About this chapter
Numerical methods finds roots that algebra cannot: where e to the power x meets a line, where a trig curve turns, or when a drug's concentration drops below a level. The marks are about precise statements, such as naming the change of sign and the continuity that makes it count, and about giving each answer to exactly the accuracy asked.
This chapter covers locating roots by a change of sign, when that method fails at an asymptote, rearranging to x = g(x), iteration, showing a root is correct to a given number of decimal places, convergence, roots that come from calculus, intersections of two curves, and answering in context with units.
The ten questions
- Q1 (5 marks): a root of x³ − 3x − 5 between 2 and 3, a rearrangement, and three iterations.
- Q2 (5 marks): a root of an exponential equation in [1, 2], then show it is 1.749 to 3 decimal places.
- Q3 (5 marks): a change of sign across an asymptote that does not show a root, and the exact root.
- Q4 (5 marks): one iteration formula that finds one root but moves away from the other, and a better rearrangement.
- Q5 (5 marks): where y = ln x meets y = 3 − x, located and then iterated.
- Q6 (11 marks): a tangent through the origin gives an equation for α, located, iterated and then confirmed to 3 decimal places.
- Q7 (10 marks): the maximum of x² cos x satisfies x = arctan(2/x): iterate, then confirm the root.
- Q8 (10 marks): a drug concentration model: the peak time, an iteration for when it falls to 2, and an answer to the nearest minute.
- Q9 (10 marks): tan x − 3x: a real root, a false sign change at π/2, and iteration.
- Q10 (11 marks): where e to the power x minus 2 meets its inverse, with a different iteration needed for each root.
Key skills tested
Locating roots. If f is continuous on an interval and changes sign across it, there is a root in between. Write all three parts: the values, the continuity, and the conclusion.
When it fails. A sign change across an asymptote, such as for 1/(x − 2) or tan x at π/2, does not show a root, because f is not continuous there.
Rearranging. Show step by step that f(x) = 0 can be written as x = g(x). For example, x³ − 3x − 5 = 0 becomes x = the cube root of (3x + 5).
Iteration. Put each value back into g(x) using the calculator's ANS key, and give each iterate to the accuracy asked.
Root to a given accuracy. For 1.749 to 3 decimal places, test the interval from 1.7485 to 1.7495: find the sign at both ends and conclude.
Convergence. Iteration converges when the gradient of g is between −1 and 1 near the root. Another root may need a different rearrangement.
Roots from calculus. A stationary point satisfies f′(x) = 0, and a tangent through the origin satisfies f(α) = α f′(α).
Intersections. Two curves meet where f(x) − g(x) = 0, so locate the roots of that new function.
Context. Answer with units and to the accuracy asked, such as a time in hours to 3 significant figures.
Worked example
Question 1 from this chapter. f(x) = x³ − 3x − 5. (a) Show that f(x) = 0 has a root between 2 and 3. (b) Show that f(x) = 0 can be written as x = the cube root of (3x + 5). (c) Starting from 2, find the next three iterates to 4 decimal places.
(a) f(2) = 8 − 6 − 5 = −3 and f(3) = 27 − 9 − 5 = 13. There is a change of sign and f is continuous, so there is a root between 2 and 3.
(b) x³ = 3x + 5, so x is the cube root of (3x + 5).
(c) The cube root of 11 is 2.2240, then 2.2684, then 2.2770.
Part (a) needs all three parts: both values, continuity, and the conclusion.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q8: a golf ball model solved with logs, a derivative rearranged into a given form, then an iteration and a maximum height with units, as in Q7 and Q8.
Where marks are lost
- Missing units. In June 2024 most students found the maximum height correctly but surprisingly often left out the units, and lost that mark.
- Writing out every iteration. Some listed many iterations for a one-mark question that only needed the final value, and ran short of the limit.
- A sign change with no reason. The conclusion needs the continuity of f and a statement that there is a root, not just two numbers.
- Degrees for trig. Any iteration with a trig function must be done in radians.
Common questions
Why does continuity matter?
A function can change sign by jumping across an asymptote without ever being zero, so the sign change only proves a root if the function is continuous.
How do I show a root is correct to 3 decimal places?
Evaluate f at the two bounds that round to that value, show the sign changes, and conclude.
Why might an iteration not converge?
If the rearranged function is too steep near the root, each step moves further away, so a different rearrangement is needed.
Where this chapter leads
- Pure 4 Chapter 5: Differentiation: more equations from calculus, some of which need numerical methods.
- Pure 4 Chapter 6: Integration: models whose values have to be found numerically.