Locating roots by sign change, iterative methods including fixed point iteration and the Newton-Raphson method, the conditions under which they succeed or fail, and the trapezium rule for approximating definite integrals.

What this dot point is asking

AQA wants you to locate roots of an equation using a change of sign, use iterative formulae including the Newton-Raphson method, understand when these methods succeed or fail, and use the trapezium rule to approximate a definite integral and judge whether it over- or under-estimates. The questions reward precise reasoning: stating continuity, justifying convergence, and explaining the direction of error.

Locating roots by sign change

A single sign change guarantees an odd number of roots in the interval (at least one), not exactly one; if the curve dips and returns, there could be more. To confirm a root to a given accuracy, evaluate $f$ at the two ends of a narrow bracketing interval and show the sign change.

Fixed point iteration

Rearrange the equation into the form $x = g(x)$, then iterate $x_{n+1} = g(x_n)$ from a starting value. The sequence may converge to a root, shown neatly on a staircase or cobweb diagram, provided the gradient of $g$ near the root has magnitude less than one. Different rearrangements of the same equation can converge, diverge, or converge to different roots, so the choice of $g$ matters.

The Newton-Raphson method

The method fails if $f'(x_n)$ is zero or near zero (the tangent is nearly horizontal and the next estimate shoots far away), or if the starting value is near a turning point so that the tangent points away from the desired root.

The trapezium rule

The rule over-estimates the area when the curve bends upwards (concave up), because the straight tops of the trapezia lie above the curve, and under-estimates when the curve bends downwards. Increasing the number of strips reduces the error. Justifying the direction of error by referring to the concavity of the curve is an examined point.

Why we need numerical methods, and how to present them

Many equations that arise in practice, such as $x^3 - 5x - 3 = 0$ or $x = \cos x$, have no neat algebraic solution, and many functions, such as $\sqrt{1 + x^3}$, have no elementary antiderivative. Numerical methods give a usable approximation in these cases, and the exam rewards a tidy, reproducible presentation. For root finding, state the function, show the sign change with continuity, then either iterate or apply Newton-Raphson, recording each estimate to the requested accuracy. For the trapezium rule, tabulate the ordinates clearly before substituting, since a single mis-evaluated ordinate ruins the answer.

To confirm a root to a stated number of decimal places, evaluate the function at the two ends of the rounding interval. For instance, to show a root is $2.49$ to two decimal places, check that $f(2.485)$ and $f(2.495)$ have opposite signs; the sign change inside that interval guarantees the rounded root. This "interval bisection of the rounding band" argument is the standard way AQA expects you to justify an accuracy claim, rather than simply asserting that successive iterates agree.