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Calculus in Twelve Short Lessons

Jan 5, 2024

This is a live draft of something that I'm currently thinking about in January 2024. I will update it lesson-by-lesson. Please let me know what you think.

In this informal guide, I will write a few short paragraphs outlining each topic in a calculus course. These “short lessons” are not meant to teach you the techniques of calculus, but rather to develop your intuition for the subject. A single paragraph isn’t enough to convey all the tips, tricks, tools, and techniques, of each topic, but a paragraph can convey the major idea of a topic.

The topics covered below are a bit personal. Whenever I teach an introduction to calculus for non-math majors, such as MAT A29 Calculus I for the Life Sciences, I tend to follow the following ordering of topics.

  1. Functions
  2. Limits
  3. Derivatives
  4. Differentiation Techniques
  5. The Chain Rule
  6. Optimization: Maxima and Minima
  7. Curve Sketching
  8. Anti-Derivatives
  9. The Fundamental Theorem of Calculus
  10. Integration Techniques
  11. Improper Integrals
  12. Area Between Curves

Each topic tends to build on the next, and creates a nice story. Of course, this order is somewhat accidental. For example, there are twelve major topics because our semesters have twelve weeks. One could add or subtract topics, but this seems to be a good amount for a single semester course.


Calculus is all about the study of quantitative relationships. Usually, these quantitative relationships come in the form of functions1. And so it is important to start off by asking: what is a function? It is anything which takes an input and returns an output. In an introductory calculus course, the inputs and outputs are always things that can be measured with plain old numbers. For example, there is a function which takes as its input a height $h$ above the surface of the earth and returns as its output the length of time $t$ it would take for an object to fall that distance. It is clunky to write these things out as words, so we write a formula $d(h) = t$ to express the fact that $d$ropping from height $h$ takes time $t$.

Understanding exactly how $h$ relates to $t$ via $d(h) = t$ was a major motivation for the development of calculus. This quantitative relationship tells us something deep and important about how gravity works near Earth2. How do we figure out a formula for $d(h)$? Calculus.

  1. One major topic, addressed under the chain rule, is what to do if the relationship under investigation is not a function. ↩︎

  2. An old legend says that Galileo measured various values of $h$ and $t$ by dropping things from the Leaning Tower of Pisa, although the exact details are disputed. ↩︎

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