DOCSLIB.ORG

Introduction to Real Analysis, Volume II

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Real Analysis, Volume II Basic Analysis II Introduction to Real Analysis, Volume II by Jiríˇ Lebl June 10, 2020 (version 2.3) 2 Typeset in LATEX. Copyright ©2012–2020 Jiríˇ Lebl This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Derivative works must be prominently marked as such. During the writing of these notes, the author was in part supported by NSF grant DMS-1362337. The date is the main identifier of version. The major version / edition number is raised only if there have been substantial changes. Each volume has its own version number. See https://www.jirka.org/ra/ for more information (including contact information, possible updates and errata). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/ra Contents Introduction 5 8 Several Variables and Partial Derivatives 7 8.1 Vector spaces, linear mappings, and convexity ....................7 8.2 Analysis with vector spaces .............................. 19 8.3 The derivative ..................................... 32 8.4 Continuity and the derivative ............................. 42 8.5 Inverse and implicit function theorems ........................ 47 8.6 Higher order derivatives ................................ 56 9 One-dimensional Integrals in Several Variables 61 9.1 Differentiation under the integral ........................... 61 9.2 Path integrals ..................................... 66 9.3 Path independence .................................. 77 10 Multivariable Integral 85 10.1 Riemann integral over rectangles ........................... 85 10.2 Iterated integrals and Fubini theorem ......................... 96 10.3 Outer measure and null sets .............................. 101 10.4 The set of Riemann integrable functions ....................... 109 10.5 Jordan measurable sets ................................ 113 10.6 Green’s theorem .................................... 117 10.7 Change of variables .................................. 121 11 Functions as Limits 125 11.1 Complex numbers ................................... 125 11.2 Swapping limits .................................... 130 11.3 Power series and analytic functions .......................... 138 11.4 The complex exponential and the trigonometric functions .............. 147 11.5 Fundamental theorem of algebra ........................... 153 11.6 Equicontinuity and the Arzelà–Ascoli theorem .................... 156 11.7 The Stone–Weierstrass theorem ............................ 162 11.8 Fourier series ..................................... 174 Further Reading 187 4 CONTENTS Index 189 List of Notation 193 Introduction About this book This second volume of “Basic Analysis” is meant to be a seamless continuation. The chapters are numbered to start where the first volume left off. The book started with my notes for a second- semester undergraduate analysis at University of Wisconsin—Madison in 2012, where I used my notes together with Rudin’s book. The choice of some of the material and some of the proofs are very similar to Rudin, though I do try to provide more detail and context. In 2016, I taught a second-semester undergraduate analysis at Oklahoma State University and heavily modified and cleaned up the notes, this time using them as the main text. In 2018, I taught this course again, this time adding chapter 11 (which I originally wrote for the Wisconsin course). I plan to eventually add some more topics. I will try to preserve the current numbering in subsequent editions as always. The new topics I have planned would add chapters onto the end of the book, or add sections to end of existing chapters, and I will try as hard as possible to leave exercise numbers unchanged. For the most part, this second volume depends on the non-optional parts of volume I, while some of the optional parts are also used. Higher order derivatives (but not Taylor’s theorem itself) are used in 8.6, 9.3, 10.6 . Exponentials, logarithms, and improper integrals are used in a few examples and exercises, and they are heavily used in chapter 11. My own plan for a two-semester course is that some bits of the first volume, such as metric spaces, are covered in the second semester, while some of the optional topics of volume I are covered in the first semester. Leaving metric spaces for the second semester makes more sense as then the second semester is the “multivariable” part of the course. Another possibility for a faster course is to leave out some of the optional parts, go quicker in the first semester including metric spaces and then arrive at chapter 11 . Several possibilities for things to cover after metric spaces, depending on time are: 1) 8.1 –8.5 , 10.1 –10.5 , 10.7 (multivariable calculus, focus on multivariable integral). 2) Chapter 8 , chapter 9 , 10.1 and 10.2 (multivariable calculus, focus on path integrals). 3) Chapters 8, 9, and 10 (multivariable calculus, path integrals, multivariable integrals). 4) Chapters 8, (maybe 9 ), and 11 (multivariable differential calculus, some advanced analysis). 5) Chapter 8 , chapter 9 , 11.1, 11.6, 11.7 (a simpler variation of the above). 6 INTRODUCTION Chapter 8 Several Variables and Partial Derivatives 8.1 Vector spaces, linear mappings, and convexity Note: 2–3 lectures 8.1.1 Vector spaces The euclidean space Rn has already made an appearance in the metric space chapter. In this chapter, we extend the differential calculus we created for one variable to several variables. The key idea in differential calculus is to approximate functions linear functions (approximating the graph by a straight line). In several variables, we must introduce a little bit of linear algebra before we can move on. We start with vector spaces and linear mappings on vector spaces. While it is common to use~v or the bold v for elements of Rn, especially in the applied sciences, we use just plain old v, which is common in mathematics. That is, v Rn is a vector, which means 2 v = (v1;v2;:::;vn) is an n-tuple of real numbers.* It is common to write and treat vectors as column vectors, that is, n-by-1 matrices: " v1 # v2 v = (v ;v ;:::;v ) = . 1 2 n . vn We will do so when convenient. We call real numbers scalars to distinguish them from vectors. We often think of vectors as a direction and a magnitude and draw the vector as an arrow. The vector (v1;v2;:::;vn) is represented by an arrow from the origin to the point (v1;v2;:::;vn), see 2 Figure 8.1 in the plane R . When we think of vectors as arrows, they are not based at the origin necessarily; a vector is simply the direction and the magnitude, and it does not know where it starts. On the other hand, each vector also represents a point in Rn. Usually, we think of v Rn as 2 a point if we are thinking of Rn as a metric space, and we think of it as an arrow if we think of the so-called vector space structure on Rn (addition and scalar multiplication). Let us define the abstract notion of a vector space, as there are many other vector spaces than just Rn. *Subscripts are used for many purposes, so sometimes we may have several vectors that may also be identified by subscript, such as a finite or infinite sequence of vectors y1;y2;:::. 8 CHAPTER 8. SEVERAL VARIABLES AND PARTIAL DERIVATIVES x2 (v1;v2) x1 Figure 8.1: Vector as an arrow. Definition 8.1.1. Let X be a set together with the operations of addition, +: X X X, and × ! multiplication, : R X X, (we usually write ax instead of a x). X is called a vector space (or a · × ! · real vector space) if the following conditions are satisfied: (i) (Addition is associative) If u;v;w X, then u + (v + w) = (u + v) + w. 2 (ii) (Addition is commutative) If u;v X, then u + v = v + u. 2 (iii) (Additive identity) There is a 0 X such that v + 0 = v for all v X. 2 2 (iv) (Additive inverse) For every v X, there is a v X, such that v + ( v) = 0. 2 − 2 − (v) (Distributive law) If a R, u;v X, then a(u + v) = au + av. 2 2 (vi) (Distributive law) If a;b R, v X, then (a + b)v = av + bv. 2 2 (vii) (Multiplication is associative) If a;b R, v X, then (ab)v = a(bv). 2 2 (viii) (Multiplicative identity) 1v = v for all v X. 2 Elements of a vector space are usually called vectors, even if they are not elements of Rn (vectors in the “traditional” sense). If Y X is a subset that is a vector space itself using the same operations, then Y is called a ⊂ subspace or a vector subspace of X. Multiplication by scalars works as one would expect. For example, 2v = (1 + 1)v = 1v + 1v = v + v, similarly 3v = v + v + v, and so on. One particular fact we often use is that 0v = 0, where the zero on the left is 0 R and the zero on the right is 0 X. To see this start with 2 2 0v = (0 + 0)v = 0v + 0v, and add (0v) to both sides to obtain 0 = 0v. Similarly v = ( 1)v, − − − which follows by ( 1)v + v = ( 1)v + 1v = ( 1 + 1)v = 0v = 0.
Load more

Recommended publications
  • Lesson 1: Solutions to Polynomial Equations
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 1: Solutions to Polynomial Equations Classwork Opening Exercise How many solutions are there to the equation 푥2 = 1? Explain how you know. Example 1: Prove that a Quadratic Equation Has Only Two Solutions over the Set of Complex Numbers Prove that 1 and −1 are the only solutions to the equation 푥2 = 1. Let 푥 = 푎 + 푏푖 be a complex number so that 푥2 = 1. a. Substitute 푎 + 푏푖 for 푥 in the equation 푥2 = 1. b. Rewrite both sides in standard form for a complex number. c. Equate the real parts on each side of the equation, and equate the imaginary parts on each side of the equation. d. Solve for 푎 and 푏, and find the solutions for 푥 = 푎 + 푏푖. Lesson 1: Solutions to Polynomial Equations S.1 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Exercises Find the product. 1. (푧 − 2)(푧 + 2) 2. (푧 + 3푖)(푧 − 3푖) Write each of the following quadratic expressions as the product of two linear factors. 3. 푧2 − 4 4. 푧2 + 4 5. 푧2 − 4푖 6. 푧2 + 4푖 Lesson 1: Solutions to Polynomial Equations S.2 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds.
    [Show full text]
  • Use of Iteration Technique to Find Zeros of Functions and Powers of Numbers
    USE OF ITERATION TECHNIQUE TO FIND ZEROS OF FUNCTIONS AND POWERS OF NUMBERS Many functions y=f(x) have multiple zeros at discrete points along the x axis. The location of most of these zeros can be estimated approximately by looking at a graph of the function. Simple roots are the ones where f(x) passes through zero value with finite value for its slope while double roots occur for functions where its derivative also vanish simultaneously. The standard way to find these zeros of f(x) is to use the Newton-Raphson technique or a variation thereof. The procedure is to expand f(x) in a Taylor series about a starting point starting with the original estimate x=x0. This produces the iteration- 2 df (xn ) 1 d f (xn ) 2 0 = f (xn ) + (xn+1 − xn ) + (xn+1 − xn ) + ... dx 2! dx2 for which the function vanishes. For a function with a simple zero near x=x0, one has the iteration- f (xn ) xn+1 = xn − with x0 given f ′(xn ) For a double root one needs to keep three terms in the Taylor series to yield the iteration- − f ′(x ) + f ′(x)2 − 2 f (x ) f ′′(x ) n n n ∆xn+1 = xn+1 − xn = f ′′(xn ) To demonstrate how this procedure works, take the simple polynomial function- 2 4 f (x) =1+ 2x − 6x + x Its graph and the derivative look like this- There appear to be four real roots located near x=-2.6, -0.3, +0.7, and +2.2. None of these four real roots equal to an integer, however, they correspond to simple zeros near the values indicated.
    [Show full text]
  • 5.2 Polynomial Functions 5.2 Polynomialfunctions
    5.2 Polynomial Functions 5.2 PolynomialFunctions This is part of 5.1 in the current text In this section, you will learn about the properties, characteristics, and applications of polynomial functions. Upon completion you will be able to: • Recognize the degree, leading coefficient, and end behavior of a given polynomial function. • Memorize the graphs of parent polynomial functions (linear, quadratic, and cubic). • State the domain of a polynomial function, using interval notation. • Define what it means to be a root/zero of a function. • Identify the coefficients of a given quadratic function and he direction the corresponding graph opens. • Determine the verte$ and properties of the graph of a given quadratic function (domain, range, and minimum/maximum value). • Compute the roots/zeroes of a quadratic function by applying the quadratic formula and/or by factoring. • !&etch the graph of a given quadratic function using its properties. • Use properties of quadratic functions to solve business and social science problems. DescribingPolynomialFunctions ' polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is the product of a number, called the coefficient of the term, and a variable raised to a non-negative integer (a natural number) power. Definition ' polynomial function is a function of the form n n ) * f(x =a n x +a n ) x − +...+a * x +a ) x+a +, − wherea ,a ,...,a are real numbers andn ) is a natural number. � + ) n ≥ In a previous chapter we discussed linear functions, f(x =mx=b , and the equation of a horizontal line, y=b .
    [Show full text]
  • Factors, Zeros, and Solutions
    Mathematics Instructional Plan – Algebra II Factors, Zeros, and Solutions Strand: Functions Topic: Exploring relationships among factors, zeros, and solutions Primary SOL: AII.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Related SOL: AII.1d, AII.4b, AII.7b Materials Zeros and Factors activity sheet (attached) Matching Cards activity sheet (attached) Sorting Challenge activity sheet (attached) Graphing utility Colored paper Graph paper Vocabulary factor, fundamental theorem of algebra, imaginary numbers, multiplicity, polynomial, rate of change, real numbers, root, slope, solution, x-intercept, y-intercept, zeros of a function Student/Teacher Actions: What should students be doing? What should teachers be doing? Time: 90 minutes 1. Have students solve the equation 3x + 5 = 11. Then, have students set the left side of the equation equal to zero. Tell students to replace zero with y and consider the line y = 3x − 6. Instruct them to graph the line on graph paper. Review the terms slope, x- intercept, and y-intercept. Then, have them perform the same procedure (i.e., solve, set left side equal to zero, replace zero with y, and graph) for the equation −2x + 6 = 4. They should notice that in both cases, the x-intercept is the same as the solution to the equation. 2. Students explored factoring of zeros in Algebra I. Use the following practice problem to review: y = x2 + 4x + 3. Facilitate a discussion with students using the following questions: o “What is the first step in graphing the given line?” o “How do you find the factors of the line?” o “What role does zero play in graphing the line?” o “In how many ‘places’ did this line cross (intersect) the x-axis?” o “When the function crosses the x-axis, what is the value of y?” o “What is the significance of the x-intercepts of the graphs?” 3.
    [Show full text]
  • Elementary Functions, Student's Text, Unit 21
    DOCUMENT RESUME BD 135 629 SE 021 999 AUTHOR Allen, Frank B.; And Others TITLE Elementary Functions, Student's Text, Unit 21. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE 61 NOTE 398p.; For related documents, see SE 021 987-022 002 and ED 130 870-877; Contains occasional light type EDRS PRICE MF-$0.83 HC-$20.75 Plus Postage. DESCRIPTO2S *Curriculum; Elementary Secondary Education; Instruction; *Instructional Materials; Mathematics Education; *Secondary School Mathematics; *Textbooks IDENTIFIERS *Functions (Mathematics); *School Mathematics Study Group ABSTRACT Unit 21 in the SMSG secondary school mathematics series is a student text covering the following topics in elementary functions: functions, polynomial functions, tangents to graphs of polynomial functions, exponential and logarithmic functions, and circular functions. Appendices discuss set notation, mathematical induction, significance of polynomials, area under a polynomial graph, slopes of area functions, the law of growth, approximation and computation of e raised to the x power, an approximation for ln x, measurement of triangles, trigonometric identities and equations, and calculation of sim x and cos x. (DT) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC makes every effort * * to obtain the best copy available. Nevertheless, items of marginal * * reproducibility are often encountered and this affects the quality * * of the microfiche and hardcopy reproductions ERIC makes available * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. Reproductions * * supplied by EDRS are the best that can be made from the original.
    [Show full text]
  • Applications of Derivatives
    5128_Ch04_pp186-260.qxd 1/13/06 12:35 PM Page 186 Chapter 4 Applications of Derivatives n automobile’s gas mileage is a function of many variables, including road surface, tire Atype, velocity, fuel octane rating, road angle, and the speed and direction of the wind. If we look only at velocity’s effect on gas mileage, the mileage of a certain car can be approximated by: m(v) ϭ 0.00015v 3 Ϫ 0.032v 2 ϩ 1.8v ϩ 1.7 (where v is velocity) At what speed should you drive this car to ob- tain the best gas mileage? The ideas in Section 4.1 will help you find the answer. 186 5128_Ch04_pp186-260.qxd 1/13/06 12:36 PM Page 187 Section 4.1 Extreme Values of Functions 187 Chapter 4 Overview In the past, when virtually all graphing was done by hand—often laboriously—derivatives were the key tool used to sketch the graph of a function. Now we can graph a function quickly, and usually correctly, using a grapher. However, confirmation of much of what we see and conclude true from a grapher view must still come from calculus. This chapter shows how to draw conclusions from derivatives about the extreme val- ues of a function and about the general shape of a function’s graph. We will also see how a tangent line captures the shape of a curve near the point of tangency, how to de- duce rates of change we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point.
    [Show full text]
  • Optimization Algorithms on Matrix Manifolds
    00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Chapter Six Newton’s Method This chapter provides a detailed development of the archetypal second-order optimization method, Newton’s method, as an iteration on manifolds. We propose a formulation of Newton’s method for computing the zeros of a vector field on a manifold equipped with an affine connection and a retrac­ tion. In particular, when the manifold is Riemannian, this geometric Newton method can be used to compute critical points of a cost function by seeking the zeros of its gradient vector field. In the case where the underlying space is Euclidean, the proposed algorithm reduces to the classical Newton method. Although the algorithm formulation is provided in a general framework, the applications of interest in this book are those that have a matrix manifold structure (see Chapter 3). We provide several example applications of the geometric Newton method for principal subspace problems. 6.1 NEWTON’S METHOD ON MANIFOLDS In Chapter 5 we began a discussion of the Newton method and the issues involved in generalizing such an algorithm on an arbitrary manifold. Sec­ tion 5.1 identified the task as computing a zero of a vector field ξ on a Riemannian manifold equipped with a retraction R. The strategy pro­ M posed was to obtain a new iterate xk+1 from a current iterate xk by the following process. 1. Find a tangent vector ηk Txk such that the “directional derivative” of ξ along η is equal to ∈ ξ.
    [Show full text]
  • Lesson 11 M1 ALGEBRA II
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1 ALGEBRA II Lesson 11: The Special Role of Zero in Factoring Student Outcomes . Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors. Students construct a polynomial function that has a specified set of zeros with stated multiplicity. Lesson Notes This lesson focuses on the first part of standard A-APR.B.3, identifying zeros of polynomials presented in factored form. Although the terms root and zero are interchangeable, for consistency only the term zero is used throughout this lesson and in later lessons. The second part of the standard, using the zeros to construct a rough graph of a polynomial function, is delayed until Lesson 14. The ideas that begin in this lesson continue in Lesson 19, in which students will be able to associate a zero of a polynomial function to a factor in the factored form of the associated polynomial as a consequence of the remainder theorem, and culminate in Lesson 39, in which students apply the fundamental theorem of algebra to factor polynomial expressions completely over the complex numbers. Classwork Scaffolding: Here is an alternative opening Opening Exercise (12 minutes) activity that may better illuminate the special role of Opening Exercise zero. ퟐ ퟐ Find all solutions to the equation (풙 + ퟓ풙 + ퟔ)(풙 − ퟑ풙 − ퟒ) = ퟎ. For each equation, list some possible values for 푥 and 푦. The main point of this opening exercise is for students to recognize and then formalize that the statement “If 푎푏 = 0, then 푎 = 0 or 푏 = 0” applies not only when 푎 and 푏 are 푥푦 = 10, 푥푦 = 1, numbers or linear functions (which we used when solving a quadratic equation), but also 푥푦 = −1, 푥푦 = 0 applies to cases where 푎 and 푏 are polynomial functions of any degree.
    [Show full text]
  • Real Zeros of Polynomial Functions
    333353_0200.qxp 1/11/07 2:00 PM Page 91 Polynomial and Chapter 2 Rational Functions y y y 2.1 Quadratic Functions 2 2 2 2.2 Polynomial Functions of Higher Degree x x x −44−2 −44−2 −44−2 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra Polynomial and rational functions are two of the most common types of functions 2.6 Rational Functions and used in algebra and calculus. In Chapter 2, you will learn how to graph these types Asymptotes of functions and how to find the zeros of these functions. 2.7 Graphs of Rational Functions 2.8 Quadratic Models David Madison/Getty Images Selected Applications Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. ■ Automobile Aerodynamics, Exercise 58, page 101 ■ Revenue, Exercise 93, page 114 ■ U.S. Population, Exercise 91, page 129 ■ Impedance, Exercises 79 and 80, page 138 ■ Profit, Exercise 64, page 145 ■ Data Analysis, Exercises 41 and 42, page 154 ■ Wildlife, Exercise 43, page 155 ■ Comparing Models, Exercise 85, page 164 ■ Media, Aerodynamics is crucial in creating racecars.Two types of racecars designed and built Exercise 18, page 170 by NASCAR teams are short track cars, as shown in the photo, and super-speedway (long track) cars. Both types of racecars are designed either to allow for as much downforce as possible or to reduce the amount of drag on the racecar. 91 333353_0201.qxp 1/11/07 2:02 PM Page 92 92 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions The Graph of a Quadratic Function What you should learn ᭿ Analyze graphs of quadratic functions.
    [Show full text]
  • Root-Finding 3.1 Physical Problems
    Physics 200 Lecture 3 Root-Finding Lecture 3 Physics 200 Laboratory Monday, February 14th, 2011 The fundamental question answered by this week's lab work will be: Given a function F (x), find some/all of the values xi for which F (xi) = 0. It's a f g modest goal, and we will use a simple method to solve the problem. But, as we shall see, there are a wide range of physical problems that have, at their heart, just such a question. We'll start in the simplest, polynomial setting, and work our way up to the \shooting" method. 3.1 Physical Problems We'll set up some direct applications of root-finding with familiar physical examples, and then shift gears and define a numerical root-finding routine that can be used to solve a very different set of problems. 3.1.1 Orbital Motion In two-dimensions, with a spherically symmetric potential (meaning here that V (x; y; z) = V (r), a function of a single variable, r x2 + y2 + z2, ≡ the distance to the origin) we can use circular coordinates to write the total p energy of a test particle moving under the influence of this potential as 1 E = m r_2 + r2 φ_2 + V (r): (3.1) 2 Conservation of momentum tells us that the z-component of angular mo- 2 mentum is conserved, with Lz = (r p) = m r φ_, so we can rewrite the × z 1 of 13 3.1. PHYSICAL PROBLEMS Lecture 3 energy as: 1 1 L2 E = m r_2 + z + V (r) (3.2) 2 2 m r2 ≡U(r) where U(r) defines an “effective potential"| { we{z have} turned a two-dimensional problem into a one-dimensional problem for the coordinate r, and an effec- tive potential that governs the motion in this setting.
    [Show full text]
  • Zeros of a Quadratic Function TEACHER NOTES MATH NSPIRED
    Zeros of a Quadratic Function TEACHER NOTES MATH NSPIRED Math Objectives • Students will determine the linear factors of a quadratic function. • Students will connect the algebraic representation to the geometric representation. • Students will discover that the zeros of the linear factors are the zeros of the quadratic function. • Students will discover that the zeros of the quadratic function are the zeros of its linear factors. TI-Nspire™ Technology Skills: • Students will use appropriate tools strategically (CCSS Download a TI-Nspire Mathematical Practice). • document • Students will look for and make use of structure (CCSS • Open a document Mathematical Practice). • Move between pages • Grab and drag a point Vocabulary • zeros Tech Tips: • Make sure the font size on your TI-Nspire handheld is About the Lesson set to Medium. • This lesson involves merging graphical and algebraic • You can hide the function representations of a quadratic function and its linear factors. entry line by pressing • As a result, students will: /G. • Manipulate the parameters of the linear functions and observe the resulting changes in the quadratic function. Lesson Materials: • Find the zeros of the quadratic function by finding the zeros of Student Activity its linear factors. Zeros_of_a_Quadratic_Function _Student.pdf • Solve quadratic equations by factoring and be able to explain Zeros_of_a_Quadratic_Function why this process is valid. _Student.doc TI-Nspire document Zeros_of_a_Quadratic_Function TI-Nspire™ Navigator™ System .tns • Use Screen Capture to examine patterns that emerge. Visit www.mathnspired.com for • Use Live Presenter to engage and focus students. lesson updates and tech tip • Use Teacher Edition computer software to review student videos. documents. ©2011 Texas Instruments Incorporated 1 education.ti.com Zeros of a Quadratic Function TEACHER NOTES MATH NSPIRED Discussion Points and Possible Answers Tech Tip: If students experience difficulty dragging a point on the slider, check to make sure that they have moved the arrow close to the point.
    [Show full text]
  • Teaching Concepts Foundational to Calculus Using Inquiry
    TEACHING CONCEPTS FOUNDATIONAL TO CALCULUS USING INQUIRY AND TECHNOLOGY A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Benjamin David Marko May, 2006 TEACHING CONCEPTS FOUNDATIONAL TO CALCULUS USING INQUIRY AND TECHNOLOGY Benjamin David Marko Thesis Approved: Accepted: Advisor Dean of the College Dr. Antonio R. Quesada Dr. Ronald F. Levant Co-Advisor Dean of the Graduate School Dr. Linda Marie Saliga George R. Newkome, Ph.D. Department Chair Date Dr. Kevin L. Kreider ii ABSTRACT There is currently a push in the state of Ohio to enhance the teaching of concepts foun- dational to Calculus in grades 8-12. Evidence of this came in the spring of 2005 when the Ohio Department of Education called for grant proposals to develop programs that are focused on topics that are foundations to Calculus. Further, the National Council of Teachers of Mathematics has stressed the effective use of technology and inquiry in their past and current standards. This thesis contains inquiry-based lessons that explore concepts that would traditionally be taught in the Calculus classroom, but can be used in lower level courses such as Algebra. The use of the graphing calculator is essential for each one of these lessons to allow students the opportunity to explore more mathematical concepts in a less time consuming manner. iii ACKNOWLEDGEMENTS I would like to take this time to thank a number of people without whom this thesis would not have been possible. Thanks to my advisors, Dr. Antonio Quesada and Dr.
    [Show full text]