Hint

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

Worked solution

(a)

\(\displaystyle - \frac{27 x^{3}}{2} + \frac{27 x^{2}}{2} + 9 x + 1\) follows by substituting \(\displaystyle u=ax\) into the rational binomial series.

Answer: \(\displaystyle - \frac{27 x^{3}}{2} + \frac{27 x^{2}}{2} + 9 x + 1\) + \(O(x^4)\)

(b)

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

Answer: \(\displaystyle \left(- \frac{1}{6}, \frac{1}{6}\right)\)

(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: \(\displaystyle \frac{18459}{16000}\) 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.

Hint

Differentiate y as a function of x. Collect \(\displaystyle (ax+2y)y'=-(2x+ay)\).

Worked solution

(a)

Differentiate to get 2x+a(y+xy′)+2yy′=0, then collect and divide where \(\displaystyle ax+2y\ne 0\).

Answer: \(\displaystyle \frac{dy}{dx}=-\frac{2x+3y}{3x+2y}\)

(b)

The point satisfies the implicit equation. Its derivative is \(\displaystyle - \frac{8}{7}\), and the tangent passes through \(\left( 2, \ 4\right)\).

Answer: \(\displaystyle y=\frac{44}{7} - \frac{8 x}{7}\)

(c)

Take the negative reciprocal of the finite nonzero tangent gradient.

\(\displaystyle \frac{7}{8}\)

Answer: \(\displaystyle \frac{7}{8}\)

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: Differentiate each term, including both product-rule terms, and isolate y′ where its denominator is nonzero.
• 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: The xy term needs the product rule.

(b) · 2 marks

• 1 mark: Evaluate the derivative at the stated point and form its tangent equation.
• 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: A zero denominator may indicate a vertical tangent.

(c) · 2 marks

• 1 mark: Take the negative reciprocal of the finite nonzero tangent gradient.
• 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.

Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.

Common error: The xy term needs the product rule.

Total: 6 marks.