Calculus of One and Many Variables Kenneth Kuttler [email protected] September 19, 2021 2 Contents I Functions of One Variable 15 1 Fundamental Concepts 17 1.1 Numbers and Simple Algebra ...................... 17 1.2 Exercises ................................. 20 1.3 Set Notation ............................... 21 1.4 Order ................................... 22 1.5 Exercises ................................. 27 1.6 The Binomial Theorem ......................... 28 1.7 Well Ordering Principle, Math Induction ................ 29 1.8 Exercises ................................. 32 1.9 Completeness of R ............................ 35 1.10 Existence of Roots ............................ 36 1.11 Completing the Square .......................... 38 1.12 Dividing Polynomials ........................... 39 1.13 The Complex Numbers ......................... 41 1.14 Polar Form of Complex Numbers .................... 44 1.15 Roots of Complex Numbers ....................... 45 1.16 Exercises ................................. 47 1.17 Videos ................................... 50 2 Functions 51 2.1 General Considerations .......................... 51 2.2 Graphs of Functions and Relations ................... 54 2.3 Circular Functions ............................ 55 2.3.1 Reference Angles and Other Identities ............. 62 2.3.2 The sin (x) =x Inequality ..................... 64 2.3.3 The Area of a Circular Sector .................. 67 2.4 Exercises ................................. 69 2.5 Exponential and Logarithmic Functions ................ 73 2.6 The Function bx ............................. 78 2.7 Applications ................................ 79 2.7.1 Interest Compounded Continuously .............. 79 2.7.2 Exponential Growth and Decay ................. 79 2.7.3 The Logistic Equation ...................... 81 2.8 Using MATLAB to Graph ........................ 82 2.9 Exercises ................................. 82 3 4 CONTENTS 2.10 Videos ................................... 84 3 Sequences and Compactness 85 3.1 Sequences ................................. 85 3.2 Exercises ................................. 86 3.3 The Limit of a Sequence ......................... 88 3.4 The Nested Interval Lemma ....................... 94 3.5 Exercises ................................. 95 3.6 Compactness ............................... 96 3.6.1 Sequential Compactness ..................... 96 3.6.2 Closed and Open Sets ...................... 97 3.7 Cauchy Sequences ............................ 100 3.8 Exercises ................................. 101 3.9 Videos ................................... 104 4 Continuous Functions and Limits of Functions 105 4.1 An Equivalent Formulation of Continuity ............... 109 4.2 Exercises ................................. 110 4.3 The Extreme Values Theorem ...................... 112 4.4 The Intermediate Value Theorem .................... 112 4.5 Continuity of the Inverse ......................... 114 4.6 Exercises ................................. 115 4.7 Uniform Continuity ............................ 116 4.8 Examples of Continuous Functions ................... 117 4.9 Sequences of Functions .......................... 118 4.10 Polynomials and Continuous Functions ................. 120 4.11 Exercises ................................. 123 4.12 Limit of a Function ............................ 124 4.13 Exercises ................................. 128 4.14 Videos ................................... 130 5 The Derivative 131 5.1 The Definition of the Derivative ..................... 131 5.2 Finding the Derivative .......................... 134 5.3 Derivatives of Inverse Functions ..................... 135 5.4 Circular Functions and Inverses ..................... 137 5.5 Exponential Functions and Logarithms ................. 139 5.6 The Complex Exponential ........................ 140 5.7 Related Rates and Implicit Differentiation ............... 141 5.8 Exercises ................................. 142 5.9 Local Extreme Points .......................... 143 5.10 Exercises ................................. 145 5.11 Mean Value Theorem ........................... 149 5.12 Exercises ................................. 150 5.13 First and Second Derivative Tests .................... 152 5.14 Exercises ................................. 153 5.15 Taylor Series Approximations ...................... 154 5.16 Exercises ................................. 156 CONTENTS 5 5.17 L'H^opital'sRule ............................. 157 5.18 Interest Compounded Continuously ................... 162 5.19 Exercises ................................. 162 5.20 Videos ................................... 164 6 Infinite Series 165 6.1 Basic Considerations ........................... 165 6.2 Absolute Convergence .......................... 168 6.3 Ratio and Root Tests .......................... 172 6.4 Exercises ................................. 173 6.5 Convergence Because of Cancellation .................. 174 6.6 Double Series ............................... 176 6.7 Exercises ................................. 179 6.8 Series of Functions ............................ 181 6.9 Exercises ................................. 182 7 The Integral 185 7.1 The Definition of the Integral ...................... 188 7.2 Uniform Convergence and the Integral ................. 192 7.3 The Riemann Darboux Integral∗ .................... 192 7.4 Exercises ................................. 200 7.5 Videos ................................... 204 8 Methods for Finding Antiderivatives 205 8.1 The Method of Substitution ....................... 205 8.2 Exercises ................................. 207 8.3 Integration by Parts ........................... 209 8.4 Exercises ................................. 211 8.5 Trig. Substitutions ............................ 213 8.6 Exercises ................................. 218 8.7 Partial Fractions ............................. 219 8.8 Rational Functions of Trig. Functions ................. 224 8.9 Using MATLAB ............................. 226 8.10 Exercises ................................. 226 8.11 Videos ................................... 228 9 A Few Standard Applications 229 9.1 Lengths of Curves and Areas of Surfaces of Revolution ........ 230 9.1.1 Lengths .............................. 230 9.1.2 Surfaces of Revolution ...................... 232 9.2 Exercises ................................. 234 9.3 Force on a Dam and Work ........................ 236 9.3.1 Force on a Dam ......................... 236 9.3.2 Work ............................... 236 9.4 Using MATLAB ............................. 238 9.5 Exercises ................................. 238 6 CONTENTS 10 Improper Integrals and Stirling's Formula 241 10.1 Stirling's Formula ............................. 241 10.2 The Gamma Function .......................... 244 10.3 Laplace Transforms ............................ 247 10.4 Exercises ................................. 251 11 Power Series 257 11.1 Functions Defined in Terms of Series .................. 257 11.2 Operations on Power Series ....................... 259 11.3 Power Series for Some Known Functions ................ 262 11.4 The Binomial Theorem ......................... 263 11.5 Exercises ................................. 265 11.6 Multiplication of Power Series ...................... 266 11.7 Exercises ................................. 268 11.8 Some Other Theorems .......................... 270 11.9 Some Historical Observations ...................... 273 12 Polar Coordinates 275 12.1 Graphs in Polar Coordinates ...................... 276 12.2 The Area in Polar Coordinates ..................... 277 12.3 The Acceleration in Polar Coordinates ................. 279 12.4 The Fundamental Theorem of Algebra ................. 281 12.5 Polar Graphing in MATLAB ...................... 283 12.6 Exercises ................................. 284 13 Algebra and Geometry of Rp 285 13.1 Rp ..................................... 285 13.2 Algebra in Rp ............................... 287 13.3 Geometric Meaning Of Vector Addition In R3 ............. 288 13.4 Lines .................................... 289 13.5 Distance in Rp .............................. 292 13.6 Geometric Meaning of Scalar Multiplication in R3 .......... 296 13.7 Exercises ................................. 296 14 Vector Products 299 14.1 The Dot Product ............................. 299 14.2 Geometric Significance of the Dot Product ............... 301 14.2.1 The Angle Between Two Vectors ................ 301 14.2.2 Work and Projections ...................... 303 14.3 Exercises ................................. 306 14.4 The Cross Product ............................ 307 14.4.1 The Box Product ......................... 310 14.5 Proof of the Distributive Law ...................... 311 14.5.1 Torque ............................... 312 14.5.2 Center of Mass .......................... 313 14.5.3 Angular Velocity ......................... 314 14.6 Vector Identities and Notation ..................... 315 14.7 Planes ................................... 318 14.8 Exercises ................................. 321 CONTENTS 7 15 Sequences, Compactness, and Continuity 325 15.1 Sequences of Vectors ........................... 325 15.2 Open and Closed Sets .......................... 326 15.3 Cartesian Products ............................ 329 15.4 Sequential Compactness ......................... 330 15.5 Vector Valued Functions ......................... 330 15.6 Continuous Functions .......................... 331 15.7 Sufficient Conditions for Continuity ................... 332 15.8 Limits of a Function of Many