What Is Calculus?: From Simple Algebra to Deep Analysis

373 Pages·2016·3 MB·English

by  R Michael Range

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World Scientific
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Published by
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Library of Congress Cataloging-in-Publication Data
Range, R. Michael.
What is calculus? : from simple algebra to deep analysis / by R. Michael Range (State University
of New York at Albany, USA).
pages cm
Includes index.
ISBN 978-9814644471 (hardcover : alk. paper) -- ISBN 978-9814644488 (pbk. : alk. paper)
1. Calculus--Textbooks. I. Title.
QA303.2.R36 2015
515--dc23
2015019347
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd.
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June15,2015 12:23 BC:9448-WhatIsCalculus? Calculus˙corrs pagev
To my grandchildren
Kareem, Kayan,Joshua, and Alexander
and all other students
who want to learn and understand calculus
v
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June15,2015 12:23 BC:9448-WhatIsCalculus? Calculus˙corrs pagevii
Contents
Preface xv
Notes for Instructors xxiii
Prelude to Calculus 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Tangents to Circles . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 8
3 Tangents to Parabolas . . . . . . . . . . . . . . . . . . . . 8
3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 10
4 Motion with Variable Speed . . . . . . . . . . . . . . . . . 11
4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 15
5 Tangents to Graphs of Polynomials . . . . . . . . . . . . . 16
5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 21
6 Rules for Differentiation . . . . . . . . . . . . . . . . . . . 21
6.1 Elementary Rules . . . . . . . . . . . . . . . . . . 22
6.2 Inverse Function Rule . . . . . . . . . . . . . . . 24
6.3 Product Rule . . . . . . . . . . . . . . . . . . . . . 26
6.4 Quotient Rule . . . . . . . . . . . . . . . . . . . . 28
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 30
7 More General Algebraic Functions . . . . . . . . . . . . . 31
7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 35
8 Beyond Algebraic Functions . . . . . . . . . . . . . . . . . 35
8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 42
I. The Cast: Functions of a Real Variable 43
I.1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 43
vii
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viii What is Calculus? From Simple Algebra toDeep Analysis
I.1.1 Rational Numbers . . . . . . . . . . . . . . . . . . 43
I.1.2 Order Properties . . . . . . . . . . . . . . . . . . . 46
I.1.3 Irrational Numbers . . . . . . . . . . . . . . . . . 47
I.1.4 Completeness of the Real Numbers . . . . . . . . 50
I.1.5 Intervals and Other Properties of R . . . . . . . . 53
I.1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . 58
I.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
I.2.1 Functions of Real Variables . . . . . . . . . . . . . 60
I.2.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 61
I.2.3 Some Simple Examples . . . . . . . . . . . . . . . 65
I.2.4 Linear Functions, Lines, and Slopes . . . . . . . . 66
I.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 70
I.3 Simple Periodic Functions . . . . . . . . . . . . . . . . . . 71
I.3.1 The Basic Trigonometric Functions . . . . . . . . 72
I.3.2 Radian Measure . . . . . . . . . . . . . . . . . . . 74
I.3.3 Simple Trigonometric Identities . . . . . . . . . . 75
I.3.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 77
I.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 78
I.4 Exponential Functions . . . . . . . . . . . . . . . . . . . . 79
I.4.1 Compound Interest . . . . . . . . . . . . . . . . . 79
I.4.2 The Functional Equation . . . . . . . . . . . . . . 80
I.4.3 Definition of Exponential Functions for Rational
Numbers . . . . . . . . . . . . . . . . . . . . . . . 81
I.4.4 Properties of Exponential Functions . . . . . . . . 83
I.4.5 Exponential Functions for Real Numbers . . . . . 86
I.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . 88
I.5 Natural Operations on Functions . . . . . . . . . . . . . . 89
I.5.1 Compositions . . . . . . . . . . . . . . . . . . . . . 89
I.5.2 Inverse Functions . . . . . . . . . . . . . . . . . . 91
I.5.3 Logarithm Functions . . . . . . . . . . . . . . . . 93
I.5.4 Inverting Functions on Smaller Domains . . . . . 97
I.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 99
I.6 Algebraic Operations and Functions . . . . . . . . . . . . 101
I.6.1 Sums and Products of Functions . . . . . . . . . . 101
I.6.2 Simple Algebraic Functions . . . . . . . . . . . . . 102
I.6.3 Local Boundedness of Algebraic Functions . . . . 104
I.6.4 Global Boundedness . . . . . . . . . . . . . . . . . 107
I.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 108
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Contents ix
II. Derivatives: How to Measure Change 111
II.1 Algebraic Derivatives by Approximation. . . . . . . . . . 112
II.1.1 From Factorization to Average Rates of Change . 112
II.1.2 From Average to Instantaneous Rates of
Change . . . . . . . . . . . . . . . . . . . . . . . 120
II.1.3 Approximation of Algebraic Derivatives . . . . . 122
II.1.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . 126
II.2 Derivatives of Exponential Functions . . . . . . . . . . . 127
II.2.1 Tangents for y =2x . . . . . . . . . . . . . . . . . 128
II.2.2 The Tangent to y =2x at x=0 . . . . . . . . . . 130
II.2.3 Other Exponential Functions . . . . . . . . . . . 136
II.2.4 The “Natural” Exponential Function . . . . . . . 137
II.2.5 The Natural Logarithm . . . . . . . . . . . . . . 139
II.2.6 The Derivative of lnx . . . . . . . . . . . . . . . 140
II.2.7 The Differential Equation of Exponential
Functions . . . . . . . . . . . . . . . . . . . . . . 141
II.2.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . 143
II.3 Differentiability and Local Linear Approximation . . . . 144
II.3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . 144
II.3.2 Continuous Functions . . . . . . . . . . . . . . . 147
II.3.3 Differentiable Functions . . . . . . . . . . . . . . 150
II.3.4 Local Linear Approximation . . . . . . . . . . . . 152
II.3.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . 157
II.4 Properties of Continuous Functions . . . . . . . . . . . . 158
II.4.1 Rules for Limits . . . . . . . . . . . . . . . . . . . 158
II.4.2 Rules for Continuous Functions . . . . . . . . . . 160
II.4.3 The Intermediate Value Theorem . . . . . . . . . 161
II.4.4 Continuity and Boundedness . . . . . . . . . . . 162
II.4.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . 164
II.5 Derivatives of Trigonometric Functions . . . . . . . . . . 165
II.5.1 Continuity of sine and cosine Functions . . . . . 165
II.5.2 The Derivative of sint at t=0 . . . . . . . . . . 166
II.5.3 The Derivative of sint . . . . . . . . . . . . . . . 169
II.5.4 The cosine Function . . . . . . . . . . . . . . . . 172
II.5.5 A Differential Equation for sine and cosine . . . 172
II.5.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . 173
II.6 Simple Differentiation Rules . . . . . . . . . . . . . . . . 175
II.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . 175