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 Contents 1 A Brief Introduction to R 1.1 Exercises 2 Describing a Graph 2.1 Exercises 3 The Function Gallery 3.1 Exercises Part I Change and the Derivative 4 How Fast is CO2 Increasing? 4.1 Exercises 5 The Idea of the Derivative 5.1 Exercises 6 Formulas Quantifying Change 6.1 Exercises 6.2 Project: Which Mountain to Climb? 7 The Microscope Equation 7.1 Exercises 8 Successive Approximations to Estimate Derivatives 8.1 Exercises 8.2 Project: Which Secant Line Approximation is Better? 8.3 Project: Estimating e 8.4 Project: Estimating π 9 The Derivative Graphically 9.1 Exercises 10 The Formal Derivative as a Limit 10.1 Exercises 10.2 Project: An Origin Story of the number e 11 Basic Derivative Rules 11.1 Exercises 12 Product Rule 12.1 Exercises 12.2 Project: Conceptual Product Rule Graphic 13 Quotient Rule 13.1 Exercises 14 Chain Rule 14.1 Exercises 15 Derivatives with R 15.1 Exercises 15.2 Project: Mauna Loa CO2 Projections 15.3 Project: Climate Change Projections 16 End Behavior of a Function - L'Hospital's Rule 16.1 Exercises 16.2 Project: Comparing Exponential and Quadratic Models ... Part II Applications of the Derivative 17 How Do We Know the Shape of a Function? 17.1 Exercises 17.2 Project: Arctic Sea Ice Analysis 18 Finding Extremes 18.1 Exercises 19 Optimization 19.1 Exercises 20 Derivatives of Functions of Two Variables 20.1 Exercises 21 Related Rates 21.1 Exercises 22 Surge Function 22.1 Exercises 23 Differential Equations - Preliminaries 23.1 Exercises 24 Differential Equations - Population Growth Models 24.1 Exercises 25 Differential Equations - Predator Prey 25.1 Exercises 26 Differential Equations - SIR Model 26.1 Exercises 27 Project: The Gini Coefficient—Prelude to Section III Part III Accumulation and the Integral 28 Area Under Curves 28.1 Exercises 29 The Accumulation Function 29.1 Exercises 29.2 Project: Hubbard Brook - The Importance of a Watershed 30 The Fundamental Theorem of Calculus 30.1 Exercises 31 Techniques of Integration - The u Substitution 31.1 Exercises 32 Techniques of Integration - Integration by Parts 32.1 Exercises Appendix A Algebra Review - Functions and Graphs A.1 Exercises Appendix B Algebra Review - Adding and Multiplying Fractions B.1 Exercises Appendix C Algebra Review - Exponents C.1 Exercises Appendix D Algebra Review - Lines D.1 Exercises Appendix E Algebra Review - Expanding, Factoring, and Roots E.1 Exercises Appendix F Algebra Review - Function Composition F.1 Exercises Appendix G R Glossary Appendix H Answers to Odd Problems Appendix Appendix I R Code for Figures Appendix References Index