involves the sum of many numbers. • Property s.1: For any constant c, n • Given a sequence of n numbers cnc {xi; i=1, …, n} i1 • The sum of these numbers • The sum of n constants (c) equals the product of n and c n xi = x1 + x2 + …. + xn i1 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 3 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 4 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation Summation Operation • Property s.2: • Property s.3: If {(xi,yi): i=1, …,n} is a set of n pairs of numbers and a and b are constants, then nn nnn cxii c x ii11 ()axii by a x i b y i iii111 The sum of c times xi equals c time the sum of xi. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 5 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 6 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation Summation Operation • Notes that the sum of ratios is not the ratio of the sums. • Note that the sum of the squares is not the square of the sum. n nn x x 22 ()x n x i ii ii1 ii11 n i1 yi yi i1 • Example: n = 2 2 2 2 • Example: n = 2 x1 + x2 (x1 + x2 ) I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 7 8 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation and Descriptive Statistics: Sample average Descriptive Statistics • When the xi are a sample of data on a particular variable, we • Given a sequence of n numbers {xi; i=1, …, n}, the average call this the sample average or sample mean. or mean can be written as 1 n • Sample average is an example of a descriptive statistic. x xi n i 1 • Sample average is a statistic that describes the central tendency of the set of n points. • Average is computed by adding them up and dividing by n I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 9 10 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Sample median Descriptive Statistics: Sample median • Other measure of central tendency is sample • Steps in finding sample median median. Step 1: order the values of the xi from smallest to largest. • Example: Given numbers, {‐4, 8, 2, 0, 21, ‐10, 18} Step 2: if n is odd, the sample median is the middle number of – Sample mean = 35/7 = 5 the ordered observations. – Sample median = 2 • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 21} Step 3: if n is even, the median is defined to be the average of the two middle values. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 11 12 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Sample median Summation Operation and Descriptive Statistics • If 21 is changed to 42 • Numbers {‐4, 8, 2, 0, 42, ‐10, 18} • Deviations – Sample mean = 56/7 = 8 – Sample median = 2 • Deviations can be found by taking each observation and subtracting off • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 42} the sample average • Sample median: dxxii Good point: it is less sensitive than sample average to changes in the extreme values in a list of numbers. Examples are median housing values or median income. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 13 14 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation and Descriptive Statistics Deviations and Demean Sample Example: n =5 • Properties d1: Given {x ; i=1, …, n}, i x = 6, x = 1, x = ‐2, x = 0, x = 5 The sum of the deviations equal zero. 1 2 3 4 5 nn x ? dxxii0 Demean sample is {4, ‐1, ‐4, ‐2, 3} ii11 n xxi 0 i1 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 15 16 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Algebraic Fact Descriptive Statistics: Algebraic Fact • Properties d2: Given {xi; i=1, …, n}, • Properties d3: Given {(xi,yi): i=1, …,n}, the sum of squared deviations is the sum of squared xi minus It can be shown that n times the squared of sample mean. nn ()()xiixy y xy ii () y ii11 nn 222 nn ()xxii x nx () ()xxyii xynxy ii () ii11 ii11 • Show! I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 17 18 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summary: Summation and Deviation Problem A.1 Summation Deviation A.1 The following table contains monthly housing s.1 d.1 expenditures for 10 families. 2 s.2 d.2 2 (i) Find the average monthly housing expenditure. [ans.] s.3 d.3 (ii) Find the median monthly housing expenditure. [ans.] I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 19 20 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Housing Problem A.1 continue Solution A.1 (i) Family Expenditures 1 300 • (iii) If monthly housing expenditures were measured in 2 440 (i) $566. 3 350 , rather than in dollars, what hundreds of dollars 4 1100 n would be the average and median expenditures? 1 5 640 x x [ans.] i 6 480 n i1 7 450 • (iv) Suppose that family number 8 increases its monthly 8 700 housing expenditure to $900 dollars, but the 9 670 expenditures of all other families remain the same. 10 530 Sum 5,660 Compute the average and median housing Mean 566 expenditures. [ans.] I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 21 22 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Solution A.1 (ii) Solution A.1 (iii) (ii) 505 (iii) Steps in finding sample mean • $566 and $505 (in dollars), respectively • Step 1: order the values of the xi from smallest to largest. • 5.66 and 5.05 (in hundreds of dollars), {300, 350, 440, 450, 480, 530, 640, 670, 700, 1100,} respectively. • Step 3: if n is even, the median is defined to be the average of the two middle values. The two middle numbers are 480 and 530; when these are averaged, we obtain 505, or $505. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 23 24 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Solution A.1 (iv) Housing Housing Family Expenditures Expenditures (iv) 1 300 300 A linear function can be written as • 2 440 440 The average increases to y = 0 + 1x $586 from $566. 3 350 350 4 1100 1100 • y and x are variables; • while the median is 5 640 640 • 0 and 1 are parameters; unchanged ($505). 6 480 480 – 0 is called the intercept; 7 450 450 – 1 is called the slope. {300, 350, 440, 450, 8 700 900 480, 530, 640, 900, 9 670 670 • We say that y is a linear function of x.
Load more

Recommended publications
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J
    Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J. Keely. All Rights Reserved. Mathematical expressions can be typed online in a number of ways including plain text, ASCII codes, HTML tags, or using an equation editor (see Writing Mathematical Notation Online for overview). If the application in which you are working does not have an equation editor built in, then a common option is to write expressions horizontally in plain text. In doing so you have to format the expressions very carefully using appropriately placed parentheses and accurate notation. This document provides examples and important cautions for writing mathematical expressions in plain text. Section 1. How to Write Exponents Just as on a graphing calculator, when writing in plain text the caret key ^ (above the 6 on a qwerty keyboard) means that an exponent follows. For example x2 would be written as x^2. Example 1a. 4xy23 would be written as 4 x^2 y^3 or with the multiplication mark as 4*x^2*y^3. Example 1b. With more than one item in the exponent you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x2n must be written as x^(2n) and NOT as x^2n. Writing x^2n means xn2 . Example 1c. When using the quotient rule of exponents you often have to perform subtraction within an exponent. In such cases you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x5 The middle step of ==xx52− 3 must be written as x^(5-2) and NOT as x^5-2 which means x5 − 2 .
    [Show full text]
  • Appendix a Short Course in Taylor Series
    Appendix A Short Course in Taylor Series The Taylor series is mainly used for approximating functions when one can identify a small parameter. Expansion techniques are useful for many applications in physics, sometimes in unexpected ways. A.1 Taylor Series Expansions and Approximations In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. A.2 Definition A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x ¼ a is given by; f 00ðÞa f 3ðÞa fxðÞ¼faðÞþf 0ðÞa ðÞþx À a ðÞx À a 2 þ ðÞx À a 3 þÁÁÁ 2! 3! f ðÞn ðÞa þ ðÞx À a n þÁÁÁ ðA:1Þ n! © Springer International Publishing Switzerland 2016 415 B. Zohuri, Directed Energy Weapons, DOI 10.1007/978-3-319-31289-7 416 Appendix A: Short Course in Taylor Series If a ¼ 0, the expansion is known as a Maclaurin Series. Equation A.1 can be written in the more compact sigma notation as follows: X1 f ðÞn ðÞa ðÞx À a n ðA:2Þ n! n¼0 where n ! is mathematical notation for factorial n and f(n)(a) denotes the n th derivation of function f evaluated at the point a.
    [Show full text]
  • How to Write Mathematical Papers
    HOW TO WRITE MATHEMATICAL PAPERS BRUCE C. BERNDT 1. THE TITLE The title of your paper should be informative. A title such as “On a conjecture of Daisy Dud” conveys no information, unless the reader knows Daisy Dud and she has made only one conjecture in her lifetime. Generally, titles should have no more than ten words, although, admittedly, I have not followed this advice on several occasions. 2. THE INTRODUCTION The Introduction is the most important part of your paper. Although some mathematicians advise that the Introduction be written last, I advocate that the Introduction be written first. I find that writing the Introduction first helps me to organize my thoughts. However, I return to the Introduction many times while writing the paper, and after I finish the paper, I will read and revise the Introduction several times. Get to the purpose of your paper as soon as possible. Don’t begin with a pile of notation. Even at the risk of being less technical, inform readers of the purpose of your paper as soon as you can. Readers want to know as soon as possible if they are interested in reading your paper or not. If you don’t immediately bring readers to the objective of your paper, you will lose readers who might be interested in your work but, being pressed for time, will move on to other papers or matters because they do not want to read further in your paper. To state your main results precisely, considerable notation and terminology may need to be introduced.
    [Show full text]
  • 9.4 Arithmetic Series Notes
    9.4 Arithmetic Series Notes Pre­AP Algebra 2 9.4 Arithmetic Series *Objective: Define arithmetic series and find their sums When you know two terms and the number of terms in a finite arithmetic sequence, you can find the sum of the terms. A series is the indicated sum of the terms of a sequence. A finite series has a first terms and a last term. An infinite series continues without end. Finite Sequence Finite Series 6, 9, 12, 15, 18 6 + 9 + 12 + 15 + 18 = 60 Infinite Sequence Infinite Series 3, 7, 11, 15, ... 3 + 7 + 11 + 15 + ... An arithmetic series is a series whose terms form an arithmetic sequence. When a series has a finite number of terms, you can use a formula involving the first and last term to evaluate the sum. The sum Sn of a finite arithmetic series a1 + a2 + a3 + ... + an is n Sn = /2 (a1 + an) a1 : is the first term an : is the last term (nth term) n : is the number of terms in the series Finding the Sum of a finite arithmetic series Ex1) a. What is the sum of the even integers from 2 to 100 b. what is the sum of the finite arithmetic series: 4 + 9 + 14 + 19 + 24 + ... + 99 c. What is the sum of the finite arithmetic series: 14 + 17 + 20 + 23 + ... + 116? Using the sum of a finite arithmetic series Ex2) A company pays $10,000 bonus to salespeople at the end of their first 50 weeks if they make 10 sales in their first week, and then improve their sales numbers by two each week thereafter.
    [Show full text]
  • Summation and Table of Finite Sums
    SUMMATION A!D TABLE OF FI1ITE SUMS by ROBERT DELMER STALLE! A THESIS subnitted to OREGON STATE COLlEGE in partial fulfillment of the requirementh for the degree of MASTER OF ARTS June l94 APPROVED: Professor of Mathematics In Charge of Major Head of Deparent of Mathematics Chairman of School Graduate Committee Dean of the Graduate School ACKOEDGE!'T The writer dshes to eicpreßs his thanks to Dr. W. E. Mime, Head of the Department of Mathenatics, who has been a constant guide and inspiration in the writing of this thesis. TABLE OF CONTENTS I. i Finite calculus analogous to infinitesimal calculus. .. .. a .. .. e s 2 Suniming as the inverse of perfornungA............ 2 Theconstantofsuirrnation......................... 3 31nite calculus as a brancn of niathematics........ 4 Application of finite 5lflITh1tiOfl................... 5 II. LVELOPMENT OF SULTION FORiRLAS.................... 6 ttethods...........................a..........,.... 6 Three genera]. sum formulas........................ 6 III S1ThATION FORMULAS DERIVED B TIlE INVERSION OF A Z FELkTION....,..................,........... 7 s urnmation by parts..................15...... 7 Ratlona]. functions................................ Gamma and related functions........,........... 9 Ecponential and logarithrnic functions...... ... Thigonoretric arÎ hyperbolic functons..........,. J-3 Combinations of elementary functions......,..... 14 IV. SUMUATION BY IfTHODS OF APPDXIMATION..............,. 15 . a a Tewton s formula a a a S a C . a e a a s e a a a a . a a 15 Extensionofpartialsunmation................a... 15 Formulas relating a sum to an ifltegral..a.aaaaaaa. 16 Sumfromeverym'thterm........aa..a..aaa........ 17 V. TABLE OFST.Thß,..,,..,,...,.,,.....,....,,,........... 18 VI. SLThMTION OF A SPECIAL TYPE OF POER SERIES.......... 26 VI BIBLIOGRAPHY. a a a a a a a a a a . a . a a a I a s .
    [Show full text]
  • List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
    List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX.
    [Show full text]
  • MATHEMATICAL INDUCTION SEQUENCES and SERIES
    MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES John J O'Connor 2009/10 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 13 Power Series 22 Taylor Series 24 Summary 29 Mathematician's pictures 30 Exercises on these topics are on the following pages: Mathematical Induction 8 Sequences 13 Series 21 Power Series 24 Taylor Series 28 Solutions to the exercises in this booklet are available at the Web-site: www-history.mcs.st-andrews.ac.uk/~john/MISS_solns/ 1 Mathematical induction This is a method of "pulling oneself up by one's bootstraps" and is regarded with suspicion by non-mathematicians. Example Suppose we want to sum an Arithmetic Progression: 1+ 2 + 3 +...+ n = 1 n(n +1). 2 Engineers' induction € Check it for (say) the first few values and then for one larger value — if it works for those it's bound to be OK. Mathematicians are scornful of an argument like this — though notice that if it fails for some value there is no point in going any further. Doing it more carefully: We define a sequence of "propositions" P(1), P(2), ... where P(n) is "1+ 2 + 3 +...+ n = 1 n(n +1)" 2 First we'll prove P(1); this is called "anchoring the induction". Then we will prove that if P(k) is true for some value of k, then so is P(k + 1) ; this is€ c alled "the inductive step". Proof of the method If P(1) is OK, then we can use this to deduce that P(2) is true and then use this to show that P(3) is true and so on.
    [Show full text]
  • 12-3 – Infinite Sequences and Series
    Notes 12-3 and 12-5: Infinite Sequences and Series, Summation Notation I. Infinite Sequences and Series A. Concept and Formula Consider the following sequence: 16, 8, 4, …. • What kind of sequence is it? • Find the 18th term. • Now find the 20th, 25th, and 50th. • So …the larger n is the more the sequence approaches what? • W=When 푟 < 1, as n increases, the terms of the sequence will decrease, and ultimately approach zero. Zero is the limit of the terms in this sequence. • What will happen to the Sum of the Series? It will reach a limit as well. The sum, Sn, of an infinite geometric series for a1 Sn which 푟 < 1 is given by 1 r the following formula: Notice that 푟 < 1. If 푟 > 1, the sum does not exist. The series must also be geometric. Why? 3 Ex 1: Find the sum of the series 21 − 3 + − ⋯ 7 푎 −3 −1 푟 = 2 = = 푎1 21 7 푎 푆 = 1 푛 1 − 푟 21 푆 = 푛 1 1 − (− ) 7 푆푛 = 18.375 Ex 2: Find the sum of the series 60 + 24 + 9.6 … 푎 24 푟 = 2 = = .4 푎1 60 푎 푆 = 1 푛 1 − 푟 60 푆 = 푛 1 − .4 푆푛 = 100 B. Applications Ex 1: Francisco designs a toy with a 푎 푆 = 1 rotary flywheel that rotates at a 푛 1 − 푟 maximum speed of 170 revolutions per minute. Suppose the flywheel is 170 푆 = operating at its maximum speed for one 푛 2 1 − minute and then the power supply to the 5 toy is turned off.
    [Show full text]
  • High School Mathematics Glossary Pre-Calculus
    High School Mathematics Glossary Pre-calculus English-Chinese BOARD OF EDUCATION OF THE CITY OF NEW YORK Board of Education of the City of New York Carol A. Gresser President Irene H. ImpeUizzeri Vice President Louis DeSario Sandra E. Lerner Luis O. Reyes Ninfa Segarra-Velez William C. Thompson, ]r. Members Tiffany Raspberry Student Advisory Member Ramon C. Cortines Chancellor Beverly 1. Hall Deputy Chancellor for Instruction 3195 Ie is the: policy of the New York City Board of Education not to discriminate on (he basis of race. color, creed. religion. national origin. age. handicapping condition. marital StatuS • .saual orientation. or sex in itS eduacional programs. activities. and employment policies. AS required by law. Inquiries regarding compiiulce with appropriate laws may be directed to Dr. Frederick..6,.. Hill. Director (Acting), Dircoetcr, Office of Equal Opporrurury. 110 Livingston Screet. Brooklyn. New York 11201: or Director. Office for o ..·ij Rights. Depa.rtmenc of Education. 26 Federal PJaz:t. Room 33- to. New York. 'Ne:w York 10278. HIGH SCHOOL MATHEMATICS GLOSSARY PRE-CALCULUS ENGLISH - ClllNESE ~ 0/ * ~l.tt Jf -taJ ~ 1- -jJ)f {it ~t ~ -ffi 'ff Chinese!Asian Bilingual Education Technical Assistance Center Division of Bilingual Education Board of Education of the City of New York 1995 INTRODUCTION The High School English-Chinese Mathematics Glossary: Pre­ calculus was developed to assist the limited English proficient Chinese high school students in understanding the vocabulary that is included in the New York City High School Pre-calculus curricu lum. To meet the needs of the Chinese students from different regions, both traditional and simplified character versions are included.
    [Show full text]
  • Exact Formulas for the Generalized Sum-Of-Divisors Functions)
    Exact Formulas for the Generalized Sum-of-Divisors Functions Maxie D. Schmidt School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 USA [email protected] [email protected] Abstract We prove new exact formulas for the generalized sum-of-divisors functions, σα(x) := dα. The formulas for σ (x) when α C is fixed and x 1 involves a finite sum d x α ∈ ≥ over| all of the prime factors n x and terms involving the r-order harmonic number P ≤ sequences and the Ramanujan sums cd(x). The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when r > 1 and are related to the generalized Bernoulli numbers when r 0 is integer-valued. ≤ A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, Φ (q), which completely factorize the Lambert series terms (1 qn) 1 into n − − irreducible polynomials in q. We focus on the computational aspects of these exact ex- pressions, including their interplay with experimental mathematics, and comparisons of the new formulas for σα(n) and the summatory functions n x σα(n). ≤ 2010 Mathematics Subject Classification: Primary 30B50; SecondaryP 11N64, 11B83. Keywords: Divisor function; sum-of-divisors function; Lambert series; cyclotomic polynomial. Revised: April 23, 2019 arXiv:1705.03488v5 [math.NT] 19 Apr 2019 1 Introduction 1.1 Lambert series generating functions We begin our search for interesting formulas for the generalized sum-of-divisors functions, σα(n) for α C, by expanding the partial sums of the Lambert series which generate these functions in the∈ form of [5, 17.10] [13, 27.7] § § nαqn L (q) := = σ (m)qm, q < 1.
    [Show full text]
  • The Summation Symbol
    THE SUMMATION SYMBOL Summary 1. Simple sum .............................................................................................................................. 1 2. Double sum .............................................................................................................................. 5 3. Double index ............................................................................................................................ 5 1. Simple sum The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum. For example, the sum of first whole numbers can be represented in the following manner: 1 2 3 ⋯ . More generally, the expression ∑ represents the sum of n terms ⋯. Example 1 Given 3, 5, 6, 2 7. Evaluate ∑ and ∑ . Solution: In the first sum, the index " i " varies from 1 to 5. We must therefore include the 5 terms in the sum. 3562723 In the second case, the index " i "varies from 2 to 4. Only the terms , and must therefore be considered. 56213 When we use the summation symbol, it is useful to remember the following rules: Example 2 Given 3, 5, 6, 2 7 et 2, 8, 3, 1 6. Verify the three preceding rules with the following sums : 4 4 Page 2 of 6 Solution : 4 4 4 4 4 4 4345464247 92 and 4 435627 42392 b) 4444445420 c) 32 58 63 21 76 43 And 35⋯7 28⋯6 232043 Attention : We must neither confound the expression with Page 3 of 6 nor the expression with Example 3 Given 3, 5, 6, 2 7 and 2, 8, 3, 1 6. a) 3 5 6 2 7 123 and 35627 23 52 123 b) 32 58 ⋯76 6 40 ⋯ 42 108 and 35⋯7 28⋯6 23 20 460 108 Page 4 of 6 2.
    [Show full text]