OT2: Mathematical problem solving: Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy.

The idea

Many equations, like x = cos x, lack exact algebraic solutions. Numerical methods provide approximate roots to a required accuracy. For f(x)=0, if f(a) and f(b) have opposite signs and f is continuous, a root lies between a and b. Iterative methods like Newton-Raphson, x_{n+1} = x_n - f(x_n)/f'(x_n), can converge rapidly. However, convergence is not guaranteed; poor starting values or turning points can cause divergence. To confirm a root to a specific accuracy (e.g. 3 decimal places), do not rely solely on the difference between successive terms. Instead, check for a sign change in f(x) between the upper and lower bounds of the rounded value (e.g. +/- 0.0005). This proves the root lies within the required tolerance.

What earns the marks

• Recognises that some equations cannot be solved analytically and selects an appropriate numerical method.
• Uses a sign-change argument to locate an interval containing a root, explicitly stating that the function must be continuous.
• Applies an iterative formula, such as Newton-Raphson, correctly evaluating successive approximations.
• Verifies the final approximation to the required accuracy by checking for a sign change between the appropriate upper and lower bounds.
• Interprets the limitations of numerical methods, identifying cases where iteration may fail to converge.

Common mistakes and how to put them right

• Assuming successive approximations being close guarantees the required accuracy.

  Put it right: Verify the accuracy by checking for a sign change between the upper and lower bounds of the rounded value.

• Applying Newton-Raphson without checking the derivative.

  Put it right: Ensure the initial guess is not at or near a stationary point, as f'(x_n) = 0 causes the method to fail.

• Forgetting to state the continuity condition when using the sign-change rule.

  Put it right: Explicitly state that f(x) is continuous on the interval [a, b] when concluding a root exists.

Exam tips

• When asked to show a root is accurate to a given number of decimal places, always evaluate the function at the half-unit bounds above and below the value.
• Write down all intermediate iterative values to at least one more decimal place than the final required accuracy to avoid premature rounding errors.