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CalcI_Complete

# 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 ......................................................................................................................... 130 Limits At Infinity, Part II ....................................................................................................................... 139 Continuity ............................................................................................................................................... 148 The Definition of the Limit .................................................................................................................... 155 Derivatives .................................................................................................................................. 170 Introduction ............................................................................................................................................ 170 The Definition of the Derivative ............................................................................................................ 172 Interpretations of the Derivative ............................................................................................................. 178 Differentiation Formulas ........................................................................................................................ 187 Product and Quotient Rule ..................................................................................................................... 195 Derivatives of Trig Functions ................................................................................................................. 201 Derivatives of Exponential and Logarithm Functions ............................................................................ 212 Derivatives of Inverse Trig Functions .................................................................................................... 217 Derivatives of Hyperbolic Functions ..................................................................................................... 223 Chain Rule .............................................................................................................................................. 225 Implicit Differentiation .......................................................................................................................... 235 Related Rates .......................................................................................................................................... 244 Higher Order Derivatives ....................................................................................................................... 258 Logarithmic Differentiation ................................................................................................................... 263 Applications of Derivatives ....................................................................................................... 266 Introduction ............................................................................................................................................ 266 Rates of Change ..................................................................................................................................... 268 Critical Points ......................................................................................................................................... 271 Minimum and Maximum Values ........................................................................................................... 277 Finding Absolute Extrema ..................................................................................................................... 285 The Shape of a Graph, Part I .................................................................................................................. 291 The Shape of a Graph, Part II ................................................................................................................. 300 The Mean Value Theorem ...................................................................................................................... 309 Optimization ........................................................................................................................................... 316 More Optimization Problems ................................................................................................................. 330 Indeterminate Forms and L’Hospital’s Rule .......................................................................................... 345
Calculus I © 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx Linear Approximations .......................................................................................................................... 351 Differentials ............................................................................................................................................ 354 Newton’s Method ................................................................................................................................... 357 Business Applications ............................................................................................................................ 362 Integrals ...................................................................................................................................... 368 Introduction ............................................................................................................................................ 368 Indefinite Integrals ................................................................................................................................. 369 Computing Indefinite Integrals .............................................................................................................. 375 Substitution Rule for Indefinite Integrals ............................................................................................... 385 More Substitution Rule .......................................................................................................................... 398 Area Problem ......................................................................................................................................... 411 The Definition of the Definite Integral ................................................................................................... 421 Computing Definite Integrals ................................................................................................................. 431 Substitution Rule for Definite Integrals ................................................................................................. 443 Applications of Integrals ........................................................................................................... 454 Introduction ............................................................................................................................................ 454 Average Function Value ......................................................................................................................... 455 Area Between Curves ............................................................................................................................. 458 Volumes of Solids of Revolution / Method of Rings ............................................................................. 469 Volumes of Solids of Revolution / Method of Cylinders ....................................................................... 479 More Volume Problems ......................................................................................................................... 487 Work ....................................................................................................................................................... 498 Extras .......................................................................................................................................... 502 Introduction ............................................................................................................................................ 502 Proof of Various Limit Properties .......................................................................................................... 503 Proof of Various Derivative Facts/Formulas/Properties ........................................................................ 514 Proof of Trig Limits ............................................................................................................................... 527 Proofs of Derivative Applications Facts/Formulas ................................................................................ 532 Proof of Various Integral Facts/Formulas/Properties ............................................................................. 543 Area and Volume Formulas ................................................................................................................... 555 Types of Infinity ..................................................................................................................................... 559 Summation Notation .............................................................................................................................. 563 Constants of Integration ......................................................................................................................... 565

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

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