A royal road to calculus

A. Ghitza

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).