This textbook provides a gentle overview of fundamental concepts related to one-variable calculus. The original approach is a result of the author’s forty years of experience in teaching the subject at universities around the world. In this book, Dr. Zalduendo makes use of the history of mathematics and a friendly, conversational approach to attract the attention of the student, emphasizing what is more conceptually relevant and putting key notions in a historical perspective. Such an approach was conceived to help them to overcome potential difficulties in teaching and learning of this subject ― caused, in many cases, by an excess of technicalities and computations.
Besides covering the core of the discipline ― real number, sequences and series, functions, derivatives, integrals, convexity and inequalities ― the book is enriched by “side trips” to relevant subjects not usually seen in traditional calculus textbooks, touching on topics like curvature, the isoperimetric inequality, Riemann’s rearrangement theorem, Snell’s law, Buffon’s needle problem, Gregory’s series, random walk and the Gauss curve, and more. An insightful collection of exercises and applications completes this book, making it ideal as a supplementary textbook for a calculus course or the main textbook for an honors course on the subject.
Author(s): Ignacio Zalduendo
Series: Springer Undergraduate Mathematics Series
Publisher: Springer
Year: 2022
Language: English
Pages: 224
City: Cham
Preface
Contents
Introduction
2400 Years of Calculus
Calculus and Education
1 The Real Numbers
The Rational Line
Density of Q
Some Basic Notions
Irrationality of Q
From Eudoxus to Dedekind
The Real Line
Dyadic Series—A Construction of R
The Scarcity of Q
The Completeness of R
Cardinality
Exercises
2 Sequences and Series
Sequences
Limits of Sequences
Cantor's Nested Intervals Theorem
Subsequences
Series
The Harmonic Series
Series of Positive Terms
Series with Positive and Negative Terms
The Riemann Series Theorem
Absolute and Unconditional Convergence
Exercises
3 Functions
The Elementary Functions
Polynomials
Circular Functions
The Exponential Function: Bernoulli's Inequality
Irrationality of e
Convergence of k=1∞(1+ak) and of k=1∞ak
Hyperbolic Functions
Injectivity and Inverse Functions
Curves in the Plane: Parametrized Curves
The Cycloid
Pythagorean Triples
Continuity
Bolzano and Weierstrass
Limits
Limits in Ancient Greece: The Area of a Circle
Three Important Limits
Exercises
4 The Derivative
Derivative
Tangents
Newton–Raphson
Derivatives of the Elementary Functions
The Chain Rule
Derivative of the Inverse Function
The Derivative of a Parametrized Curve
First Derivative, Tangent Line, and Growth
The Mean Value Theorems
L'Hôpital's Rule
Snell's Law
The Brachistochrone
Exercises
5 The Integral
Measure and Integral
The Fundamental Theorem of Calculus
A Pause for Comments
Buffon's Needle
Irrationality of π
Improper Integrals
Integration and Sums: Linearity of the Integral
Uniform Convergence—The Weierstrass M-Test
Gregory's Series
Integration and Products: Integration by Parts
Stirling's Formula
Integration and Composition: Integration by Substitution
A Note on Notation
Length of Curves. The Catenary
Area Enclosed by a Simple Closed Curve
Exercises
6 More Derivatives
Second Derivative, Best-Fitting Parabola, and Curvature
The Taylor Polynomial of Order Two
Curvature
Random Walk and the Gauss Curve
The Taylor Series
Exercises
7 Convexity and the Isoperimetric Inequality
The Arithmetic-Geometric Inequality
Convexity
Young, Hölder, Jensen, Cauchy–Schwarz…
The Isoperimetric Inequality
Exercises
8 More Integrals
Volume
Double Integrals
The Basel Problem
Solids of Revolution
Integration of e
Density Functions, Barycenter, and Expectation
Center of Mass or Barycenter
Pappus' Theorem
The Method
Surface Area
Normal Distribution. Gauss, Laplace, and Stirling
Exercises
9 The Gamma Function
The Gamma Function
Weierstrass' Formula
Growth of the Harmonic Series, Again
Exercises
Bibliography
Index
