I: Numerical methods: Use numerical methods to solve problems in context.

The idea

Numerical methods approximate solutions when exact algebra is impractical. You must choose a suitable method, carry it out accurately, and interpret the result in the problem's context. For example, to solve x^3 - 2x - 5 = 0, use sign changes to locate a root, then apply interval bisection or the Newton-Raphson iteration x_{n+1} = x_n - f(x_n)/f'(x_n). Continue until the required accuracy is reached, then round appropriately and state the solution with units or meaning. In context, you may need to set up the equation from given information, decide on a sensible starting value, and explain why the approximation is valid. Always relate the final value back to the original problem, checking it lies within the domain and makes sense practically.

What earns the marks

• Correctly rearrange or define a function f(x) so that the equation to solve is f(x) = 0, showing clear working.
• Use a sign change over an interval to establish a root lies between two values, stating the interval.
• Apply a valid iterative formula (such as interval bisection or Newton-Raphson) accurately for at least two iterations, showing substituted values.
• Continue the method until the required degree of accuracy is achieved, then round and state the final approximate solution.
• Interpret the numerical answer in the context of the problem, including correct units or a concluding statement.

Common mistakes and how to put them right

• Using degrees instead of radians when applying trigonometric functions in numerical methods.

  Put it right: Check the mode and use radians for calculus-based iterations unless the context explicitly requires degrees.

• Stopping after one iteration and assuming the answer is accurate enough.

  Put it right: Continue until two successive approximations agree to the required accuracy, or until the interval width is sufficiently small.

• Ignoring the context and giving a purely mathematical answer without units or explanation.

  Put it right: Relate the final value back to the original problem, stating what it represents and including units where appropriate.

Exam tips

• Show every step of your iteration, including the formula used and the substituted values, so that method marks can be awarded even if an arithmetic slip occurs.
• When using Newton-Raphson, clearly state your starting value and derivative, and be prepared to justify why that starting value is suitable.
• If the question asks for a solution to a given accuracy, state the final answer explicitly and indicate the interval or error bound to demonstrate the accuracy.