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Unformatted text preview: Introduction to Calculus Contents 1 Introduction to Calculus 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Origin of Calculus . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 The Two Branches of Calculus . . . . . . . . . . . . . . 4 1.2 Secant and Tangent Lines . . . . . . . . . . . . . . . . . . . . 5 1.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Definition of the Derivative . . . . . . . . . . . . . . . 14 1.4.2 Rules for Calculating Derivatives . . . . . . . . . . . . 16 1.5 Applications of Derivatives . . . . . . . . . . . . . . . . . . . . 18 1.5.1 Rates of Change . . . . . . . . . . . . . . . . . . . . . 18 1.5.2 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.3 Graphing Polynomials . . . . . . . . . . . . . . . . . . 24 1.5.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . 27 A Answers to All Exercises 31 2 Chapter 1 Introduction to Calculus 1.1 Introduction 1.1.1 Origin of Calculus The development of Calculus by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibnitz (1646–1716) is one of the most important achievements in the history of science and mathematics. Newton is without doubt one of the greatest mathematicians of all time. In his efforts to find a mathematical method that could explain universal gravitation, he devised what he called the method of fluxions . Today we call it differential and integral calculus . Newton was a private and secretive man, and for the most part kept his monumental discoveries for himself. He did not publish much, and the majority of his great works, like his famous Philosophiae Naturalis Principia Mathematica , had to be dragged out of him by the persistence of his friends. It is now well established that Newton and Leibnitz developed their own form of calculus independently, that Newton was first by about 10 years but did not publish, and that Leibnitz’s papers of 1684 and 1686 were the earliest publications on the subject. If you are interested in finding out more about Newton and Leibnitz, or the history of mathematics in general, consult the following website: http://wwwhistory.mcs.stand.ac.uk/history 3 4 CHAPTER 1. INTRODUCTION TO CALCULUS 1.1.2 The Two Branches of Calculus There are two basic geometric problems that call for the use of calculus: • Finding the slope of the tangent line to a curve at a given point. • Finding the area between a curve and the xaxis for a ≤ x ≤ b . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 12/05/2011 for the course MATHEMATIC 101 taught by Professor Dr.author during the Fall '11 term at Camosun College.
 Fall '11
 Dr.Author
 Calculus, Limits

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