Ask of any rule how fast it changes and how much it adds up to, find that these are the same question asked in two directions, and be able to answer either one by hand, by approximation or by stepping it forward with a known error

Change and Accumulation

Calculus as two questions about a rule, a rate and a total, which turn out to be inverse to each other. Every figure is either stepped forward and checked against a formula or holds a quantity fixed.

8 lessons, written and corrected before you arrived. Reading them here needs no account. The first reads the whole way through; the others open and then stop, because a page nobody owns cannot tell who is reading it. Starting the course gives you your own copy, where every idea has problems standing under it and you can ask about any sentence.

Start reading

01A Rule, and the Picture of ItA function is a rule that answers one number for each number given. The graph is a record of what the rule did, and a table of answers never determines the rule.
02How Fast, Exactly Hereopening onlyA rate over an interval is a division anyone can do. The rate at a single instant is what that division settles on as the interval shrinks, and it usually exists.
03One Rule Inside Anotheropening onlyMost rules worth asking about are built out of simpler ones. When one rule is fed into another the rates multiply, and the reason is that a small nudge is scaled twice on its way through.
04Adding Up Thin Slicesopening onlyA total is built by cutting the thing into slices thin enough that each one can be treated as constant, adding them, and asking what the sums settle on as the slices thin.
05The Two Questions Are One Questionopening onlyAccumulating a rate gives back the quantity it came from. That single fact turns a limit of sums into a subtraction, and it is why rates and totals are studied together.
06The Straight Line That Fits, Then the Parabolaopening onlyNear a point, a rule can be replaced by a straight line that matches its value and rate, then by a parabola that also matches its bend, with an error that shrinks sharply.
07Doing It With Arithmetic, and the Error You Buyopening onlyWhen nothing can be written down, a known rate can be marched forward in small steps using only arithmetic, and the error that buys is predictable and controllable.
08When the Rate Depends on the Amountopening onlyA quantity whose rate is proportional to how much of it there is multiplies by a fixed factor in every fixed stretch of time, and the rule describing it is its own rate.