Applied Calculus with R

This textbook integrates scientific programming with the use of R and uses it both as a tool for applied problems and to aid in learning calculus ideas. Adding R, which is free and used widely outside academia, introduces students to programming and expands the types of problems students can engage. There are no expectations that a student has any coding experience to use this text. While this is an applied calculus text including real world data sets, a student that decides to go on in mathematics should develop sufficient algebraic skills so that they can be successful in a more traditional second semester calculus course. Hopefully, the applications provide some motivation to learn techniques and theory and to take additional math courses. The book contains chapters in the appendix for algebra review as algebra skills can always be improved. Exercise sets and projects are included throughout with numerous exercises based on graphs. Preface 6 Contents 10 1 A Brief Introduction to R 14 1.1 Exercises 29 2 Describing a Graph 33 2.1 Exercises 36 3 The Function Gallery 40 3.1 Exercises 53 Part I Change and the Derivative 57 4 How Fast is CO2 Increasing? 58 4.1 Exercises 64 5 The Idea of the Derivative 68 5.1 Exercises 70 6 Formulas Quantifying Change 73 6.1 Exercises 75 6.2 Project: Which Mountain to Climb? 84 7 The Microscope Equation 85 7.1 Exercises 88 8 Successive Approximations to Estimate Derivatives 98 8.1 Exercises 107 8.2 Project: Which Secant Line Approximation is Better? 113 8.3 Project: Estimating e 113 8.4 Project: Estimating π 114 9 The Derivative Graphically 115 9.1 Exercises 121 10 The Formal Derivative as a Limit 140 10.1 Exercises 145 10.2 Project: An Origin Story of the number e 147 11 Basic Derivative Rules 149 11.1 Exercises 157 12 Product Rule 160 12.1 Exercises 165 12.2 Project: Conceptual Product Rule Graphic 170 13 Quotient Rule 171 13.1 Exercises 175 14 Chain Rule 177 14.1 Exercises 181 15 Derivatives with R 187 15.1 Exercises 192 15.2 Project: Mauna Loa CO2 Projections 196 15.3 Project: Climate Change Projections 198 16 End Behavior of a Function - L'Hospital's Rule 200 16.1 Exercises 206 16.2 Project: Comparing Exponential and Quadratic Models ... 208 Part II Applications of the Derivative 210 17 How Do We Know the Shape of a Function? 211 17.1 Exercises 219 17.2 Project: Arctic Sea Ice Analysis 225 18 Finding Extremes 227 18.1 Exercises 231 19 Optimization 236 19.1 Exercises 238 20 Derivatives of Functions of Two Variables 241 20.1 Exercises 246 21 Related Rates 248 21.1 Exercises 252 22 Surge Function 255 22.1 Exercises 260 23 Differential Equations - Preliminaries 267 23.1 Exercises 269 24 Differential Equations - Population Growth Models 272 24.1 Exercises 283 25 Differential Equations - Predator Prey 288 25.1 Exercises 295 26 Differential Equations - SIR Model 298 26.1 Exercises 304 27 Project: The Gini Coefficient—Prelude to Section III 308 Part III Accumulation and the Integral 310 28 Area Under Curves 311 28.1 Exercises 318 29 The Accumulation Function 324 29.1 Exercises 329 29.2 Project: Hubbard Brook - The Importance of a Watershed 338 30 The Fundamental Theorem of Calculus 341 30.1 Exercises 347 31 Techniques of Integration - The u Substitution 351 31.1 Exercises 353 32 Techniques of Integration - Integration by Parts 355 32.1 Exercises 357 Appendix A Algebra Review - Functions and Graphs 359 A.1 Exercises 363 Appendix B Algebra Review - Adding and Multiplying Fractions 376 B.1 Exercises 377 Appendix C Algebra Review - Exponents 379 C.1 Exercises 382 Appendix D Algebra Review - Lines 385 D.1 Exercises 387 Appendix E Algebra Review - Expanding, Factoring, and Roots 389 E.1 Exercises 391 Appendix F Algebra Review - Function Composition 393 F.1 Exercises 394 Appendix G R Glossary 396 Appendix H Answers to Odd Problems 402 Appendix Appendix I R Code for Figures 496 Appendix References 516 516 Index 519