Method guideCalculus IIntermediate10 min

Newton's method: approximating roots with tangent lines

Use the tangent line at a current guess to generate a better root estimate, then monitor whether the iteration is actually improving.

LaTeX article Updated July 13, 2026

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}

A tangent line predicts where the graph reaches zero

At xₙ, the tangent line approximates the graph. Solving that line for its x-intercept produces the next estimate xₙ₊₁.

The formula is therefore a geometric construction, not an unexplained numerical trick. Its quality depends on how well the tangent represents the function nearby.

0\approx f(x_n)+f'(x_n)(x_{n+1}-x_n)

A starting value can decide success

A guess close to a simple root often converges quickly. A poor guess can jump to another root, cycle, or move into a region where the derivative vanishes.

A graph or sign-changing interval helps select a sensible start. Newton's method should not be treated as guaranteed for every differentiable-looking formula.

f'(x_n)=0\quad\Rightarrow\quad\text{the Newton step is undefined}

Stop with a stated accuracy rule

Common stopping rules compare successive estimates or check the residual |f(xₙ)|. State the tolerance instead of stopping because a decimal display looks stable.

Round only the reported result. Carry extra digits during iteration so rounding error does not steer later steps.

|x_{n+1}-x_n|<\varepsilon\quad\text{or}\quad|f(x_n)|<\varepsilon

Worked example

Common mistakes

Using f(xₙ) where f′(xₙ) belongs.
Starting where the derivative is zero or nearly zero.
Rounding every intermediate estimate aggressively.

Keep these ideas

Each step is a tangent-line x-intercept.
Convergence depends on the starting point.
Use an explicit stopping rule.