CALCULUS I Paul Dawkins Calculus I Table of Contents Preface iii Outline iv Review 2 Introduction 2 Review Functions 4 Review Inverse Functions 14 Review Trig Functions 21 Review Solving Trig Equations 28 Review Solving Trig Equations with Calculators Part I 37 Review Solving Trig Equations with Calculators Part II 48 Review Exponential Functions 53 Review Logarithm Functions 56 Review Exponential and Logarithm Equations 62 Review Common Graphs 68 Limits 80 Introduction 80 Rates of Change and Tangent Lines 82 The Limit 91 One Sided Limits 101 Limit Properties 107 Computing Limits 113 Infinite Limits 121 Limits At Infinity Part I 131 Limits At Infinity Part II 140 Continuity 150 The Definition of the Limit 157 Derivatives 172 Introduction 172 The Definition of the Derivative 174 Interpretations of the Derivative 180 Differentiation Formulas 189 Product and Quotient Rule 197 Derivatives of Trig Functions 203 Derivatives of Exponential and Logarithm Functions 214 Derivatives of Inverse Trig Functions 219 Derivatives of Hyperbolic Functions 225 Chain Rule 227 Implicit Differentiation 237 Related Rates 246 Higher Order Derivatives 260 Logarithmic Differentiation 265 Applications of Derivatives 268 Introduction 268 Rates of Change 270 Critical Points 273 Minimum and Maximum Values 279 Finding Absolute Extrema 287 The Shape of a Graph Part I 293 The Shape of a Graph Part II 302 The Mean Value Theorem 311 Optimization 318 More Optimization Problems 332 Indeterminate Forms and L Hospital s Rule 347 2007 Paul Dawkins i http tutorial math lamar edu terms aspx Calculus I Linear Approximations 354 Differentials 357 Newton s Method 360 Business Applications 365 Integrals 371 Introduction 371 Indefinite Integrals 372 Computing Indefinite Integrals 378 Substitution Rule for Indefinite Integrals 388 More Substitution Rule 401 Area Problem 414 The Definition of the Definite Integral 424 Computing Definite Integrals 434 Substitution Rule for Definite Integrals 446 Applications of Integrals 457 Introduction 457 Average Function Value 458 Area Between Curves 461 Volumes of Solids of Revolution Method of Rings 472 Volumes of Solids of Revolution Method of Cylinders 482 More Volume Problems 490 Work 501 Extras 508 Introduction 508 Proof of Various Limit Properties 509 Proof of Various Derivative Facts Formulas Properties 525 Proof of Trig Limits 538 Proofs of Derivative Applications Facts Formulas 543 Proof of Various Integral Facts Formulas Properties 554 Area and Volume Formulas 566 Types of Infinity 570 Summation Notation 574 Constants of Integration 576 2007 Paul Dawkins ii http tutorial math lamar edu terms aspx Calculus I Preface Here are my online notes for my Calculus I course that I teach here at Lamar University Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus I or needing a refresher in some of the early topics in calculus I ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from an Algebra or Trig class or contained in other sections of the notes Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed 1 Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here You will need to find one of your fellow class mates to see if there is something in these notes that wasn t covered in class 2 Because I want these notes to provide some more examples for you to read through I don t always work the same problems in class as those given in the notes Likewise even if I do work some of the problems in here I may work fewer problems in class than are presented here 3 Sometimes questions in class will lead down paths that are not covered here I try to anticipate as many of the questions as possible when writing these up but the reality is that I can t anticipate all the questions Sometimes a very good question gets asked in class that leads to insights that I ve not included here You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are 4 This is somewhat related to the previous three items but is important enough to merit its own item THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS Using these notes as a substitute for class is liable to get you in trouble As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class 2007 Paul Dawkins iii http tutorial math lamar edu terms aspx Calculus I Outline Here is a listing and brief description of the material in this set of notes Review Review Functions Here is a quick review of functions function notation and a couple of fairly important ideas about functions Review Inverse Functions A quick review of inverse functions and the notation for inverse functions Review Trig Functions A review of trig functions evaluation of trig functions and the unit circle This section usually gets a quick review in my class Review Solving Trig Equations A reminder on how to solve trig equations This section is always covered in my class Review Solving Trig Equations with Calculators Part I The previous section worked problems whose answers were always the standard angles In this section we work some problems whose answers are not standard and so a calculator is needed This section is always covered in my class as most trig equations in the remainder will need a calculator Review Solving Trig Equations with Calculators Part II Even more trig equations requiring a calculator to solve Review Exponential Functions A review of exponential functions This section usually gets a quick review in my class Review Logarithm Functions A review of logarithm functions and logarithm properties This section usually gets a quick review in my class Review Exponential and Logarithm Equations How to solve exponential and logarithm equations This section is always covered in my class Review Common Graphs This section isn t much It s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class Limits Tangent Lines and Rates of Change In this section we will take a look at two problems that we will see time and
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