Author Profile

Joseph C. Mazur is Professor Emeritus of Mathematics at Marlboro College, in Marlboro, Vermont. He holds a B.S. from Pratt Institute, where he first studied architecture. He spent his junior year in Paris, studying and returned to Pratt to earn a B.S. in mathematics. From there he went directly to M.I.T to receive his Ph.D. in mathematics (algebraic geometry) in 1972. He has held a visiting scholar position at M.I.T and several visiting professor positions at The Mathematics Institute of the University of Warwick. In 2006 he was awarded a Guggenheim Fellowship for work on mathematical narrative. In 2008 he was awarded a Bellagio Fellowship from the Rockefeller Foundation, and in 2009 was elected to Fellow of the Vermont Academy of Arts and Sciences. In 2011, 2013, and 2019 he was awarded Bogliasco Fellowships.

About The Book

Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behaviour of a function at points ever closer to a particular point, but without ever actually reaching that point. Calculus has two basic applications: differential calculus and integral calculus. The main purpose of this volume is to provide an introduction to calculus in its many forms. Give some presentations to illustrate how powerful calculus is as a mathematical tool for solving a variety of scientific problems, introduce concepts from a variety of application areas, such as biology, chemistry, economics, physics and engineering, to demonstrate applications of calculus. This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This book emphasizes the fundamental concepts from calculus and analytic geometry and the application of these concepts to selected areas of science and engineering. Topics covered includes: Sets, Functions, Graphs and Limits, Differential Calculus, Integral Calculus, Sequences, Summations and Products and Applications of Calculus. In this book, much emphasis is put on explanations of concepts and solutions to examples. By reading the book carefully, students should be able to understand the concepts introduced and know how to answer questions with justification. Students should bear in mind that the main purpose of learning calculus is not just knowing how to perform differentiation and integration but also knowing how to apply differentiation and integration to solve problems. For that, one must understand the concepts.

Table of Contents

Preface............................................................................................................ v
Chapter 1. Foundations of Calculus....................................................................... 1
ƒ The Need for Calculus in Science and Engineering................................... 3
ƒ Intervals, Subsets and Real Numbers...................................................... 18
ƒ Functions and Their Representations......................................................21
ƒ Types of Functions: Polynomial, Rational, Trigonometric...................... 24
ƒ Domain, and Range of a Function........................................................... 28
Chapter 2. Limits and Continuity....................................................................... 34
ƒ Formal Definition of a Limit.....................................................................36
ƒ One-Sided Limits........................................................................................41
ƒ Infinite Limits and Vertical Asymptotes..................................................43
ƒ Continuity and Types of Discontinuities..................................................45
ƒ Theorems on Limits..................................................................................48
ƒ Limits at Infinity and Horizontal Asymptotes......................................... 53
ƒ Intermediate Value Theorem................................................................... 55
Chapter 3. Derivatives: Basic Concepts............................................................. 59
ƒ Tangent Lines and Slopes........................................................................ 61
ƒ The Definition of the Derivative............................................................... 72
ƒ Derivative as a Function...........................................................................79
ƒ Physical Interpretation: Velocity and Rate of Change............................ 81
Chapter 4. Techniques of Differentiation.......................................................... 86
ƒ Polynomial and Rational Functions.........................................................87
ƒ Chain Rule.................................................................................................98
ƒ Product and Quotient Rules................................................................... 101
ƒ Implicit Differentiation........................................................................... 104
ƒ Parametric and Polar Differentiation Basics......................................... 106
Chapter 5. Applications of Derivatives..............................................................110
ƒ Increasing/Decreasing Functions............................................................ 112
ƒ Relative Extrema and First Derivative Test............................................ 115
ƒ Curve Sketching.......................................................................................120
ƒ Concavity and Inflection Points..............................................................123
Contents
ƒ Optimization Problems...........................................................................125
ƒ Linear Approximation and Differentials................................................. 127
ƒ Related Rates..........................................................................................133
Chapter 6. Integration: Antiderivatives and Definite Integrals...................... 137
ƒ Antiderivatives and Indefinite Integrals.................................................139
ƒ Basic Integration Rules...........................................................................155
ƒ Area Under a Curve.................................................................................158
ƒ The Definite Integral...............................................................................165
ƒ Properties of the Definite Integral........................................................ 166
Chapter 7. Techniques of Integration............................................................... 170
ƒ Integration by Parts................................................................................ 171
ƒ Substitution Method............................................................................... 172
ƒ Trigonometric Integrals..........................................................................175
ƒ Partial Fraction Decomposition..............................................................176
ƒ Numerical Integration: Trapezoidal and Simpson’s Rule......................178
Chapter 8. Applications of the Integral............................................................184
ƒ Area Between Curves............................................................................. 186
ƒ Defining Disc Integration ...................................................................... 189
ƒ Volume by Shell Method........................................................................ 190
ƒ Arc Length and Surface Area..................................................................192
ƒ Applications in Economics and Biology..................................................195
Chapter 9. Differential Equations.....................................................................199
ƒ Introduction to Differential Equations...................................................201
ƒ Separable Differential Equations........................................................... 207
ƒ Linear First-Order Equations..................................................................209
ƒ Exponential Growth and Decay Models................................................. 211
ƒ Applications in Physics and Population Dynamics.................................216
ƒ Euler’s Method........................................................................................219
Chapter 10. Sequences and Series....................................................................225
ƒ Introduction to Sequences.................................................................... 226
ƒ Limits of Sequences............................................................................... 236
ƒ Geometric Series.................................................................................... 238
ƒ Power Series...........................................................................................246
ƒ Taylor Series............................................................................................ 253
Bibliography................................................................................................257