Numerical Methods Topic Review summary

Key learning points

• Change of sign on $[a,b]$ suggests a root if $f$ is continuous.
• Decimal search / bisection narrow the interval systematically.
• Fixed-point iteration $x_{n+1}=g(x_n)$ from a rearrangement of $f(x)=0$.
• Newton–Raphson: $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$.
• Justify accuracy (e.g. to 3 d.p.) with a sign check on a tight interval.

Worked example 1 — Sign change

Show $x^3-x-1=0$ has a root in $[1,2]$.

Working

Continuous ⇒ a root exists in between.

Worked example 2 — Newton

One NR step for $x^2-2$ from 1.5.

Working

$x-\frac{f}{f'}$.

Worked example 3 — Change of sign on [a,b] suggests a root if

Change of sign on $[a,b]$ suggests a root if $f$ is continuous.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Decimal search / bisection narrow the inte

Apply this idea from the Topic Review: Decimal search / bisection narrow the interval systematically.

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.