Calculus from the Ground Up

Description

Calculus is one of humanity's greatest achievements. With calculus, you can analyze the infinitely large, the infinitely small, and everything in-between. Calculus gives you the tools to develop your own mathematics, your own formulas, and, most importantly, your own imagination. Calculus is not just a subject. It is an invitation to think differently about the way that the world works. Calculus has a reputation for being a dry, dreary, difficult subject. However, the problem generally lies with tedious, lifeless books that present calculus with all the vigor of a wilted salad. Calculus was not discovered as a series of dull proofs about the real number line. It was discovered by curious people who took the time to look at the world in a new way. Calculus from the Ground Up invites readers to not just read about mathematics but to become active participants--making numbers and symbols the servants of their minds as they grow their imaginations in ways they didn't think possible. Calculus from the Ground Up isn't your typical mathematics book. It is a guidebook for students to learn not only the bare subject of calculus, but also to discover how its artistry impacts other areas of life. Calculus from the Ground Up includes sections on how to solve impossible problems, how to break problems into manageable sizes, how to develop new mathematical formulas, and even how to live a more ethical life, all through lessons gained from calculus. Title Pages Table of Contents Chapter 1: Introduction Part I: Preliminaries Chapter 2: A General Method for Solving Mathematics Problems Chapter 3: Basic Tools: Lines Chapter 4: Basic Tools: Variables and Functions Chapter 5: Basic Tools: Graphs Chapter 6: Calculus: The Big Picture Part II: The Derivative Chapter 7: The Derivative Chapter 8: Basic Derivative Rules for Polynomials Chapter 9: Basic Uses of the Derivative Chapter 10: The Differential Chapter 11: Differentials of Composite Functions Chapter 12: Additional Differential Rules Chapter 13: Multivariable Differentials Chapter 14: Relating Rates Using Differentials Chapter 15: Working with Problems Involving Differentials Part III: The Integral Chapter 16: The Integral as an Antidifferential Chapter 17: The Integral as an Infinite Sum Chapter 18: Using the Integral to Find the Area Under the Curve Chapter 19: Integration and u-Substitution Chapter 20: Other Geometric Uses of the Integral Chapter 21: Transforming Equations for Integration Chapter 22: Integration by Parts Chapter 23: Problem-Solving Using the Integral Chapter 24: Numeric Integration Techniques Part IV: Manipulating Infinity Chapter 25: Modeling Functions with Polynomial Series Chapter 26: Advanced Topics in Polynomial Series Chapter 27: Thinking About Infinity Chapter 28: Limits: Finding Impossible and Non-Existent Values of Functions Chapter 29: Limits: Finding Results Near Infinity Chapter 30: Limits: Difficult Limits with L'Hospital Chapter 31: Conclusion Part V: Appendices Appendix A: Guide for Instructors Appendix B: Liebniz Notation of Higher Order Differentials Appendix C: Well-Known Calculus Theorems Appendix D: Additional Uses of the Derivative Appendix E: Additional Uses of the Integral Appendix F: Proofs and Derivations Appendix G: Additional Things to Know for Standardized Tests Appendix H: Mathematical Formulas, Relationships, and Tables