CALCULUS   I 
                              
                         Paul Dawkins 
                              
          
                                                                Calculus I 
                                                          Table of Contents 
                     
                    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 
                    © 2007 Paul Dawkins                         i               http://tutorial.math.lamar.edu/terms.aspx 
                     
                                                                                                         Calculus I 
                                     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 
                                 © 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