CALCULUS I

CALCULUS IPaul Dawkins

Calculus ITable of ContentsPreface . iiiOutline . ivReview. 2Introduction . 2Review : Functions . 4Review : Inverse Functions . 10Review : Trig Functions . 17Review : Solving Trig Equations . 24Review : Solving Trig Equations with Calculators, Part I . 33Review : Solving Trig Equations with Calculators, Part II . 44Review : Exponential Functions . 49Review : Logarithm Functions . 52Review : Exponential and Logarithm Equations . 58Review : Common Graphs . 64Limits . 76Introduction . 76Rates of Change and Tangent Lines . 78The Limit . 87One‐Sided Limits . 97Limit Properties .103Computing Limits .109Infinite Limits .117Limits At Infinity, Part I .126Limits At Infinity, Part II .135Continuity.144The Definition of the Limit .151Derivatives . 166Introduction .166The Definition of the Derivative .168Interpretations of the Derivative .174Differentiation Formulas .179Product and Quotient Rule .187Derivatives of Trig Functions .193Derivatives of Exponential and Logarithm Functions .204Derivatives of Inverse Trig Functions .209Derivatives of Hyperbolic Functions .215Chain Rule .217Implicit Differentiation .227Related Rates .236Higher Order Derivatives .250Logarithmic Differentiation .255Applications of Derivatives . 258Introduction .258Rates of Change.260Critical Points .263Minimum and Maximum Values .269Finding Absolute Extrema .277The Shape of a Graph, Part I.283The Shape of a Graph, Part II .292The Mean Value Theorem .301Optimization .308More Optimization Problems .322 2007 Paul Dawkinsihttp://tutorial.math.lamar.edu/terms.aspx

Calculus IIndeterminate Forms and L’Hospital’s Rule .336Linear Approximations .342Differentials .345Newton’s Method.348Business Applications .353Integrals. 359Introduction .359Indefinite Integrals .360Computing Indefinite Integrals .366Substitution Rule for Indefinite Integrals .376More Substitution Rule .389Area Problem .402The Definition of the Definite Integral .412Computing Definite Integrals .422Substitution Rule for Definite Integrals .434Applications of Integrals . 445Introduction .445Average Function Value .446Area Between Curves .449Volumes of Solids of Revolution / Method of Rings .460Volumes of Solids of Revolution / Method of Cylinders .470Work .478Extras. 482Introduction .482Proof of Various Limit Properties .483Proof of Various Derivative Facts/Formulas/Properties .494Proof of Trig Limits .507Proofs of Derivative Applications Facts/Formulas .512Proof of Various Integral Facts/Formulas/Properties .523Area and Volume Formulas .535Types of Infinity .539Summation Notation .543Constants of Integration .545 2007 Paul Dawkinsiihttp://tutorial.math.lamar.edu/terms.aspx

Calculus IPrefaceHere are my online notes for my Calculus I course that I teach here at Lamar University. Despitethe fact that these are my “class notes” they should be accessible to anyone wanting to learnCalculus 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 toread through them is either from an Algebra or Trig class or contained in other sections of thenotes.Here are a couple of warnings to my students who may be here to get a copy of what happened ona day that you missed.1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learncalculus I have included some material that I do not usually have time to cover in classand because this changes from semester to semester it is not noted here. You will need tofind one of your fellow class mates to see if there is something in these notes that wasn’tcovered in class.2. Because I want these notes to provide some more examples for you to read through, Idon’t always work the same problems in class as those given in the notes. Likewise, evenif I do work some of the problems in here I may work fewer problems in class than arepresented here.3. Sometimes questions in class will lead down paths that are not covered here. I try toanticipate as many of the questions as possible when writing these up, but the reality isthat I can’t antici

Derivatives of Trig Functions – We’ll give the derivatives of the trig functions in this section. Derivatives of Exponential and Logarithm Functions – In this section we will get the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions – Here we will look at the derivatives of inverse trig functions.