Chapter 2 — Further Reading

An annotated guide. Start with the textbook section mapping if you want more drill problems; branch into the history, modeling, or Python sources as your interests and portfolio track dictate. You do not need all of these — pick what serves you.

Matching Textbook Sections

This chapter corresponds to the standard "review of functions" material that opens most calculus texts. Use these to find parallel explanations and additional exercises.

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. The reference we benchmark against. Chapter 1 ("Functions and Models") is the direct parallel: §1.1 four ways to represent a function, §1.2 essential functions, §1.3 transformations and combinations (our §2.3, §2.5), §1.4 exponential functions, §1.5 inverses and logarithms (our §2.5). Exhaustive exercise sets if you want more drill. (List price $280+; library copies and used editions are common.) A section-by-section map lives in appendices/appendix-h-stewart-chapter-mapping.md.

OpenStax, Calculus, Volume 1 (Strang & Herman). Free at https://openstax.org/details/books/calculus-volume-1. Our §2.1–§2.5 map onto OpenStax §1.1 (Review of Functions), §1.2 (Basic Classes of Functions, including transformations and piecewise), §1.3 (Trigonometric Functions), §1.4 (Inverse Functions), and §1.5 (Exponential and Logarithmic Functions). Free, well-edited, with worked examples and answers. Full mapping in appendices/appendix-i-openstax-chapter-mapping.md.

For a precalculus refresher below this chapter's level, see Appendix A — Precalculus Review in this book before reaching for an outside text.

On the Function Concept and Its History

Dunham, W. (2018). The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press. Traces the function concept from Newton's "fluents," through Euler's "analytic expressions," to the modern set-theoretic definition — illuminating the very idea we sharpened in §2.1.

Maor, E. (1998). Trigonometric Delights. Princeton University Press. A lively tour of trigonometry — history, identities, surprising connections. A good companion to the trig review in §2.2.

Khan Academy — Trigonometry & Precalculus. https://www.khanacademy.org/math/trigonometry Free videos and graded exercises on the unit circle, identities, and special values. The fastest way to refresh §2.2 mechanics.

On Exponentials and Logarithms

Maor, E. (1994). e: The Story of a Number. Princeton University Press. The history and meaning of Euler's number. Sets up why $e^x$ is "the right" exponential — the property $\frac{d}{dx}e^x = e^x$ that Chapter 7 proves.

Strogatz, S. (2014). The Joy of x. Houghton Mifflin Harcourt. Conversational chapters on exponentials, logarithms, and modeling; ideal light reading alongside §2.2 and §2.7.

On Mathematical Modeling (§2.7)

Edwards, D., and Hamson, M. (2007). Guide to Mathematical Modelling (2nd ed.). Industrial Press. A practical introduction to building models, with worked examples across physics, biology, and business. Directly extends the modeling discipline of §2.7.

Bender, E. A. (2000). An Introduction to Mathematical Modeling. Dover. Affordable and broad; good for self-learners who want to see the propose-fit-critique cycle across many domains.

Otto, S. P., and Day, T. (2007). A Biologist's Guide to Mathematical Modeling in Ecology and Evolution. Princeton University Press. If your portfolio track is Biology, this shows what serious biological modeling — including the growth models behind Case Study 1 — looks like in practice.

Polya, G. (1954). Mathematics and Plausible Reasoning, Vol. 1. Princeton University Press. On the style of modeling — reasoning by analogy and educated guessing — rather than the mechanics. Useful for thinking about how a good model is chosen.

On Python, matplotlib, and sympy (§2.6)

VanderPlas, J. (2016). Python Data Science Handbook. O'Reilly. Free at https://jakevdp.github.io/PythonDataScienceHandbook/. The standard reference for numpy and matplotlib. Chapter 2 covers numpy vectorization (the engine behind our §2.6 plots); Chapter 4 covers matplotlib in depth.

matplotlib documentation. https://matplotlib.org/stable/tutorials/index.html Official tutorials; the "Quick start" and "Pyplot" guides are enough to follow every code block in this book. Environment-setup notes are in Appendix C.

SymPy documentation. https://docs.sympy.org/latest/tutorials/intro-tutorial/index.html The symbolic-algebra tool we used in §2.6 to verify inverses. The introductory tutorial covers symbols, simplify, and subs — all you need here.

For the Case Studies

Case Study 1 (outbreak modeling). - Our World in Data — Coronavirus. https://ourworldindata.org/coronavirus. Clean data for the mini-project. - Murray, J. D. (2002). Mathematical Biology (3rd ed., Vol. I). Springer. Chapter 10 develops the SIR model that Chapter 19 builds toward. - Brauer, F., and Castillo-Chavez, C. (2012). Mathematical Models in Population Biology and Epidemiology. Springer. Graduate-level but accessible; comprehensive on epidemic models.

Case Study 2 (the sigmoid). - Hastie, Tibshirani & Friedman (2009). The Elements of Statistical Learning (2nd ed.). Springer. Free at https://hastie.su.domains/ElemStatLearn/. Chapter 4: logistic regression in full. - Goodfellow, Bengio & Courville (2016). Deep Learning. MIT Press. Free at https://www.deeplearningbook.org/. Chapter 6: activation functions (sigmoid, ReLU). - Cox, D. R. (1958). "The regression analysis of binary sequences." J. Royal Statistical Society B 20(2). The original logistic-regression paper.

How to Use These References