Newton iteration and an inadmissible start

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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A-level core. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Newton-Raphson iteration. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

Use Newton’s method for f(x) = $x^{2}$ − 3.

Axes for a curve-and-tangent sketch.

Part a

Derive the iteration formula and state a value for which it is undefined.

Part b

Starting at $x_{0}$ = 1, find $x_{1}$ exactly.

Part c

Sketch the tangent interpretation and explain why $x_{0}$ = 0 fails.

Part d

A student says a small step proves an accurate root. Comment.

Model solution and review criteria

Part a

x_next = x − ( $x^{2}$ −3)/(2x) = (x+3/x)/2. It is undefined at x = 0 because f′(0)=0.

Use f/f′.
Simplify without losing the zero-denominator condition.

Part b

$x_{1}$ = (1+3)/2 = 2.

Substitute into the derived update.

Part c

The next iterate is the x-intercept of the tangent at the current point. At zero the tangent is horizontal and does not meet the axis for c>0, so it provides no next iterate.

Show the curve and a tangent from a nonzero start.
Connect the zero derivative to the missing intercept.

Part d

A small step alone need not imply a small residual or a nearby root in general. Check |f(x)| as well as the step and the problem’s conditioning.

Distinguish change in iterate from residual.
Avoid claiming a universal convergence guarantee.

Compare your own reasoning. No automatic examination marks are awarded.