Calculus
How fast is something changing at this instant, and how much has piled up over time? Why do those two questions turn out to undo each other?
Where you are 12 inside
What this area is about
The derivative measures a rate at an instant; the integral adds up infinitely many thin pieces. Both are limits, and the fundamental theorem of calculus shows that each undoes the other.
How these fit together
Foundation. Limits say where a function is heading.
Change. Derivatives define the rate and give the rules for finding it. Applications of derivatives use it for graphs, motion, estimates and optimization.
Accumulation. Integrals build the definite integral and the techniques for evaluating it. Applications of integrals turn it into areas, volumes, lengths and work.
Further tools. Sequences and series write functions as infinite polynomials. Differential equations find a function from a rule about its rate. Parametric, polar and vector valued functions reach curves that are not graphs.
College areas, not yet written. Partial derivatives (functions of several variables and their extremes), multiple integrals (over regions in the plane and in space), vectors and space curves (lines, planes and curves in three dimensions) and vector calculus (fields, flow, and the theorems of Green, Stokes and Gauss).
A path through it
Limits first. Then derivatives and their applications together, since each borrows lessons from the other. Then integrals. The first-course lessons of differential equations come before applications of integrals, whose motion lesson solves initial value problems. Series, then parametric and polar calculus.
The college areas follow: partial derivatives, then multiple integrals and space curves, then vector calculus. The college parts of differential equations need series and partial derivatives.