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Analysis of Functions of a Single Variable a Detailed Development
ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 2 For Christy My Light i PREFACE I have written this book primarily for serious and talented mathematics scholars , seniors or first-year graduate students, who by the time they finish their schooling should have had the opportunity to study in some detail the great discoveries of our subject. What did we know and how and when did we know it? I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis. I have tried to write a complete and thorough account of the elementary theories of functions of a single real variable and functions of a single complex variable. Separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to me to be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time. Everyone is taught (told) in school that the area of a circle is given by the formula A = πr2: We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational. Students of natural sciences learn that eiπ = 1 and that sin2 + cos2 = 1: More sophisticated students are taught the Fundamental− Theorem of calculus and the Fundamental Theorem of Algebra. -
Variables in Mathematics Education
Variables in Mathematics Education Susanna S. Epp DePaul University, Department of Mathematical Sciences, Chicago, IL 60614, USA http://www.springer.com/lncs Abstract. This paper suggests that consistently referring to variables as placeholders is an effective countermeasure for addressing a number of the difficulties students’ encounter in learning mathematics. The sug- gestion is supported by examples discussing ways in which variables are used to express unknown quantities, define functions and express other universal statements, and serve as generic elements in mathematical dis- course. In addition, making greater use of the term “dummy variable” and phrasing statements both with and without variables may help stu- dents avoid mistakes that result from misinterpreting the scope of a bound variable. Keywords: variable, bound variable, mathematics education, placeholder. 1 Introduction Variables are of critical importance in mathematics. For instance, Felix Klein wrote in 1908 that “one may well declare that real mathematics begins with operations with letters,”[3] and Alfred Tarski wrote in 1941 that “the invention of variables constitutes a turning point in the history of mathematics.”[5] In 1911, A. N. Whitehead expressly linked the concepts of variables and quantification to their expressions in informal English when he wrote: “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters. it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics.”[6] There is a question, however, about how to describe the use of variables in mathematics instruction and even what word to use for them. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Leibniz and the Infinite
Quaderns d’Història de l’Enginyeria volum xvi 2018 LEIBNIZ AND THE INFINITE Eberhard Knobloch [email protected] 1.-Introduction. On the 5th (15th) of September, 1695 Leibniz wrote to Vincentius Placcius: “But I have so many new insights in mathematics, so many thoughts in phi- losophy, so many other literary observations that I am often irresolutely at a loss which as I wish should not perish1”. Leibniz’s extraordinary creativity especially concerned his handling of the infinite in mathematics. He was not always consistent in this respect. This paper will try to shed new light on some difficulties of this subject mainly analysing his treatise On the arithmetical quadrature of the circle, the ellipse, and the hyperbola elaborated at the end of his Parisian sojourn. 2.- Infinitely small and infinite quantities. In the Parisian treatise Leibniz introduces the notion of infinitely small rather late. First of all he uses descriptions like: ad differentiam assignata quavis minorem sibi appropinquare (to approach each other up to a difference that is smaller than any assigned difference)2, differat quantitate minore quavis data (it differs by a quantity that is smaller than any given quantity)3, differentia data quantitate minor reddi potest (the difference can be made smaller than a 1 “Habeo vero tam multa nova in Mathematicis, tot cogitationes in Philosophicis, tot alias litterarias observationes, quas vellem non perire, ut saepe inter agenda anceps haeream.” (LEIBNIZ, since 1923: II, 3, 80). 2 LEIBNIZ (2016), 18. 3 Ibid., 20. 11 Eberhard Knobloch volum xvi 2018 given quantity)4. Such a difference or such a quantity necessarily is a variable quantity. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Math 35: Real Analysis Winter 2018 Chapter 2
Math 35: Real Analysis Winter 2018 Monday 01/22/18 Lecture 8 Chapter 2 - Sequences Chapter 2.1 - Convergent sequences Aim: Give a rigorous denition of convergence for sequences. Denition 1 A sequence (of real numbers) a : N ! R; n 7! a(n) is a function from the natural numbers to the real numbers. Though it is a function it is usually denoted as a list (an)n2N or (an)n or fang (notation from the book) The numbers a1; a2; a3;::: are called the terms of the sequence. Example 2: Find the rst ve terms of the following sequences an then sketch the sequence a) in a dot-plot. n a) (−1) . n n2N b) 2n . n! n2N c) the sequence (an)n2N dened by a1 = 1; a2 = 1 and an = an−1 + an−2 for all n ≥ 3: (Fibonacci sequence) Math 35: Real Analysis Winter 2018 Monday 01/22/18 Similar as for functions from R to R we have the following denitions for sequences: Denition 3 (bounded sequences) Let (an)n be a sequence of real numbers then a) the sequence (an)n is bounded above if there is an M 2 R, such that an ≤ M for all n 2 N : In this case M is called an upper bound of (an)n. b) the sequence (an)n is bounded below if there is an m 2 R, such that m ≤ an for all n 2 N : In this case m is called a lower bound of (an)n. c) the sequence (an)n is bounded if there is an M~ 2 R, such that janj ≤ M~ for all n 2 N : In this case M~ is called a bound of (an)n. -
Lecture Notes
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 1 18.01 Fall 2006 Unit 1: Derivatives A. What is a derivative? • Geometric interpretation • Physical interpretation • Important for any measurement (economics, political science, finance, physics, etc.) B. How to differentiate any function you know. d • For example: �e x arctan x �. We will discuss what a derivative is today. Figuring out how to dx differentiate any function is the subject of the first two weeks of this course. Lecture 1: Derivatives, Slope, Velocity, and Rate of Change Geometric Viewpoint on Derivatives y Q Secant line Tangent line P f(x) x0 x0+∆x Figure 1: A function with secant and tangent lines The derivative is the slope of the line tangent to the graph of f(x). But what is a tangent line, exactly? 1 Lecture 1 18.01 Fall 2006 • It is NOT just a line that meets the graph at one point. • It is the limit of the secant line (a line drawn between two points on the graph) as the distance between the two points goes to zero. Geometric definition of the derivative: Limit of slopes of secant lines P Q as Q ! P (P fixed). The slope of P Q: Q (x0+∆x, f(x0+∆x)) Secant Line ∆f (x0, f(x0)) P ∆x Figure 2: Geometric definition of the derivative Δf f(x0 + Δx) − f(x0) 0 lim = lim = f (x0) Δx!0 Δx Δx!0 Δx | {z } | {z } \derivative of f at x0 " “difference quotient" 1 Example 1. -
Gottfried Wilhelm Leibniz (1646-1716)
Gottfried Wilhelm Leibniz (1646-1716) • His father, a professor of Philosophy, died when he was small, and he was brought up by his mother. • He learnt Latin at school in Leipzig, but taught himself much more and also taught himself some Greek, possibly because he wanted to read his father’s books. • He studied law and logic at Leipzig University from the age of fourteen – which was not exceptionally young for that time. • His Ph D thesis “De Arte Combinatoria” was completed in 1666 at the University of Altdorf. He was offered a chair there but turned it down. • He then met, and worked for, Baron von Boineburg (at one stage prime minister in the government of Mainz), as a secretary, librarian and lawyer – and was also a personal friend. • Over the years he earned his living mainly as a lawyer and diplomat, working at different times for the states of Mainz, Hanover and Brandenburg. • But he is famous as a mathematician and philosopher. • By his own account, his interest in mathematics developed quite late. • An early interest was mechanics. – He was interested in the works of Huygens and Wren on collisions. – He published Hypothesis Physica Nova in 1671. The hypothesis was that motion depends on the action of a spirit ( a hypothesis shared by Kepler– but not Newton). – At this stage he was already communicating with scientists in London and in Paris. (Over his life he had around 600 scientific correspondents, all over the world.) – He met Huygens in Paris in 1672, while on a political mission, and started working with him. -
Introduction to Real Analysis I
ROWAN UNIVERSITY Department of Mathematics Syllabus Math 01.330 - Introduction to Real Analysis I CATALOG DESCRIPTION: Math 01.330 Introduction to Real Analysis I 3 s.h. (Prerequisites: Math 01.230 Calculus III and Math 03.150 Discrete Math with a grade of C- or better in both courses) This course prepares the student for more advanced courses in analysis as well as introducing rigorous mathematical thought processes. Topics included are: sets, functions, the real number system, sequences, limits, continuity and derivatives. OBJECTIVES: Students will demonstrate the ability to use rigorous mathematical thought processes in the following areas: sets, functions, sequences, limits, continuity, and derivatives. CONTENTS: 1.0 Introduction 1.1 Real numbers 1.1.1 Absolute values, triangle inequality 1.1.2 Archimedean property, rational numbers are dense 1.2 Sets and functions 1.2.1 Set relations, cartesian product 1.2.2 One-to-one, onto, and inverse functions 1.3 Cardinality 1.3.1 One-to-one correspondence 1.3.2 Countable and uncountable sets 1.4 Methods of proof 1.4.1 Direct proof 1.4.2 Contrapositive proof 1.4.3 Proof by contradiction 1.4.4 Mathematical induction 2.0 Sequences 2.1 Convergence 2.1.1 Cauchy's epsilon definition of convergence 2.1.2 Uniqueness of limits 2.1.3 Divergence to infinity 2.1.4 Convergent sequences are bounded 2.2 Limit theorems 2.2.1 Summation/product of sequences 2.2.2 Squeeze theorem 2.3 Cauchy sequences 2.2.3 Convergent sequences are Cauchy sequences 2.2.4 Completeness axiom 2.2.5 Bounded monotone sequences are -
Real Analysis Mathematical Knowledge for Teaching: an Investigation
IUMPST: The Journal. Vol 1 (Content Knowledge), February 2021 [www.k-12prep.math.ttu.edu] ISSN 2165-7874 REAL ANALYSIS MATHEMATICAL KNOWLEDGE FOR TEACHING: AN INVESTIGATION Blain Patterson Virginia Military Institute [email protected] The goal of this research was to investigate the relationship between real analysis content and high school mathematics teaching so that we can ultimately better prepare our teachers to teach high school mathematics. Specifically, I investigated the following research questions. (1) What connections between real analysis and high school mathematics content do teachers make when solving tasks? (2) What real analysis content is potentially used by mathematics teachers during the instructional process? Keywords: Teacher Content Knowledge, Real Analysis, Mathematical Understanding for Secondary Teachers (MUST) Introduction What do mathematics teachers need to know to be successful in the classroom? This question has been at the forefront of mathematics education research for several years. Clearly, high school mathematics teachers should have a deep understanding of the material they teach, such as algebra, geometry, functions, probability, and statistics (Association of Mathematics Teacher Educators, 2017). However, only having knowledge of the content being taught may lead to various pedagogical difficulties, such as primarily focusing on procedural fluency rather than conceptual understanding (Ma, 1999). The general perception by mathematicians and mathematics educators alike is that teachers should have some knowledge of mathematics beyond what they teach (Association of Mathematics Teacher Educators, 2017; Conference Board of the Mathematical Sciences, 2012, Wasserman & Stockton, 2013). The rationale being that concepts from advanced mathematics courses, such as abstract algebra and real analysis, are connected to high school mathematics content (Wasserman, Fukawa-Connelly, Villanueva, Meja-Ramos, & Weber, 2017). -
Mathematical Analysis, Second Edition
PREFACE A glance at the table of contents will reveal that this textbooktreats topics in analysis at the "Advanced Calculus" level. The aim has beento provide a develop- ment of the subject which is honest, rigorous, up to date, and, at thesame time, not too pedantic.The book provides a transition from elementary calculusto advanced courses in real and complex function theory, and it introducesthe reader to some of the abstract thinking that pervades modern analysis. The second edition differs from the first inmany respects. Point set topology is developed in the setting of general metricspaces as well as in Euclidean n-space, and two new chapters have been addedon Lebesgue integration. The material on line integrals, vector analysis, and surface integrals has beendeleted. The order of some chapters has been rearranged, many sections have been completely rewritten, and several new exercises have been added. The development of Lebesgue integration follows the Riesz-Nagyapproach which focuses directly on functions and their integrals and doesnot depend on measure theory.The treatment here is simplified, spread out, and somewhat rearranged for presentation at the undergraduate level. The first edition has been used in mathematicscourses at a variety of levels, from first-year undergraduate to first-year graduate, bothas a text and as supple- mentary reference.The second edition preserves this flexibility.For example, Chapters 1 through 5, 12, and 13 providea course in differential calculus of func- tions of one or more variables. Chapters 6 through 11, 14, and15 provide a course in integration theory. Many other combinationsare possible; individual instructors can choose topics to suit their needs by consulting the diagram on the nextpage, which displays the logical interdependence of the chapters. -
Lesson 4-1 Polynomials
Lesson 4-1: Polynomial Functions L.O: I CAN determine roots of polynomial equations. I CAN apply the Fundamental Theorem of Algebra. Date: ________________ 풏 풏−ퟏ Polynomial in one variable – An expression of the form 풂풏풙 + 풂풏−ퟏ풙 + ⋯ + 풂ퟏ풙 + 풂ퟎ where the coefficients 풂ퟎ,풂ퟏ,…… 풂풏 represent complex numbers, 풂풏 is not zero, and n represents a nonnegative integer. Degree of a polynomial in one variable– The greatest exponent of the variable of the polynomial. Leading Coefficient – The coefficient of the term with the highest degree. Polynomial Function – A function y = P(x) where P(x) is a polynomial in one variable. Zero– A value of x for which f(x)=0. Polynomial Equation – A polynomial that is set equal to zero. Root – A solution of the equation P(x)=0. Imaginary Number – A complex number of the form a + bi where 풃 ≠ ퟎ 풂풏풅 풊 풊풔 풕풉풆 풊풎풂품풊풏풂풓풚 풖풏풊풕 . Note: 풊ퟐ = −ퟏ, √−ퟏ = 풊 Complex Number – Any number that can be written in the form a + bi, where a and b are real numbers and 풊 is the imaginary unit. Pure imaginary Number – The complex number a + bi when a = 0 and 풃 ≠ ퟎ. Fundamental Theorem of Algebra – Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. Corollary to the Fundamental Theorem of Algebra – Every polynomial P(x) of degree n ( n>0) can be written as the product of a constant k (풌 ≠ ퟎ) and n linear factors. 푷(풙) = 풌(풙 − 풓ퟏ)(풙 − 풓ퟐ)(풙 − 풓ퟑ) … .