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. To get that slope, differentiate term by term: the derivative of x squared is 2x and the derivative of a constant is zero. 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.
More to read
4.9 Newton’s Method
Some equations cannot be rearranged to give x directly, yet a computer still needs their roots. Newton starts with a rough guess, draws the tangent at that point, and takes where the tangent hits the axis as the next guess. Each round uses the same rule, and good starts home in on the root fast. The price is that bad starts can cycle, stall, or fly away. You need to know both how fast it is and how it can fail. The rest is practice with the formula.
4.2 Linear Approximations and Differentials
Newton works because a curve looks straight when you zoom in close. The tangent line is that zoomed in view, and following it to the axis is nearly as good as following the curve itself. Linear approximation makes this precise: near a point, the function value is roughly the tangent value. The error shrinks fast as you close in, which explains the doubling of correct digits. Read this page to see why the tangent crossing is such a clever next guess.
3.1 Defining the Derivative
Every Newton step divides by the slope at the current guess, so slopes matter. The derivative is exactly that slope: the limit of secant gradients as the second point slides in. A steep slope gives a confident step, a shallow one gives a wild leap, and a zero slope breaks the method outright. Revising what the derivative means on a graph turns the formula from a mystery into common sense. If you can picture tangents, you can predict what Newton will do.
2 · Watch
En anglaisTake it off screen
Where it sits
14 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.