Numerical Solution of Equations Topic Review summary

Key learning points

• Rearrange $f(x)=0$ into $x=g(x)$ for fixed-point iteration; sketch to see convergence.
• Newton–Raphson needs $f$ and $f'$ and a sensible starting value.
• Justify accuracy to $n$ d.p. with a change-of-sign on a tight interval.
• Failure modes: $f'=0$, divergence, or wrong root attraction.
• Compare methods’ speed qualitatively when asked.

Worked example 1 — NR

Newton step formula.

Working

Tangent intersection with axis.

Worked example 2 — Accuracy

Show root is 1.23 to 2 d.p.

Working

Standard exam justification.

Worked example 3 — Practice point 1

Rearrange $f(x)=0$ into $x=g(x)$ for fixed-point iteration; sketch to see convergence.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Newton–Raphson needs $f$ and $f'$ and a sensible starting value.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.