The Newton-Raphson Method
Follow the tangent at the current guess down to the x-axis to get the next guess, which is x - f(x)/f'(x). Note when it fails: a nearly flat tangent throws the next guess far away.
What a learner can do afterwards
- Apply one step of the formula from a given starting value
- Explain the formula as the x-intercept of a tangent
- Describe a starting value for which the method fails and say why
1 · Read
Start with a rough guess x0. Draw the tangent to the curve at x0 and take where it meets the x-axis as your next guess x1. The tangent at x0 has equation y = f(x0) + f prime(x0)(x - x0), and setting y = 0 gives x1 = x0 - f(x0)/f prime(x0). Repeat from the new point.
For f(x) = x squared minus 2 from x0 = 1, f(1) = -1 and the slope f prime(x) = 2x gives 2. The step is 1 - (-1)/2 = 1.5, already much closer to the root near 1.414. Near a simple root with a good start, each step roughly doubles the correct digits.
A flat tangent never meets the axis, and the formula divides by zero, so a zero derivative breaks the method outright. Bad starts can also overshoot far away or cycle between values without settling. A steep slope gives a confident step while a shallow one gives a wild leap.
Pick the start from a sketch near the root you want, and check each new guess stays nearby. If guesses jump far away or repeat old values, stop and restart closer. Speed means nothing without a start the geometry can work with.
Follow each tangent down to the axis for the next guess, and restart closer whenever the steps misbehave.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.