Calculus

Last modified: 2026-11-05 14:20:19 (PST)

The calculus notes are split across these pages:

• Derivatives: limits and derivatives, derivative rules, tangent lines, and linear approximation.
• Taylor Series: Taylor and Maclaurin polynomials, and Taylor series.
• Integrals and the Fundamental Theorem of Calculus: antiderivatives, regularity conditions, and the Fundamental Theorem of Calculus.
• Double Integrals and Fubini’s Theorem: double integrals over rectangles and non-rectangular regions, and Fubini’s and Tonelli’s theorems.

A link to a section, theorem or definition on this page from before the split goes on to the page that now holds it.

1 Further reading

• Kaplan [1]
• Khuri [2]
• Banner [3]
• Larson and Edwards [4]
• Miller [5] The calculus review exercises and worked solutions across these pages and in Vector Calculus are adapted from this supplemental review chapter.
  • http://www.youtube.com/watch?v=xYzQL0TUtBA
  • http://www.youtube.com/watch?v=Ps2SBo_WjoE
• Grinberg [6] (the rigorous foundations behind these results)
• Nield [7] is a short, code-driven review of the calculus used in data science (Chapter 1), with Python examples and little rigor.
• Essence of Calculus is a 12-video YouTube playlist by Grant Sanderson (3Blue1Brown) that develops geometric and visual intuition for limits, derivatives, product and chain rules, implicit differentiation, integration, and Taylor series.

References

[1]

D. Kaplan, MOSAIC calculus. www.mosaic-web.org, 2022. Available: www.mosaic-web.org

[2]

A. I. Khuri, Advanced calculus with applications in statistics. John Wiley & Sons, 2003. doi: 10.1002/0471394882.

[3]

A. D. Banner, The calculus lifesaver : All the tools you need to excel at calculus. in A princeton lifesaver study guide. Princeton, New Jersey: Princeton University Press, 2007. Available: https://press.princeton.edu/books/paperback/9780691130880/the-calculus-lifesaver

[4]

R. Larson and B. H. Edwards, Calculus, 11th ed. Boston, MA: Cengage Learning, 2018. Available: https://www.cengage.com/c/calculus-11e-larson/

[5]

S. J. Miller, The probability lifesaver: Calculus review problems. 2016. Available: https://web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/index.htm#:~:text=http%3A//web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/supplementalchap_calcreview.pdf

[6]

R. Grinberg, The real analysis lifesaver: All the tools you need to understand proofs, 1st ed. in Princeton lifesaver study guides. Princeton: Princeton University Press, 2017. Available: https://press.princeton.edu/books/paperback/9780691172934/the-real-analysis-lifesaver

[7]

T. Nield, Essential math for data science: Take control of your data with fundamental linear algebra, probability, and statistics. Sebastopol, CA: O’Reilly Media, 2022.