Teaching yourself calculus

Who this is for

This is for adults and independent learners who want to learn single-variable calculus: limits, derivatives and integrals. You need a textbook or free online course with exercises and answers, a notebook, and about five to seven hours a week. This plan makes no promises about how it compares to a class, and it does not replace any course you need for credit.

The plan at a glance

• Check the prerequisites with a short diagnostic before week 1.
• Take one topic a week, and do not move on until you can solve a mixed set from memory.
• Choose problems over videos: watch or read for about a third of the time, and solve for the rest.
• Keep an error log of every mistake and the reason for it.
• Review last week's topic for 20 minutes at the start of each new week.

Step by step

1. 1. Run a prerequisite check

   Solve a dozen problems on: simplifying algebraic fractions, solving quadratics, function notation, exponents and logarithms, and the sine, cosine and tangent of common angles. If you miss several in one area, spend your first week there. Calculus mistakes are most often algebra mistakes.

2. 2. Pick a main source

   Use one textbook or one course as your spine so the notation and order stay consistent. Use other sources only to get a second explanation of a hard idea.

3. 3. Work the weekly loop

   Read or watch a section, close it, and do the exercises. Check the answers, and for each wrong answer find out why. Redo it the next day.

4. 4. Explain it aloud

   At the end of each week, explain the main idea in your own words, as if to a friend, in under two minutes. If you cannot, that is next week's first review.

5. 5. Test with something checkable

   Calculus offers many ways to check yourself: differentiate and integrate to get back where you started, compare a limit definition to a rule, or plug in numbers.

Worked example: a 10-week self-study plan

Here is a week 3 check you can do by hand. The definition of the derivative is f'(x) = lim (h → 0) of [f(x + h) − f(x)] / h. Take f(x) = x² at x = 3.

1. 1.Compute f(3 + h) = (3 + h)² = 9 + 6h + h².
2. 2.Subtract f(3) = 9: the difference is 6h + h².
3. 3.Divide by h: (6h + h²) / h = 6 + h.
4. 4.Let h approach 0: the limit is 6, so f'(3) = 6.

Now check it with the power rule from week 4: if f(x) = x², then f'(x) = 2x, and 2 × 3 = 6. The two methods agree. Do the same with a number: with h = 0.01, the quotient is 6.01, with h = 0.001 it is 6.001, and the values close in on 6. If you ever get a different answer from the rule and the definition, one of them has an arithmetic error, and finding which is where the learning happens.

If something goes wrong

• You fall a week behind. Merge two light weeks, or drop the least central applications. Do not skip the definition weeks.
• You keep making algebra errors. Go back to a prerequisite set and do ten more problems on the exact skill.
• An idea will not click. Draw it. Limits, slopes and areas all have pictures, and sketching a curve with a tangent line often unlocks the formula.
• You are tempted to skip to the rules. Do the definition work at least once. Rules without meaning are easy to forget.
• Motivation drops. Set a fixed time and a short minimum, such as three problems, so that you never have a zero day.

Keep it going

After ten weeks, read about the fundamental theorem of calculus again and notice how much more it says. Then choose a use for it: a physics course, a data project or a programming language. If you want a study partner for the stuck moments, DuckyHelper can see the problem on your screen and work it with you on its blackboard, but do the attempt first so the learning stays yours.