Maths · Calculus · Derivative · Integral · Limit
Three classic pictures: a tangent line as the derivative, rectangles stacking toward an integral, and values closing in on a limit. Return to the maths hub.
The derivative f′(a) is the slope of the tangent at x = a. Drag a along the curve (or use the slider). Choose a simple function.
Definition intuition: f′(a) ≈ [f(a+h) − f(a)] / h for small h. The line shown uses the exact derivative for these functions.
Approximate ∫ from a to b of f(x) dx with n rectangles (right endpoints here). Slide n up and watch the sum approach the true area under y = 1 + 0.3x² on [0, 2].
For g(x) = (x² − 1)/(x − 1) when x ≠ 1 (and undefined at x = 1), values approach 2 as x → 1. Slide ε-closeness: only show sample points within δ of 1.
Educational calculus sketches — original explanations of standard ideas. Not a substitute for a full calculus course.