# A First Course in Analysis

Springer Science & Business Media, Mar 11, 1994 - Mathematics - 279 pages
The first course in Analysis, which follows calculus, along with other courses, such as differential equations and elementary linear algebra, in the curricu lum, presents special pedagogical challenges. There is a change of stress from computational manipulation to "proof. " Indeed, the course can become more a course in Logic than one in Analysis. Many students, caught short by a weak command of the means of mathematical discourse and unsure of what is expected of them, what "the game" is, suffer bouts of a kind of mental paralysis. This text attempts to address these problems in several ways: First, we have attempted to define "the game" as that of "inquiry," by using a form of exposition that begins with a question and proceeds to analyze, ultimately to answer it, bringing in definitions, arguments, conjectures, exam ples, etc. , as they arise naturally in the course of a narrative discussion of the question. (The true, historical narrative is too convoluted to serve for first explanations, so no attempt at historical accuracy has been made; our narra tives are completely contrived. ) Second, we have kept the logic informal, especially in the course of preliminary speculative discussions, where common sense and plausibility tempered by mild skepticism-serve to energize the inquiry.

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### Contents

 Approximation The Real Numbers 35 2 Completeness Nested Intervals 38 3 Bounded Monotonic Sequences 40 4 Cauchy Sequences 45 5 The Real Number System 48 6 Countability 50 Appendix The Fundamental Theorem of Algebra Complex Numbers 53 The ExtremeValue Problem 62
 1 Differential and Derivative Tangent Line 133 2 The Foundations of Differentiation 138 3 Curve Sketching The MeanValue Theorem 144 4 Taylors Theorem 152 5 Functions Defined Implicitly 158 Integration 167 1 Definitions Darboux Theorem 169 2 Foundations of Integral Calculus The Fundamental Theorem of Calculus 177

 1 Continuity Compactness and the ExtremeValue Theorem 63 2 Continuity of Rational Functions Limits of Sequences 70 Completion of the Proof of the Fundamental Theorem of Algebra 74 3 Sequences and Series of Reals The Number e 76 4 Sets of Reals Limits of Functions 88 Continuous Functions 95 2 Inverse Functions 𝑥ʳ for 𝑟 ℚ 99 3 Continuous Extension Uniform Continuity The Exponential and Logarithm 102 4 The Elementary Functions 107 5 Uniformity The HeineBorel Theorem 110 6 Uniform Convergence A Nowhere Differentiable Continuous Function 116 7 The Weierstrass Approximation Theorem 122 Summary of the Main Properties of Continuous Functions 126 FOUNDATIONS OF CALCULUS 129 Differentiation 131
 3 The Nature of Integrability Lebesgues Theorem 187 4 Improper Integral 195 5 Arclength Bounded Variation 204 A Word About the Stieltjes Integral and Measure Theory 213 Infinite Series 216 1 The Vibrating String 217 General Considerations 222 Series of Positive Terms 228 4 Computation with Series 235 5 Power Series 242 6 Fourier Series 252 Bibliography 264 Index 267 Copyright

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Page 3 - RC, the total sample size less the total number of cells in the design represented as the product of the number of rows by the number of columns (RC).