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E-grāmata: Introductory Analysis: An Inquiry Approach [Taylor & Francis e-book]

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  • Formāts: 250 pages, 22 Illustrations, black and white
  • Izdošanas datums: 10-Feb-2020
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781351246743
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  • Taylor & Francis e-book
  • Cena: 128,96 €*
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  • Standarta cena: 184,22 €
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Introductory Analysis: An Inquiry ApproachPalielināt
  • Formāts: 250 pages, 22 Illustrations, black and white
  • Izdošanas datums: 10-Feb-2020
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781351246743
Citas grāmatas par šo tēmu:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Rokasgrāmatas mājas lapa: https://www.taylorfrancis.com/books/9781351246743
"Introductory Analysis: An Inquiry Approach aims to provide a self-contained, inquiry-oriented approach to undergraduate-level real analysis. The presentation of the material in the book is intended to be "inquiry-oriented'" in that as each major topic is discussed, details of the proofs are left to the student in a way that encourages an active approach to learning. The book is "self-contained" in two major ways: it includes scaffolding (i.e., brief guiding prompts marked as Key Steps in the Proof) for many of the theorems. Second, it includes preliminary material that introduces students to the fundamental framework of logical reasoning and proof-writing techniques. Students will be able to use the guiding prompts (and refer to the preliminary work) todevelop their proof-writing skills. Features Structured in such a way that approximately one week of class can be devoted to each chapter Suitable as a primary text for undergraduates, or as a supplementary text for some postgraduate courses Strikes a unique balance between enquiry-based learning and more traditional approaches to teaching"--

Introductory Analysis: An Inquiry Approach aims to provide a self-contained, inquiry-oriented approach to undergraduate-level real analysis.

The presentation of the material in the book is intended to be "inquiry-oriented'" in that as each major topic is discussed, details of the proofs are left to the student in a way that encourages an active approach to learning. The book is "self-contained" in two major ways: it includes scaffolding (i.e., brief guiding prompts marked as Key Steps in the Proof) for many of the theorems. Second, it includes preliminary material that introduces students to the fundamental framework of logical reasoning and proof-writing techniques. Students will be able to use the guiding prompts (and refer to the preliminary work) to develop their proof-writing skills.

Features

  • Structured in such a way that approximately one week of class can be devoted to each chapter

  • Suitable as a primary text for undergraduates, or as a supplementary text for some postgraduate courses

    • Strikes a unique balance between enquiry-based learning and more traditional approaches to teaching

    • Preface ix
    Prerequisites 1(2)
    P1 Exploring Mathematical Statements 3(18)
    P1.1 What is a mathematical statement?
    		3(2)
    P1.2 Basic set theory
    		5(5)
    P1.3 Quantifiers, both existential and universal
    		10(2)
    P1.4 Implication: the heart of a "provable" mathematical statement
    		12(1)
    P1.5 Negations
    		13(4)
    P1.6 Statements related to implication
    		17(4)
    P2 Proving Mathematical Statements 21(14)
    P2.1 Using definitions
    		21(1)
    P2.2 Proving a basic statement with an existential quantifier
    		22(1)
    P2.3 Proving a basic statement with a universal quantifier
    		23(1)
    P2.4 Proving an implication directly
    		24(2)
    P2.5 Proof by contrapositive
    		26(1)
    P2.6 Proof involving cases
    		27(1)
    P2.7 Proof by contradiction
    		28(1)
    P2.8 Proof by induction
    		29(4)
    P2.9 Proving that one of two (or one of several) conclusions is true
    		33(2)
    P3 Preliminary Content 35(10)
    P3.1 Relations and equivalence
    		35(2)
    P3.2 Functions
    		37(4)
    P3.3 Inequalities and epsilons
    		41(4)
    1 Properties of R 45(14)
    1.1 Preliminary work
    		46(2)
    1.2 Main Theorems
    		48(8)
    1.3 Follow-up work
    		56(3)
    2 Accumulation Points and Closed Sets 59(6)
    2.1 Preliminary work
    		59(1)
    2.2 Main Theorems
    		60(2)
    2.3 Follow-up work
    		62(3)
    3 Open Sets and Open Covers 65(6)
    3.1 Preliminary work
    		65(2)
    3.2 Main Theorems
    		67(2)
    3.3 Follow-up work
    		69(2)
    4 Sequences and Convergence 71(12)
    4.1 Preliminary work
    		71(4)
    4.2 Main Theorems
    		75(3)
    4.3 Follow-up work
    		78(5)
    5 Subsequences and Cauchy Sequences 83(10)
    5.1 Preliminary Work
    		83(3)
    5.2 Main Theorems
    		86(1)
    5.3 Follow-up Work
    		87(6)
    6 Functions, Limits, and Continuity 93(8)
    6.1 Preliminary Work
    		94(2)
    6.2 Main Theorems
    		96(2)
    6.3 Follow-up Work
    		98(3)
    7 Connected Sets and the Intermediate Value Theorem 101(8)
    7.1 Preliminary Work
    		101(1)
    7.2 Main Theorems
    		102(2)
    7.3 Follow-up Work
    		104(5)
    8 Compact Sets 109(6)
    8.1 Preliminary Work
    		109(3)
    8.2 Main Theorems
    		112(1)
    8.3 Follow-up Work
    		113(2)
    9 Uniform Continuity 115(4)
    9.1 Preliminary Work
    		115(1)
    9.2 Main Theorems
    		116(2)
    9.3 Follow-up Work
    		118(1)
    10 Introduction to the Derivative 119(6)
    10.1 Preliminary Work
    		119(2)
    10.2 Main Theorems
    		121(2)
    10.3 Follow-up Work
    		123(2)
    11 The Extreme and Mean Value Theorems 125(8)
    11.1 Preliminary Work
    		125(1)
    11.2 Main Theorems
    		126(2)
    11.3 Follow-up Work
    		128(5)
    12 The Definite Integral: Part I 133(10)
    12.1 Preliminary Work
    		134(4)
    12.2 Main Theorems
    		138(1)
    12.3 Follow-up Work
    		139(4)
    13 The Definite Integral: Part II 143(8)
    13.1 Preliminary Work
    		143(2)
    13.2 Main Theorems
    		145(3)
    13.3 Follow-up Work
    		148(3)
    14 The Fundamental Theorem(s) of Calculus 151(8)
    14.1 Preliminary Work
    		151(1)
    14.2 Main Theorems
    		152(4)
    14.3 Follow-up Work
    		156(3)
    15 Series 159(16)
    15.1 Preliminary work
    		160(2)
    15.2 Main Theorems
    		162(8)
    15.3 Follow-up work
    		170(5)
    Extended Explorations 175(54)
    E1 Function Approximation
    		177(10)
    E1.1 Taylor Polynomials and Taylor's Theorem
    		177(3)
    E1.2 Interpolation
    		180(4)
    E1.3 Divided Differences
    		184(1)
    E1.4 A Hybrid Approach
    		185(2)
    E2 Power Series
    		187(12)
    E2.1 Introduction to Power Series
    		187(1)
    E2.2 Differentiation of a Power Series
    		188(5)
    E2.3 Taylor Series
    		193(6)
    E3 Sequences and Series of Functions
    		199(8)
    E3.1 Pointwise Convergence
    		199(1)
    E3.2 Uniform Convergence and Uniformly Cauchy Sequences of Functions
    		200(2)
    E3.3 Consequences of Uniform Convergence
    		202(5)
    E4 Metric Spaces
    		207(12)
    E4.1 What is a Metric Space? Examples
    		207(3)
    E4.2 Metric Space Completeness
    		210(3)
    E4.3 Metric Space Compactness
    		213(6)
    E5 Iterated Functions and Fixed Point Theorems
    		219(10)
    E5.1 Iterative Maps and Fixed Points
    		219(4)
    E5.2 Contraction Mappings
    		223(2)
    E5.3 Newton's Method
    		225(4)
    Appendix 229(4)
    A Brief Summary of Ordered Field Properties
    		231(2)
    Bibliography 233(2)
    Index 235
    John Ross is an Assistant Professor of Mathematics at Southwestern University. He earned his Ph.D. and M.A. in Mathematics from Johns Hopkins University, and his B.A. in Mathematics from St. Mary's College of Maryland. His research is in geometric analysis, answering questions about manifolds that arise under curvature flows. He enjoys overseeing undergraduate research, teaching in an inquiry-based format, biking to work, and hiking in Central Texas.

    Kendall Richards is a Professor of Mathematics at Southwestern University. He earned his B.S. and M.A. in Mathematics from Eastern New Mexico University and his Ph.D. in Mathematics from Texas Tech University. He is inspired by working with students and the process of learning. His research pursuits have included questions involving special functions, inequalities, and complex analysis. He also enjoys long walks and a strong cup of coffee.
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