Analysis

The following sections are included:

• DEDICATION
• Contents
• PREFACE

Using quadratic equations as an example, we see how one might approach an unknown formula in mathematics.

To further develop an intuitive approach, prior to a formal investigation, we sketch and modify the graphs of some basic functions.

We define functions rigorously and discuss their basic properties.

We define and discuss special collections of functions: surjective, injective, bijective, inverse, strictly increasing, strictly decreasing, indicator and convex functions.

We examine the natural numbers N, the integers Z and the rational numbers Q. Properties of these collections suggest germs of ideas that will make later developments appear natural and almost inevitable.

We consider properties of sets relating to size: small (finite), medium (countable), large (uncountable); and properties of dispersion such as boundedness or being an interval.

We discuss convergence of increasing and decreasing sequences and of series with positive entries. We consider absolute convergence of series, the geometric and exponential series, and the rearrangement and multiplication of series.

We give a general definition of convergent sequence using increasing and decreasing sequences. We define power series and apply the ratio test to find the radius of convergence of certain power series.

We define continuous functions and discuss two fundamental theorems: the Intermediate Value Theorem and the Fundamental Existence Theorem for Maxima and Minima. We show that the exponential and log functions are continuous.

We define and prove basic results about differentiable functions such as the Product Rule, the Chain Rule and the Mean Value Theorem. We prove that the exponential and log functions are differentiable, relate the sign of the derivative to the direction of increase and show how to locate local maxima and minima.

We continue developing the methods of the differential calculus and obtain new approaches to sequences, series, limits and curve sketching.

We construct the real numbers and discuss recent notions of rigor and logic.

We make general remarks about mathematics as a prelude to considering the background to integration theory. We prove the Fundamental Theorem of Calculus and discuss methods of integration.

We select a small number of applications from the large number available. These include integral inequalities, infinite integrals, a test for series convergence, a definition of the Gamma Function, density functions and power series expansions. Our aim is to use integrals to shed further light on what we already know and, in this way, to emphasize the unity of mathematics.

The following sections are included:

• SOLUTIONS
• BIBLIOGRAPHY
• INDEX