Differential Calculus and Sage
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Differential Calculus and Sage William Granville and David Joyner Contents ii CONTENTS 0 Preface ix 1 Variables and functions 1 1.1 Variablesandconstants . 1 1.2 Intervalofavariable ........................ 2 1.3 Continuous variation . 2 1.4 Functions .............................. 2 1.5 Notation of functions . 3 1.6 Independent and dependent variables . 5 1.7 Thedomainofafunction . 6 1.8 Exercises .............................. 7 2 Theory of limits 9 2.1 Limitofavariable.......................... 9 2.2 Division by zero excluded . 12 2.3 Infinitesimals ............................ 12 2.4 The concept of infinity ( )..................... 13 2.5 Limiting value of a function∞ . 14 2.6 Continuous and discontinuous functions . 14 2.7 Continuity and discontinuity of functions illustrated by their graphs 18 2.8 Fundamental theorems on limits . 26 2.9 Speciallimitingvalues . 31 sin x 2.10 Show that limx 0 = 1 ..................... 32 → x 2.11 The number e ............................ 34 2.12 Expressions assuming the form ∞ ................. 36 2.13Exercises ..............................∞ 37 iii CONTENTS 3 Differentiation 41 3.1 Introduction............................. 41 3.2 Increments.............................. 42 3.3 Comparison of increments . 43 3.4 Derivative of a function of one variable . 44 3.5 Symbols for derivatives . 45 3.6 Differentiable functions . 46 3.7 General rule for differentiation . 46 3.8 Exercises .............................. 50 3.9 Applications of the derivative to geometry . 52 3.10Exercises .............................. 55 4 Rules for differentiating standard elementary forms 59 4.1 Importance of General Rule . 59 4.2 Differentiation of a constant . 63 4.3 Differentiation of a variable with respect to itself . ...... 63 4.4 Differentiation of a sum . 64 4.5 Differentiation of the product of a constant and a function . 64 4.6 Differentiation of the product of two functions . 65 4.7 Differentiation of the product of any finite number of functions . 65 4.8 Differentiation of a function with a constant exponent . ...... 66 4.9 Differentiation of a quotient . 67 4.10Examples .............................. 68 4.11 Differentiation of a function of a function . 72 4.12 Differentiation of inverse functions . 74 4.13 Differentiation of a logarithm . 77 4.14 Differentiation of the simple exponential function . ....... 78 4.15 Differentiation of the general exponential function . ........ 79 4.16 Logarithmic differentiation . 81 4.17Examples .............................. 83 4.18 Differentiation of sin v ....................... 86 4.19 Differentiation of cos v ....................... 87 4.20 Differentiation of tan v ....................... 87 4.21 Differentiation of cot v ....................... 88 4.22 Differentiation of sec v ....................... 88 4.23 Differentiation of csc v ....................... 88 4.24Exercises .............................. 89 4.25 Differentiation of arcsin v ...................... 93 iv CONTENTS 4.26 Differentiation of arccos v ..................... 95 4.27 Differentiation of arctan v ..................... 96 4.28 Differentiation of arccot v ..................... 97 4.29 Differentiation of arcsec v ..................... 98 4.30 Differentiation of arccsc v .....................100 4.31 Example...............................101 4.32 Implicit functions . 107 4.33 Differentiation of implicit functions . 107 4.34 Exercises ..............................108 4.35 MiscellaneousExercises . 110 5 Simple applications of the derivative 115 5.1 Directionofacurve. 115 5.2 Exercises ..............................121 5.3 Equations of tangent and normal lines . 123 5.4 Exercises ..............................126 5.5 Parametric equations of a curve . 130 5.6 Exercises ..............................136 5.7 Angle between radius vector and tangent . 140 5.8 Lengths of polar subtangent and polar subnormal . 144 5.9 Examples ..............................145 5.10 Solution of equations having multiple roots . 148 5.11 Examples ..............................149 5.12 Applications of the derivative in mechanics . 152 5.13 Component velocities. Curvilinear motion . 154 5.14 Acceleration. Rectilinear motion . 156 5.15 Component accelerations. Curvilinear motion . 156 5.16 Examples ..............................157 5.17 Application: Newton’s method . 163 5.17.1 Description of the method . 163 5.17.2 Analysis ..........................164 5.17.3 Fractals ...........................165 6 Successive differentiation 167 6.1 Definition of successive derivatives . 167 6.2 Notation...............................167 6.3 The n-thderivative .........................168 6.4 Leibnitz’s Formula for the n-th derivative of a product . 169 v CONTENTS 6.5 Successive differentiation of implicit functions . 170 6.6 Exercises ..............................172 7 Maxima, minima and inflection points 177 7.1 Introduction.............................177 7.2 Increasing and decreasing functions . 183 7.3 Tests for determining when a function is increasing or decreasing . 185 7.4 Maximum and minimum values of a function . 186 7.5 Examining a function for extremal values: first method . 189 7.6 Examining a function for extremal values: second method . 191 7.7 Problems ..............................194 7.8 Pointsofinflection . 213 7.9 Examples ..............................214 7.10 Curve plotting . 216 7.11 Exercises ..............................217 8 Application to arclength and rates 221 8.1 Introduction.............................221 8.2 Definitions..............................222 8.3 Derivative of the arclength in rectangular coordinates .......223 8.4 Derivative of the arclength in polar coordinates . 225 8.5 Exercises ..............................227 8.6 The derivative considered as the ratio of two rates . 228 8.7 Exercises ..............................230 9 Change of variable 239 9.1 Interchange of dependent and independent variables . 239 9.2 Change of the dependent variable . 241 9.3 Change of the independent variable . 242 9.4 Change of independent and dependent variables . 243 9.5 Exercises ..............................245 10 Applications of higher derivatives 249 10.1 Rolle’sTheorem. 249 10.2 Themeanvaluetheorem . 251 10.3 The extended mean value theorem . 253 10.4 Exercises ..............................254 10.5 Maxima and minima treated analytically . 255 vi CONTENTS 10.6 Exercises ..............................257 10.7 Indeterminate forms . 258 10.8 Evaluation of a function taking on an indeterminate form . 259 0 10.9 Evaluation of the indeterminate form 0 ...............260 0 10.9.1 Rule for evaluating the indeterminate form 0 .......264 10.9.2 Exercises ..........................266 10.10Evaluation of the indeterminate form ∞ ..............267 10.11Evaluation of the indeterminate form 0∞ ............268 · ∞ 10.12Evaluation of the indeterminate form ...........269 ∞ − ∞ 10.12.1Exercises . 270 0 0 10.13Evaluation of the indeterminate forms 0 , 1∞, .........272 ∞ 10.14Exercises ..............................273 10.15Application: Using Taylor’s Theorem to approximate functions. 275 10.16Example/Application: finite difference methods . 281 11 Curvature 285 11.1 Curvature ..............................285 11.2 Curvature of a circle . 285 11.3 Curvature at a point . 287 11.4 Formulas for curvature . 288 11.5 Radius of curvature . 291 11.6 Circle of curvature . 293 11.7 Exercises ..............................296 11.8 Circle of curvature . 301 11.9 Second method for finding center of curvature . 304 11.10Centerofcurvature . 306 11.11Evolutes...............................308 11.12Properties of the evolute . 312 11.13Exercises ..............................313 12 Appendix: Collection of formulas 317 12.1 Formulas for reference . 317 12.2 Greekalphabet ...........................322 12.3 Rules for signs of the trigonometric functions . 322 12.4 Natural values of the trigonometric functions . 322 vii CONTENTS 13 Appendix: A mini-Sage tutorial 325 13.1 Ways to Use Sage .........................326 13.2 Longterm Goals for Sage .....................327 13.3 The Sage commandline......................329 13.4 The Sage notebook ........................330 13.5 Aguidedtour ............................333 13.5.1 Assignment, Equality, and Arithmetic . 333 13.5.2 Getting Help . 335 13.5.3 Basic Algebra and Calculus . 337 13.5.4 Plotting ...........................340 13.5.5 Some common issues with functions . 343 13.6Tryit! ................................347 14 Appendix: GNU Free Documentation License 349 1. APPLICABILITY AND DEFINITIONS . 349 2.VERBATIMCOPYING . 351 3.COPYINGINQUANTITY . 351 4.MODIFICATIONS. 352 5.COMBININGDOCUMENTS. 354 6.COLLECTIONSOFDOCUMENTS . 355 7. AGGREGATION WITH INDEPENDENT WORKS . 355 8.TRANSLATION ............................355 9.TERMINATION ............................356 10. FUTURE REVISIONS OF THIS LICENSE . 356 11.RELICENSING............................356 ADDENDUM: How to use this License for your documents . 357 viii CHAPTER ZERO Preface This is a free and open source differential calculus book. The “free and open source” part means you, as a student, can give digital versions of this book to any- one you want (for free). It means that if you are a teacher, you can (a) give or print or xerox copies for your students, (b) use potions for your own class notes (if they are published then you might need to add some acknowledgement, depending on which parts you copied), and you can xerox even very large portions of it to your hearts content. The “differential calculus” part means it covers derivatives and applications