Newton's Method — Finding Roots

What just happened?

We are looking for the root of the function $f(x) = x^2 - 2$ — that is, the number $\sqrt{2}$. Starting from $x_0 = 2$, we draw a tangent to the graph at that point and see where it crosses the x-axis. This gives a better approximation $x_1$. We repeat — and after just a few steps we hit the root with enormous accuracy.

Newton's method (Newton–Raphson):

It is one of the most important numerical algorithms — it quickly finds approximate solutions of equations $f(x)=0$ that cannot be solved exactly by a formula. Isaac Newton developed it around 1669 (in "De analysi"), and Joseph Raphson simplified its notation in 1690. The idea: at each step we replace the complicated function by its tangent (the best local linear approximation) and take the tangent's root as a better approximation of the root.

Iteration formula:

For $f(x)=x^2-2$ we have $f'(x)=2x$, so $x_{n+1}=\tfrac12\!\left(x_n+\tfrac{2}{x_n}\right)$. In order: $x_0=2$, $x_1=1.5$, $x_2=1.41\overline{6}$, $x_3=1.4142157\dots$, $x_4=1.41421356\dots$

The invariant — the tangent sets the direction:

Every step uses the slope of the tangent $f'(x_n)$ at the current point — that is what shows where the next, better approximation lies. The method converges quadratically: the number of correct digits roughly doubles with each iteration, so after only a few steps we reach enormous accuracy — far faster than, e.g., the bisection (halving) method.

Why this matters on the exam:

Newton's method is a practical application of differential calculus (the derivative as the slope of the tangent) and the basis of numerical computation — it works in calculators, computers, graphics and optimization. The trick: the derivative $f'(x)$ is the slope of the tangent, and its root is the next approximation $x_{n+1}$.

Remember: the tangent method turns the hard equation $f(x)=0$ into a sequence of simple steps $x_{n+1}=x_n-\tfrac{f(x_n)}{f'(x_n)}$, which converges to the root in a flash — digit by digit, faster and faster.