Short - Calculus

A Summary of Calculus
Karl Heinz Dovermann
Professor of Mathematics University of Hawaii

July 28, 2003

c Copyright 2003 by the author. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of theauthor. Printed in the United States of America. This publication was typeset using AMS-TEX, the American MathematA ical Society’s TEX macro system, and L TEX 2ε . The graphics were produced 1. with the help of Mathematica This is an incomplete draft which will undergo further changes.

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Mathematica Version 2.2, Wolfram Research, Inc., Champaign, Illinois (1993).

Contents
Preface 1 BasicConcepts 1.1 Real Numbers and Functions . . . . . . . . 1.2 Limits . . . . . . . . . . . . . . . . . . . . . 1.2.1 Two important estimates . . . . . . 1.3 More Limits . . . . . . . . . . . . . . . . . . 1.4 Continuous Functions . . . . . . . . . . . . 1.5 Lines . . . . . . . . . . . . . . . . . . . . . . 1.6 Tangent Lines and the Derivative . . . . . . 1.6.1 Derivatives without Limits . . . . . 1.7Secant Lines and the Derivative . . . . . . . 1.8 Differentiability implies Continuity . . . . . 1.9 Basic Examples of Derivatives . . . . . . . . 1.10 The Exponential and Logarithm Functions . 1.11 Differentiability on Closed Intervals . . . . . 1.12 Other Notations for the Derivative . . . . . 1.13 Rules of Differentiation . . . . . . . . . . . 1.13.1 Linearity of the Derivative . . . . . . 1.13.2Product and Quotient Rules . . . . . 1.13.3 Chain Rule . . . . . . . . . . . . . . 1.13.4 Hyperbolic Functions . . . . . . . . 1.13.5 Derivatives of Inverse Functions . . . 1.13.6 Implicit Differentiation . . . . . . . . 1.14 Related Rates . . . . . . . . . . . . . . . . . 1.15 Exponential Growth and Decay . . . . . . . 1.16 More Exponential Growth and Decay . . . 1.17 The Second and HigherDerivatives . . . . . 1.18 Numerical Methods . . . . . . . . . . . . . . i v 1 1 2 4 6 8 9 10 12 13 14 15 18 20 21 22 22 23 26 29 30 34 38 41 43 48 48

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1.18.1 Approximation by Differentials 1.18.2 Newton’s Method . . . . . . . . 1.18.3 Euler’s Method . . .. . . . . . 1.19 Table of Important Derivatives . . . .

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48 51 53 63 65 65 67 69 69 75 76 77 80 81 83 88 90 92 98 105 105 107 108 111 112 112 115 116 118 121 122 124 126 128 131 133 136 142

2 Global Theory 2.1 Cauchy’s Mean Value Theorem . . . . . .2.2 Unique Solutions of Differential Equations 2.3 The First Derivative and Monotonicity . . 2.3.1 Monotonicity on Intervals . . . . . 2.3.2 Monotonicity at a Point . . . . . . 2.4 The Second Derivative and Concavity . . 2.4.1 Concavity on Intervals . . . . . . . 2.4.2 Concavity at a Point . . . . . . . . 2.5 Local Extrema and Inflection Points . . . 2.6 Detection of Local Extrema . . . . . . . . 2.7Detection of Inflection Points . . . . . . . 2.8 Absolute Extrema of Functions . . . . . . 2.9 Optimization Story Problems . . . . . . . 2.10 Sketching Graphs . . . . . . . . . . . . . . 3 Integration 3.1 Properties of Areas . . . . . . . . . . . . . 3.2 Partitions and Sums . . . . . . . . . . . . 3.2.1 Upper and Lower Sums . . . . . . 3.2.2 Riemann Sums . . . . . . . . . . . 3.3 Limits and...