Link : http://rapidshare.com/files/115287749/CalcI_Complete.pdf

Preface ........................................................................................................................................... iii 

Outline ........................................................................................................................................... iv 

Review............................................................................................................................................. 2 

Introduction .............................................................................................................................................. 2 

Review : Functions ................................................................................................................................... 4 

Review : Inverse Functions ....................................................................................................................  10 

Review : Trig Functions ......................................................................................................................... 17 

Review : Solving Trig Equations ............................................................................................................  24 

Review : Solving Trig Equations with Calculators, Part I .................................................................... 33 

Review : Solving Trig Equations with Calculators, Part II ................................................................... 44 

Review : Exponential Functions ............................................................................................................ 49 

Review : Logarithm Functions ...............................................................................................................  52 

Review : Exponential and Logarithm Equations .................................................................................. 58 

Review : Common Graphs ......................................................................................................................  64 

Limits ............................................................................................................................................ 76 

Introduction ............................................................................................................................................ 76 

Rates of Change and Tangent Lines ...................................................................................................... 78 

The Limit ................................................................................................................................................. 87 

One?Sided Limits .................................................................................................................................... 97 

Limit Properties .....................................................................................................................................103 

Computing Limits ..................................................................................................................................109 

Infinite Limits ........................................................................................................................................117 

Limits At Infinity, Part I .........................................................................................................................126 

Limits At Infinity, Part II .......................................................................................................................135 

Continuity ...............................................................................................................................................144 

The Definition of the Limit ....................................................................................................................151 

Derivatives .................................................................................................................................. 166 

Introduction ...........................................................................................................................................166 

The Definition of the Derivative ...........................................................................................................168 

Interpretations of the Derivative .........................................................................................................174 

Differentiation Formulas ......................................................................................................................179 

Product and Quotient Rule ...................................................................................................................187 

Derivatives of Trig Functions ...............................................................................................................193 

Derivatives of Exponential and Logarithm Functions ........................................................................204 

Derivatives of Inverse Trig Functions ..................................................................................................209 

Derivatives of Hyperbolic Functions ....................................................................................................215 

Chain Rule ..............................................................................................................................................217 

Implicit Differentiation .........................................................................................................................227 

Related Rates .........................................................................................................................................236 

Higher Order Derivatives ......................................................................................................................250 

Logarithmic Differentiation ..................................................................................................................255 

Applications of Derivatives ....................................................................................................... 258 

Introduction ...........................................................................................................................................258 

Rates of Change......................................................................................................................................260 

Critical Points .........................................................................................................................................263 

Minimum and Maximum Values ...........................................................................................................269 

Finding Absolute Extrema ....................................................................................................................277 

The Shape of a Graph, Part I ..................................................................................................................283 

The Shape of a Graph, Part II ................................................................................................................292 

The Mean Value Theorem .....................................................................................................................301 

Optimization ..........................................................................................................................................308 

More Optimization Problems ...............................................................................................................322 Calculus I

© 2007 Paul Dawkins  ii  http://tutorial.math.lamar.edu/terms.aspx

 

Indeterminate Forms and L’Hospital’s Rule ........................................................................................336 

Linear Approximations .........................................................................................................................342 

Differentials ...........................................................................................................................................345 

Newton’s Method ...................................................................................................................................348 

Business Applications ...........................................................................................................................353 

Integrals ...................................................................................................................................... 359 

Introduction ...........................................................................................................................................359 

Indefinite Integrals ................................................................................................................................360 

Computing Indefinite Integrals ............................................................................................................366 

Substitution Rule for Indefinite Integrals ............................................................................................376 

More Substitution Rule .........................................................................................................................389 

Area Problem .........................................................................................................................................402 

The Definition of the Definite Integral .................................................................................................412 

Computing Definite Integrals ...............................................................................................................422 

Substitution Rule for Definite Integrals ...............................................................................................434 

Applications of Integrals ........................................................................................................... 445 

Introduction ...........................................................................................................................................445 

Average Function Value ........................................................................................................................446 

Area Between Curves ............................................................................................................................449 

Volumes of Solids of Revolution / Method of Rings ............................................................................460 

Volumes of Solids of Revolution / Method of Cylinders .....................................................................470 

Work .......................................................................................................................................................478 

Extras .......................................................................................................................................... 482 

Introduction ...........................................................................................................................................482 

Proof of Various Limit Properties ........................................................................................................483 

Proof of Various Derivative Facts/Formulas/Properties ...................................................................494 

Proof of Trig Limits ...............................................................................................................................507 

Proofs of Derivative Applications Facts/Formulas .............................................................................512 

Proof of Various Integral Facts/Formulas/Properties .......................................................................523 

Area and Volume Formulas ..................................................................................................................535 

Types of Infinity .....................................................................................................................................539 

Summation Notation .............................................................................................................................543 

Constants of Integration .......................................................................................................................545