What exactly is calculus of a single variable?

Answer 1

It is the calculus, both differential and integral, of functions (algebraic and transcendental) of one variable. That is, the range set, the values of the function, are mapped from a single set of domain values. All such functions can be expressed in the the general form y = f(x).

This contrasts with functions of several variables that are expressed in the form y = f(x1, x2, x3...., xN).

From an analytic geometrical perspective, the calculus of one variable deals with functions that can be graphed in a two-dimensional space (X-Y plane); from this geometric perspective multivariate calculus requires higher-order spaces of 3, 4,...., N dimensions.

Answer 2

I think of it this way: you know those algebra problems you had that read like " a boat is going upstream at a certain speed, and the current is such and such a speed, and the time for the round trip is whatever......"? In all those problems, the speeds were constant. The boat never speeds up or slows down. Neither does the river.

What if the speed varies? Then the old formula "distance = rate times time" makes no sense any more. Which rate would you use, if the rate is changing over time?

Algebra can no longer solve the problem.

Calculus is the study of variable rates of change.