Newton's Method Root Approximator

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published December 23, 2025 Updated February 24, 2026

6. Newton's Method Root Approximator

Approximate roots of \(f(x)=0\) using Newton’s iteration \(\;x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}\;\) with convergence checks and a step-by-step table.

Inputs

Function \(f(x)\)

Supported: + − * / ^, parentheses, variable x, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication: 2x, (x+1)(x-1), 3sin(x).

Initial guess \(x_0\)

Choose a starting value near the expected root.

Tolerance

Used by the stop condition.

Max steps

Upper limit on iterations.

Stop condition

Controls when the method stops.

Display Show tangent steps Show grid Auto-fit view on compute

Advanced Basins of attraction (1D preview)

Shows which initial guesses converge to which roots (rough preview).

Output format

Graph center \(c\)

Auto-fit recommended.

Graph window half-width \(w\)

Graph uses \(x\in[c-w,c+w]\).

Ready

Graph

Drag to pan • wheel/pinch to zoom • \(f(x)\) curve + Newton steps (tangents) + root marker.

x: 0, y: 0, zoom(px/unit): 80

Result

Enter \(f(x)\), choose \(x_0\), then click Compute.

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6. Newton's Method Root Approximator — Theory

Newton’s Method is an iterative technique for approximating solutions to \(f(x)=0\). Starting from an initial guess \(x_0\), we repeatedly replace the curve by its tangent line at the current point and use the tangent’s x-intercept as the next guess.

\[ \textbf{Newton iteration:}\qquad x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. \]

Geometric meaning

At \(x_n\), the tangent line to \(y=f(x)\) is \[ y \approx f(x_n)+f'(x_n)(x-x_n). \] Setting \(y=0\) gives the x-intercept: \[ 0=f(x_n)+f'(x_n)(x_{n+1}-x_n) \;\Rightarrow\; x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. \]

Stopping criteria

Common ways to stop include:

\(|x_{n+1}-x_n|<\text{tol}\) (the iterate stops changing much)
\(|f(x_n)|<\text{tol}\) (the function value is close to 0)
\(\max(|x_{n+1}-x_n|,\;|f(x_n)|)<\text{tol}\) (stronger combined condition)

When Newton’s Method works well

If \(f\) is smooth and the initial guess is sufficiently close to a simple root (where \(f'(r)\neq 0\)), Newton’s Method often converges very fast (typically “quadratically” near the root).

Common failure cases

Derivative near zero: If \(f'(x_n)\approx 0\), the update \(\dfrac{f(x_n)}{f'(x_n)}\) can explode or become unstable.
Bad initial guess: The method can diverge or jump to a different root.
Oscillation/cycles: Some functions cause iterates to bounce between values.
Non-smooth points: If \(f\) is not differentiable, Newton’s formula may fail.

Worked example

Find \(\sqrt{2}\) by solving \(f(x)=x^2-2=0\), starting at \(x_0=1\).

\[ f(x)=x^2-2,\quad f'(x)=2x. \] \[ x_1 = 1-\frac{f(1)}{f'(1)} = 1-\frac{1-2}{2} = 1.5 \] \[ x_2 = 1.5-\frac{2.25-2}{3} \approx 1.416666\ldots \] \[ x_3 \approx 1.414215686\ldots \] which converges to \(\sqrt{2}\approx 1.414213562\ldots\).

Basins of attraction (idea)

If a function has multiple roots, different initial guesses can converge to different roots. The set of starting values that converge to a particular root is called that root’s basin of attraction. This calculator provides a lightweight 1D preview across the current x-window.

Frequently Asked Questions

What does Newton's method compute in this calculator?

It computes successive approximations x0, x1, x2, ... that aim to solve f(x)=0. Each update uses the tangent line at the current point through x_{n+1} = x_n - f(x_n)/f'(x_n).

Which stop condition should I use for Newton's method?

|x_{n+1}-x_n| < tol stops when the iterates stop changing much, while |f(x_n)| < tol stops when the function value is close to zero. The combined max(|Delta x|, |f(x_n)|) < tol is stricter and can be more reliable.

Why does Newton's method fail when f'(x) is near zero?

The update divides by f'(x_n), so if the derivative is very small the step can become extremely large or unstable. This can cause divergence, oscillation, or jumping to an unintended root.

How do I pick a good initial guess x0?

Choose x0 close to the desired root and avoid places where the derivative is zero or the function is not smooth. Using the graph window and tangent-step display can help you see whether the iterates are moving toward a root.

What are basins of attraction in the Newton's method preview?

When a function has multiple roots, different starting guesses can converge to different roots. A basin of attraction is the set of starting values that converge to the same root, and the 1D preview shows this behavior across the current x-window.

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Frequently Asked Questions

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