Newton's method: follow the tangent to the root

Starting from x0 = 2.5 on f(x) = x^2 - 2, this animation draws each tangent line and follows it down to the x-axis to find the next estimate. Four iterations are shown, with the approximation rapidly tightening until it matches the square root of 2 to six decimal places. Useful for students learning root-finding algorithms and seeing why Newton's method converges so quickly near a simple root.

16:9 · every frame verified for overlaps, spacing and edges before rendering

The prompt that made it

Starting at x0 = 2.5 on f(x) = x^2 - 2, each tangent line's x-intercept becomes the next estimate; four steps reach sqrt(2) to six decimals.

Make your own version

Make the next one in this series

• Compare Newton's method convergence speed to the bisection method on the same function
• Show what happens when Newton's method starts near a point with zero derivative
• Animate Newton's method failing or diverging for a poorly chosen starting point

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