I intend this book to be, firstly, a introduction to calculus based on the hyperreal number system. In other words, I will use infinitesimal and infinite numbers freely. Just as most beginning calculus books provide no logical justification for the real number system, I will provide none for the hyperreals. The reader interested in questions of foundations should consult books such as Abraham Robinson's Non-standard Analysis or Robert Goldblatt's Lectures on the Hyperreals. Secondly, I have aimed the text primarily at readers who already have some familiarity with calculus. Although the book does not explicitly assume any prerequisites beyond basic algebra and trigonometry, in practice the pace is too fast for most of those without some acquaintance with the basic notions of calculus.

Table of Contents

1 Derivatives

1.1 The arrow paradox

1.2 Rates of change

1.3 The hyperreals

1.4 Continuous functions

1.5 Properties of continuous functions

1.6 The derivative

1.7 Properties of derivatives

1.8 A geometric interpretation of the derivative

1.9 Increasing, decreasing, and local extrema

1.10 Optimization

1.11 Implicit differentiation and rates of change

1.12 Higher-order derivatives

2 Integrals

2.1 Integrals

2.2 Definite integrals

2.3 Properties of definite integrals

2.4 The fundamental theorem of integrals

2.5 Applications of definite integrals

2.6 Some techniques for evaluating integrals

2.7 The exponential and logarithm functions

Answers to Exercises

Index

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