# CalcI_Complete.pdf

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Published on 30 Oct 2014
School
Miami University
Department
Mathematics
Course
MTH 151
Professor
CALCULUS I
Paul Dawkins
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Calculus I
© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
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’Hospitals Rule .......................................................................................... 345
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Calculus I
© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx
Linear Approximations .......................................................................................................................... 351
Differentials............................................................................................................................................ 354
Newtons Method ................................................................................................................................... 357
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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