A royal road to calculus

2023-12-04

OpenCourseWare has a few versions of their single-variable calculus subject. I looked at the 2010 one, because it seemed more complete.

It might be a little overwhelming (101 sessions!). But you don't absolutely have to go through it in a linear way; you can do some of the parts first to get the fundamental principles and maybe some applications, and then return to look at some of the other parts (or not).

Here's what I have in mind for a fairly minimalist introduction (I kept the original section numbers). The topics on integration start on this list around section 37, so you can stop before that if you're primarily focused on derivatives at the moment.

1. Introduction to derivatives

2. Examples of derivatives

3. Derivatives as rate of change

4. Limits and continuity

6. Calculating derivatives

9. Product rule

10. Quotient rule

11. Chain rule

23. Linear approximation

24. Examples of linear approximation

27. Sketching graphs I

28. Sketching graphs II

29. Optimization problems

30. Optimization problems II

34. Introduction to the Mean Value Theorem

36. Differentials

37. Antiderivatives

38. Integration by substitution

43. Definite integrals

44. Adding areas of rectangles

45. Some easy integrals

46. Riemann sums

47. Introduction to the Fundamental Theorem of Calculus

48. The Fundamental Theorem of Calculus

49. Applications of the Fundamental Theorem of Calculus

60. Integrals and averages

A couple of remarks

1. I'm not saying that the other sections are not important, only that they can fairly safely be ignored on a first go.
2. Skipping stuff means that there will be the occasional point where they use (in a more or less essential way) something that you didn't look at yet. It's not the end of the world. Often you can get away with looking up only what they use (for instance, that the derivative of sin(x) is cos(x)), without delving into why or how. If you really get thrown off by this, you can always work through the relevant skipped part.
3. The beauty of online content is that you can work through it at your own pace; one strategy I recommend is to stop video recordings whenever they start discussing examples or problems, and spend a bit of time thinking about how you would approach them yourself. Then resume the recording and see what they do. Whether you had "gotten it right" or not is irrelevant; in both cases the act of trying will make it easier to absorb the material.
4. They have exercises. Exercises are a good thing, so you should try to do the ones that don't involve stuff you skipped (most of the time it should be possible to identify this without too much trouble).