Calculus: an interactive calculus course

Calculus is an interactive calculus course that builds from limits to derivatives and integrals, then uses them on curves and motion. Its 29 short sections cover limits, the gradient of a curve, differentiation from first principles, the power, chain, product and quotient rules, stationary points, integration, definite integrals, the fundamental theorem of calculus, motion and the area between curves.

Every big idea comes with an explorer you can drag, such as a secant line shrinking into a tangent or rectangles filling the area under a curve, so you see why a rule works before you use it.

What the calculus course covers

Differentiation

The course starts with limits because a derivative is a limit: the gradient of a secant as the gap h shrinks to zero. You work through that once, from first principles, then meet the shortcuts. The power rule handles any xⁿ, and the chain, product and quotient rules handle functions inside, times and divided by other functions. Stationary points then show what derivatives are for: finding maximums and minimums.

Integration

Integration starts as area, approximated by rectangles, then becomes the reverse of differentiation: raise the power by one and divide by the new power, adding a constant C. The fundamental theorem of calculus ties the two halves together, and definite integrals give exact areas.

Using calculus

The last sections apply both halves. Velocity is the derivative of position and acceleration the derivative of velocity, so differentiating s(t) = 20t − 5t² tells you when a thrown ball stops rising. Area between curves uses intersections as the limits of an integral.

Try a question from the course

The power rule says: multiply by the power, then take one off the power. So for f(x) = 4x³ − 5x² + 7x − 2, differentiate term by term: 4x³ becomes 12x², −5x² becomes −10x, 7x becomes 7, and the constant −2 becomes 0. The answer is f'(x) = 12x² − 10x + 7.

Now differentiate a new polynomial, then press Mark to check your answer in the course.

Who this course is for

• Students starting calculus, in AP Calculus AB, A-level Maths or a first university course, who want the ideas explained visually.
• Students revising the differentiation rules, stationary points or definite integrals.
• Physics, engineering, economics and computer science students who use derivatives and integrals in other subjects. The AI Models course, for example, uses derivatives and the chain rule to explain how neural networks learn.

What you need to know first

Confident algebra, including factoring and working with powers, and a feel for graphs. The Algebra and Graphing Functions courses cover both. The chain rule section is easier if you are happy with functions inside functions.

How long it takes

There are 29 sections and most take a few minutes: a short explanation with an explorer, or a practice question with fresh numbers each time. Learners without a plan can open two sections of a course a day, in order, which takes 15 days. A plan removes the daily limit and lets you go straight to any topic, such as integration. For more on single topics, see the chain rule and product rule guides.

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