A-Level Maths numerical methods: sign change, iteration and the trapezium rule

Where numerical methods sit in each specification

The key ideas

If f is continuous on an interval and f(a) and f(b) have opposite signs, then f(x) = 0 has a root between a and b. To prove a root to a given accuracy, test the bounds of the rounding interval: to show a root is 2.09 to 2 decimal places, show f(2.085) and f(2.095) have opposite signs.

Iteration rearranges f(x) = 0 into x = F(x), then repeatedly applies x(n+1) = F(x(n)) from a starting value. If the values settle down, the limit is a root of the original equation. Not every rearrangement converges; some move away from the root, which is why exam questions usually give you the rearrangement.

The trapezium rule splits the area under a curve into n strips of equal width h = (b - a)/n and adds the trapezium areas: area ≈ (h/2)[y0 + yn + 2(y1 + y2 + ... + y(n-1))]. If the curve bends upwards (convex) the chords lie above it and the rule overestimates; if it bends downwards (concave) the rule underestimates. More strips give a better estimate.

Worked example 1: show that x^3 - 2x - 5 = 0 has a root between 2 and 3, then find it by iteration

1. Let f(x) = x^3 - 2x - 5. f(2) = 8 - 4 - 5 = -1 and f(3) = 27 - 6 - 5 = 16.
2. There is a sign change and f is continuous, so there is a root between 2 and 3.
3. Rearrange: x^3 = 2x + 5, so x = cube root of (2x + 5). Use x(n+1) = (2x(n) + 5)^(1/3) with x1 = 2.
4. x2 = 9^(1/3) = 2.08008, x3 = 2.09235, x4 = 2.09422, x5 = 2.09450, x6 = 2.09454, x7 = 2.09455.
5. To confirm the root is 2.095 to 3 d.p., check f(2.0945) = -0.0006 and f(2.0955) = 0.0105. Opposite signs, so the root is 2.095 to 3 decimal places.

Worked example 2: use the trapezium rule with 4 strips to estimate the integral of ln x from 1 to 3

1. h = (3 - 1)/4 = 0.5, so the x values are 1, 1.5, 2, 2.5 and 3.
2. y values: ln 1 = 0, ln 1.5 = 0.405465, ln 2 = 0.693147, ln 2.5 = 0.916291, ln 3 = 1.098612.
3. Area ≈ (0.5/2)[0 + 1.098612 + 2(0.405465 + 0.693147 + 0.916291)] = 0.25 × 5.128418 = 1.282 (3 d.p.).
4. The exact value is 3 ln 3 - 2 = 1.296, so the estimate is too small.
5. This is expected: y = ln x bends downwards (it is concave), so each chord lies below the curve and the trapezium rule underestimates.

Common mistakes that cost marks

• Showing a sign change but not writing the conclusion 'so there is a root between a and b'.
• Testing the wrong bounds: to confirm 2.095 to 3 d.p. you must test 2.0945 and 2.0955.
• Rounding iterates too early, so the sequence appears to converge to the wrong value.
• Confusing the number of strips with the number of ordinates: 4 strips need 5 y-values.
• Using degrees instead of radians for trig functions in the trapezium rule.
• Saying 'overestimate' or 'underestimate' with no reason. Refer to the shape of the curve.

Exam technique and how a tutor helps

Numerical methods questions are some of the most reliable marks on the pure papers because the method never changes. Lay out iterations in a list with 4 or 5 decimal places, give your final answer to the accuracy asked, and if a question asks you to show that the limit satisfies an equation, substitute x for both x(n) and x(n+1) and rearrange back to f(x) = 0.

For the trapezium rule, put your x and y values in a table first. It makes the formula substitution clean and earns the method mark even if one value is wrong.

A one-to-one tutor usually spends little time on the method itself and more on the written conclusions and the calculator routine, since that is where marks leak. Students practise with the calculator's answer key to run iterations quickly, then complete past-paper questions against the mark scheme wording.

Self-check: can you do these?

• Show that x^3 + x - 3 = 0 has a root between 1 and 2. (f(1) = -1, f(2) = 7, sign change)
• The iteration x(n+1) = (2x(n) + 5)^(1/3) converges to α. What equation does α satisfy? (Answer: α^3 - 2α - 5 = 0)
• How many ordinates are needed for 6 strips? (Answer: 7)
• Does the trapezium rule overestimate or underestimate the integral of e^x from 0 to 1? (Answer: overestimates, because e^x is convex)
• Which method from Further Pure 1 is not needed for Cambridge 9709? (Answer: Newton-Raphson)