Calculus begins with one question: how fast is something changing right now? Answering it takes a new idea, the limit, and it leads to the derivative, a machine for turning any curve into its slope — 25 interactive demonstrations. Shrink a secant line until it becomes a tangent and zoom into a curve until it turns straight. Play the ε–δ game, hunt roots by bisection, and steady a wobbly table. Draw any curve and watch its derivative appear, type any formula and watch a machine differentiate it, slide a ladder down a wall, fold the biggest box, and race Newton’s method to a root, or watch it paint a fractal.
See also: Integral Calculus for area, the Fundamental Theorem that ties it to slope, and Taylor series, Vector Calculus for the same ideas in several variables, and Real Analysis for the rigorous foundations underneath.
How fast is it changing right now? Secant lines that become tangents, a car’s speedometer and odometer, and curves that turn straight when you zoom in
What a function is heading towards, whether or not it gets there. Jumps, holes, wild oscillation, the ε–δ game, and the squeeze theorem
Functions you can draw without lifting the pen. Hunting roots by bisection, a wobbly table you can always fix, and behaviour at infinity
Slope at every point becomes a new function. Trace it, draw your own curve and watch its derivative appear, and read the shape of a graph from f′ and f″
Why the product rule has two terms and the chain rule multiplies. Growing rectangles, linked number lines, the number e, and a machine that differentiates anything you type
Slopes of curves that are not graphs, a ladder sliding down a wall, and water rising in tanks of every shape
The biggest box from a sheet of card, the fastest route to a swimmer, and the theorem that catches speeding drivers
Tangent lines as the best local guess, Newton’s method racing to a root and painting fractals, and l’Hôpital’s rule seen up close