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Integrated Calculus/Pre-Calculus Furman University Department of Mathematics Greenville, SC, 29613 Integrated Calculus/Pre-Calculus Abstract Mark R. Woodard Contents JJ II J I Home Page Go Back Close Quit Abstract This work presents the traditional material of calculus I with some of the material from a traditional precalculus course interwoven throughout the discussion. Pre- calculus topics are discussed at or soon before the time they are needed, in order to facilitate the learning of the calculus material. Miniature animated demonstra- tions and interactive quizzes will be available to help the reader deepen his or her understanding of the material under discussion. Integrated This project is funded in part by the Mellon Foundation through the Mellon Furman- Calculus/Pre-Calculus Wofford Program. Many of the illustrations were designed by Furman undergraduate Mark R. Woodard student Brian Wagner using the PSTricks package. Thanks go to my wife Suzan, and my two daughters, Hannah and Darby for patience while this project was being produced. Title Page Contents JJ II J I Go Back Close Quit Page 2 of 191 Contents 1 Functions and their Properties 11 1.1 Functions & Cartesian Coordinates .................. 13 1.1.1 Functions and Functional Notation ............ 13 1.2 Cartesian Coordinates and the Graphical Representa- tion of Functions .................................. 20 1.3 Circles, Distances, Completing the Square ............ 22 1.3.1 Distance Formula and Circles ................. 22 Integrated 1.3.2 Completing the Square ...................... 23 Calculus/Pre-Calculus 1.4 Lines ............................................ 24 Mark R. Woodard 1.4.1 General Equation and Slope-Intercept Form .... 24 1.4.2 More On Slope ............................. 24 1.4.3 Parallel and Perpendicular Lines .............. 25 Title Page 1.5 Catalogue of Familiar Functions ..................... 27 1.5.1 Polynomials ................................ 27 Contents 1.5.2 Rational Functions .......................... 28 JJ II 1.5.3 Algebraic Functions ......................... 28 1.5.4 The Absolute Value and Other Piecewise De- J I fined Functions ............................. 29 Go Back 1.5.5 The Greatest Integer Function ................ 30 Close 1.6 New Functions From Old ........................... 31 1.6.1 New Functions Via Composition .............. 31 Quit 1.7 New Functions via Algebra ......................... 34 Page 3 of 191 1.8 Shifting and Scaling ............................... 35 2 Limits 38 2.1 Secant Lines, Tangent Lines, and Velocities ........... 39 2.1.1 The Tangent Line ........................... 39 2.1.2 Instantaneous Velocity ....................... 41 2.2 Simplifying Difference Quotients .................... 43 2.3 Limits (Intuitive Approach)......................... 45 2.4 Intervals via Absolute Values and Inequalities ......... 50 2.5 If – Then Statements .............................. 52 2.6 Rigorous Definition of Limit ........................ 54 2.7 Continuity ........................................ 56 2.7.1 Definitions Related to Continuity ............. 56 Integrated 2.7.2 Basic Theorems ............................. 57 Calculus/Pre-Calculus 2.7.3 The Intermediate Value Theorem ............. 59 Mark R. Woodard 3 The Derivative 61 3.1 Definition of Derivative ............................ 63 3.1.1 Slopes and Velocities Revisited ............... 63 Title Page 3.1.2 The Derivative ............................. 64 Contents 3.1.3 The Derivative as a Function ................. 64 3.2 Rules for Taking Derivatives ........................ 65 JJ II 3.3 Differentiability ................................... 70 J I 3.4 Applications of Derivatives as Rates of Change ........ 72 3.4.1 Overview .................................. 72 Go Back 3.4.2 Some Specific Applications ................... 73 Close Rectilinear Motion .......................... 73 Biology Applications ........................ 74 Quit Chemistry Applications ...................... 75 Page 4 of 191 Economics ................................. 75 3.5 The Chain Rule ................................... 76 3.6 Implicit Differentiation ............................. 78 3.7 Higher Derivatives ................................. 81 3.7.1 The Second Derivative ....................... 81 3.7.2 Third and Higher Derivatives ................. 81 3.8 Related Rates ..................................... 83 3.9 Linearization ..................................... 85 3.9.1 Mathematicizing the main idea ............... 85 4 An Interlude – Trigonometric Functions 87 4.1 Definition of Trigonometric Functions ................ 88 Integrated 4.1.1 Radian Measure ............................ 90 Calculus/Pre-Calculus 4.2 Trigonometric Functions at Special Values ............ 91 Mark R. Woodard 4.3 Properties of Trigonometric Functions................ 93 4.4 Graphs of Trigonometric Functions .................. 94 4.4.1 Graph of sin(x) and cos(x) ................... 94 Title Page 4.4.2 Graph of tan(x) and cot(x) .................. 94 4.4.3 Graph of sec(x) and csc(x) ................... 95 Contents 4.5 Trigonometric Identities ............................ 96 4.6 Derivatives of Trigonometric Functions ............... 97 JJ II 4.6.1 Two Important Trigonometric Limits .......... 97 J I 4.6.2 Derivatives of other Trigonometric Functions ... 99 Go Back 5 Applications of the Derivative 102 Close 5.1 Definition of Maximum and Minimum Values ......... 104 5.2 Overview ......................................... 110 Quit 5.3 Rolle’s Theorem ................................... 111 Page 5 of 191 5.4 The Mean Value Theorem .......................... 113 5.5 An Application of the Mean Value Theorem .......... 115 5.6 Monotonicity ..................................... 117 5.7 Concavity ........................................ 121 5.8 Vertical Asymptotes ............................... 124 5.9 Limits at Infinity; Horizontal Asymptotes ............ 126 5.9.1 A Simplification Technique ................... 127 5.10 Graphing ......................................... 129 5.11 Optimization ..................................... 133 5.12 Newton’s Method ................................. 137 5.13 Antiderivatives .................................... 140 5.14 Rectilinear Motion ................................ 142 Integrated 5.15 Antiderivatives Mini-Quiz .......................... 143 Calculus/Pre-Calculus Mark R. Woodard 6 Areas and Integrals 145 6.1 Summation Notation............................... 147 6.2 Computing Some Sums ............................ 149 6.3 Areas of Planar Regions ............................ 154 Title Page 6.3.1 Approximating with Rectangles ............... 155 Contents 6.4 The Definite Integral .............................. 159 6.4.1 Definition and Notation ...................... 159 JJ II 6.4.2 Properties of R b f(x) dx. ..................... 162 a J I 6.4.3 Relation of the Definite Integral to Area ....... 162 6.4.4 Another Interpretation of the Definite Integral .. 164 Go Back 6.5 The Fundamental Theorem ........................ 166 Close 6.5.1 The Fundamental Theorem................... 166 6.5.2 The Fundamental Theorem – Part II .......... 168 Quit 6.6 The Indefinite Integral and Substitution .............. 170 Page 6 of 191 6.6.1 The Indefinite Integral ....................... 170 6.6.2 Substitution in Indefinite Integrals ............ 173 6.6.3 Substitution in Definite Integrals .............. 176 6.7 Areas Between Curves ............................. 179 6.7.1 Vertically Oriented Areas .................... 179 6.7.2 Horizontally Oriented Areas .................. 181 7 Appendix 184 7.1 3 – Minute Reviews ................................ 185 7.1.1 The Real Numbers .......................... 185 7.1.2 Factoring .................................. 186 7.1.3 Solving Equations ........................... 186 Integrated 7.1.4 Solving Systems of Equations ................. 187 Calculus/Pre-Calculus 7.1.5 Absolute Value ............................. 188 Mark R. Woodard 7.1.6 Quadratics ................................. 189 7.1.7 Exponents ................................. 190 7.1.8 Inequalities ................................ 190 7.1.9 Setting Up Word Problems ................... 190 Title Page Solutions to Exercises .............................. 192 Contents Solutions to Quizzes ............................... 309 JJ II J I Go Back Close Quit Page 7 of 191 Preface Every teacher of calculus has at one time or another remarked about a particular student or group of students “I think they understand the cal- culus concepts, but their algebra skills are so weak, they can’t finish the problems.” There are many reasons that explain why this problem is so pre- velant: some students never learned the precalculus material, some learned it but forgot most of it, others are just a little rusty and need some extra practice. For these reasons, some colleges and universities are starting so called “integrated” precalculus/calculus courses. This text is designed to Integrated be used in exactly those kinds of courses, although it could also be used in Calculus/Pre-Calculus a traditional calculus course, with the precalculus material omitted by the Mark R. Woodard instructor, but available nonetheless to the motivated and interested stu- dent who needs or desires some extra amplifications of, reminders about, or practice with the precalculus material. An attempt is made to have the precalculus material appear “just in time”, that is, directly before it is Title Page needed for the first time. Contents In addition, this document is unusual in that the online version
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