Intro
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

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