Hint
Use the real cube root in this iteration. Apply the given rearrangement to \(\displaystyle x_{0}\).
Worked solution
(a)
Isolate \(\displaystyle x^{3}\) and take its real cube root.
Answer: \(x=\sqrt[3]{12-x}\)
(b)
Use the recurrence in order, with guard digits retained.
Answer: \(2.15443469003188\), \(2.14328645394556\), \(2.14409510399536\)
(c)
The root is 2.144040. The original increasing cubic changes sign between 2.1440395 and 2.1440405, which are the half-unit rounding boundaries.
Answer: \(2.14404\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Rearrange the cubic into the specified real cube-root fixed-point rule.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not round every iterate to the final requested accuracy.
(b) · 2 marks
- 1 mark: Apply the recurrence three times starting at the stated initial value.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Solving a rearranged equation does not guarantee convergence of every iteration.
(c) · 2 marks
- 1 mark: Obtain a converged approximation and verify signs on a bracket supporting six-decimal-place rounding.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not round every iterate to the final requested accuracy.
Total: 6 marks.