Topic
Calculus
7 articles on calculus: the maths or physics explained, ready-to-teach lesson plans, and how to build each slide in Tracé.
The Brachistochrone: Why the Fastest Path Is a CurveWhy a ball on a cycloid beats one on a straight ramp (3.17 s against 3.76 s for a 20 m drop), the 1696 challenge behind it, and how to animate the race.
Fourier Series: How Sine Waves Build a Square WaveAdd sin x, sin 3x / 3, sin 5x / 5… and a square wave appears. The formula, why only odd harmonics, the 9% Gibbs overshoot, and a click-by-click slide.
The Derivative as the Slope of a Tangent: Lesson PlanIntroduce the derivative as the slope of the tangent line: secants that shrink, the limit (f(x+h) − f(x))/h, a worked example with x², mistakes and checks.
Riemann Sums: Teaching the Area Under a CurveA Riemann sums lesson: left, right and midpoint rectangles under y = x², exact numbers for 8 rectangles, how the error shrinks, mistakes and checks.
Taylor Series: Approximating Functions with PolynomialsHow Taylor series build eˣ and sin x from polynomials term by term, how accurate a few terms are, the small-angle approximation, and a click-by-click slide.
Newton's Method, Animated: One Tangent per ClickTeach Newton's method visually: follow each tangent down to the x-axis, compare it with bisection, and see why it converges so fast, or fails.
Euler's Method and Direction Fields, VisuallyA visual lesson on differential equations: draw the direction field, follow solution curves, then watch Euler’s method step away from the exact solution.