Hint

Expand about \(\displaystyle x=0\). Use ra.

Worked solution

(a)

\(\displaystyle - \frac{135 x^{3}}{16} + \frac{27 x^{2}}{8} - \frac{3 x}{2} + 1\) follows by substituting \(\displaystyle u=ax\) into the rational binomial series.

Answer: \(\displaystyle - \frac{135 x^{3}}{16} + \frac{27 x^{2}}{8} - \frac{3 x}{2} + 1\) + \(O(x^4)\)

(b)

The condition |ax|<1 gives the strict bounds.

Answer: \(\displaystyle \left(- \frac{1}{3}, \frac{1}{3}\right)\)

(c)

Here \(\displaystyle ax= \frac{1}{10}\) is inside the validity interval. Substitute into the cubic polynomial; the omitted higher terms make the value an approximation.

Answer: \(\displaystyle \frac{3051}{3200}\) approximately

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: Apply r(r−1)… coefficients with all powers of the inner scale through degree three.
• 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 use the finite positive-integer restriction rule here.

(b) · 2 marks

• 1 mark: Apply |ax|<1 and solve both endpoint inequalities.
• 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: Remember each power of a when substituting \(\displaystyle u=ax\).

(c) · 2 marks

• 1 mark: Substitute the given input into the cubic truncation and distinguish an estimate from an exact 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: Do not use the finite positive-integer restriction rule here.

Total: 6 marks.