Mathematics · Beginner
Calculus
Rediscovering the rules rather than memorising them — three hours that make a whole textbook make sense.
- Lessons
- 22 lessons
- Total runtime
- 35 hours of video
- Modules
- 9 modules
- Optional lessons
- 11 optional
Free. No account.
About this course
Most calculus courses hand you a list of rules and then test whether you can apply them. The rules are correct and the tests are passable, and at the end a great many people can differentiate a function without being able to say what a derivative is for. This course inverts that. Its organising claim is that every formula in a first calculus course is something you could have invented yourself, given the right picture, and it spends three hours demonstrating that on the formulas that matter.
The order is the series' own and it is worth keeping. Chapter one builds the whole subject out of the area of a circle, which sounds like a party trick until you notice that integration, differentiation and the relationship between them have all appeared before any notation has. Everything afterwards is that first idea in more detail.
What this is not is a substitute for practice. Calculus is a skill as well as a set of ideas, and skills need repetition that no video can provide. Watch this alongside a problem set, or before a course you are about to take, or after one you passed without enjoying. Any of those three works well; watching it alone and expecting to be able to sit an exam does not.
Chapter seven, on limits, is the one to take slowly — epsilon-delta is the definition that makes students give up on mathematics, and this is the gentlest correct treatment of it available. Chapter eleven on Taylor series is the best single video in the series and the natural place to finish.
That sequence is about three hours. Everything after it is optional and a great deal longer: two full college courses, a set of worked-problem videos for when you need practice rather than intuition, and university lectures from MIT and Professor Leonard. Around thirty-five hours in the pool as a whole, though nobody should take all of it — the series teaches you what the rules mean, and one of the longer courses teaches you to use them.
What you'll be able to do
- Say what a derivative measures, and why the definition involves a limit at all
- Derive the power, product and chain rules from a picture rather than recalling them
- Explain the fundamental theorem of calculus in terms of area and slope
- Understand what a limit is, including the epsilon-delta definition
- See a Taylor series as a way of approximating anything with polynomials
- Follow a calculus textbook with a mental picture behind each rule
Curriculum
22 lessons · 35 hours
The Big Picture
1 lesson · 17m
Derivatives
3 lessons · 49m
- 02CompletedThe paradox of the derivative3Blue1BrownVideoHow something can have a rate of change at an instant, when nothing changes during an instant. Taking the paradox seriously is what makes the limit definition feel necessary rather than pedantic.17m
- 03CompletedDerivative formulas through geometry3Blue1BrownVideoThe table of derivatives everyone memorises, derived instead from squares and rectangles. Afterwards you can reconstruct any line of that table on paper, which is more useful than remembering it.18m
- 04CompletedVisualizing the chain rule and product rule3Blue1BrownVideoThe two rules that get applied mechanically for years, shown as pictures. The chain rule in particular stops being a mnemonic here.14m
Where the Rules Come From
3 lessons · 47m
- 05CompletedWhat's so special about Euler's number e?3Blue1BrownVideoWhy one particular irrational number keeps appearing, and why its exponential is its own derivative. This is also the cleanest explanation of what a natural logarithm is naturally the logarithm of.14m
- 06CompletedImplicit differentiation, what's going on here?3Blue1BrownVideoA technique usually taught as a procedure with an unexplained step. Here the step is explained, and it turns out to be about curves rather than about algebra.15m
- 07CompletedLimits, L'Hôpital's rule, and epsilon delta definitions3Blue1BrownVideoThe chapter to slow down for. Epsilon-delta is where a lot of people quietly decide mathematics is not for them, and this is the gentlest correct treatment of it anywhere.18m
Integration
2 lessons · 32m
- 08CompletedIntegration and the fundamental theorem of calculus3Blue1BrownVideoWhy the two halves of calculus are the same subject. The theorem is stated everywhere and made obvious almost nowhere; this does the second thing.20m
- 09CompletedWhat does area have to do with slope?3Blue1BrownVideoAverages of continuous things, and a second angle on the same theorem. Short, and it fixes the lingering sense that the connection was a coincidence.12m
Approximation
2 lessons · 27m
- 10CompletedHigher order derivatives3Blue1BrownVideoFive minutes on what the second derivative means physically. Included mainly because the next chapter needs it.5m
- 11CompletedTaylor series3Blue1BrownVideoThe best chapter in the series and the right place to finish. Approximating any function with polynomials is the single most useful idea in applied mathematics, and it is built here from nothing but derivatives.22m
Another Way to See It
1 lesson · 14m
Learning to Compute
3 lessons · 1h 30m
- 13CompletedIntroduction to LimitsOptionalThe Organic Chemistry TutorVideoUnderstanding what a limit means and being able to evaluate one are separate skills, and this course only teaches the first. Twenty minutes of straightforward technique.20m
- 14CompletedUnderstand Calculus in 35 MinutesOptionalThe Organic Chemistry TutorVideoA rapid pass over the mechanics — the rules, applied to problems, without the derivations. Useful as revision or as a preview of what the exercises will look like.36m
- 15CompletedUnderstanding Calculus (for Engineers)OptionalThe Efficient EngineerVideoCalculus framed by what it is for — rates, accumulation and the physical quantities each one corresponds to. Half an hour, and unusually good at motivating the subject.34m
The Full Courses
4 lessons · 27h 10m
- 16CompletedCalculus VisualizedOptionalDennis DavisVideoThree hours in the same spirit as the series above but covering a full first course. The natural next step if you liked the visual approach and want more of it.3h 1m
- 17CompletedYou Can Learn Calculus 1 in One VideoOptionalThe Math SorcererVideoFive hours of worked examples, essentially a textbook read aloud with the problems done. Dull and extremely effective.5h 22m
- 18CompletedCalculus 1 — Full College CourseOptionalfreeCodeCamp.orgVideoTwelve hours matching a standard first university course, with exercises. The version to take if you need to pass something.11h 54m
- 19CompletedCalculus 2 — Full College CourseOptionalfreeCodeCamp.orgVideoIntegration techniques, sequences and series. The natural continuation, and where the Taylor series chapter above gets its formal treatment.6h 53m
University Lectures
3 lessons · 3h 12m
- 20CompletedDerivatives and Rates of Change — MIT 18.01 Lecture 1OptionalMIT OpenCourseWareVideoA real lecture, chalk and all. Worth one hour of your time simply to see how the same material is delivered in a mathematics department.52m
- 21CompletedLimits and Continuity — MIT 18.01 Lecture 2OptionalMIT OpenCourseWareVideoThe rigorous treatment of limits, which is what chapter seven above is preparing you for.53m
- 22CompletedAn Introduction to LimitsOptionalProfessor LeonardVideoNinety minutes on limits alone, at the pace of someone teaching a room of students who need it to land. The most patient explanation of this topic anywhere.1h 27m
Credits
Every lesson in this course was made by one of these channels. Tubeversity sequenced them; it did not make them. If a course is useful to you, the people below are the ones who earned it.