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CalcI_Complete - CALCULUS I Paul Dawkins Calculus I Table...

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CALCULUS I Paul Dawkins
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Calculus I © 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx 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
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Calculus I © 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx Linear Approximations .......................................................................................................................... 353 Differentials ............................................................................................................................................ 356 Newton’s Method ................................................................................................................................... 359 Business Applications ............................................................................................................................ 364 Integrals ...................................................................................................................................... 370 Introduction ............................................................................................................................................ 370 Indefinite Integrals ................................................................................................................................. 371 Computing Indefinite Integrals .............................................................................................................. 377 Substitution Rule for Indefinite Integrals ............................................................................................... 387 More Substitution Rule .......................................................................................................................... 400 Area Problem ......................................................................................................................................... 413 The Definition of the Definite Integral ................................................................................................... 423 Computing Definite Integrals ................................................................................................................. 433 Substitution Rule for Definite Integrals ................................................................................................. 445 Applications of Integrals ........................................................................................................... 456 Introduction ............................................................................................................................................ 456 Average Function Value ......................................................................................................................... 457 Area Between Curves ............................................................................................................................. 460 Volumes of Solids of Revolution / Method of Rings ............................................................................. 471 Volumes of Solids of Revolution / Method of Cylinders ....................................................................... 481 More Volume Problems ......................................................................................................................... 489 Work ....................................................................................................................................................... 500 Extras .......................................................................................................................................... 504 Introduction ............................................................................................................................................ 504 Proof of Various Limit Properties .......................................................................................................... 505 Proof of Various Derivative Facts/Formulas/Properties ........................................................................ 521 Proof of Trig Limits ............................................................................................................................... 534 Proofs of Derivative Applications Facts/Formulas ................................................................................ 539 Proof of Various Integral Facts/Formulas/Properties ............................................................................. 550 Area and Volume Formulas ................................................................................................................... 562 Types of Infinity ..................................................................................................................................... 566 Summation Notation .............................................................................................................................. 570 Constants of Integration ......................................................................................................................... 572
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Calculus I © 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx 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.
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