No. 03November 2025.
Calculus II, MT2103. Rendered with Manim.

Extrema and double integrals, animated in Manim

Two short animations from Calculus II. In the first, a tangent slides along f(x) = x³ − 3x + 1, its slope taken from f′(x) = 3x² − 3, and reads zero exactly at the turning points. In the second, columns rise under z = x + 2y over a rectangle until their summed volume stands in for the double integral.

Maxima and minima, 15 seconds, rendered from my Manim scene.

Double integral as columns, 20 seconds, rendered from my Manim scene.

f′(x) = 3x² − 3 = 0⇒x = ±1

∬R f dA ≈ Σ f(xi, yj) ΔA

what I don’t understand yet

How to show the limit rather than assert it. The animation draws one grid; shrinking ΔA step by step and printing the sum each time would let the columns converge on screen.