Hint
Use the same parameter for x, y and the gradient. Substitute t into x(t).
Worked solution
(a)
\(\displaystyle \frac{dx}{dt}= 2t\) and \(\displaystyle \frac{dy}{dt}= 3t^{2}+ 4\). Divide only where \(\displaystyle 2t\ne 0\).
Answer: \(\displaystyle \frac{3 t^{2} + 4}{2 t}\), \(\displaystyle t\ne 0\).
(b)
The point is \(\left( 7, \ 16\right)\) and the gradient is \(4\). Use the point-gradient equation.
Answer: \(y=4 x - 12\)
(c)
Differentiate the quotient gradient with respect to t, then divide by 2t once more.
Answer: \(\displaystyle \frac{3}{4 t} - \frac{1}{t^{3}}\); value \(\displaystyle \frac{1}{4}\).
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: Divide \(\displaystyle \frac{dy}{dt}\) by \(\displaystyle \frac{dx}{dt}\) and exclude the zero denominator.
- 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 confuse \(\displaystyle \frac{dy}{dt}\) with \(\displaystyle \frac{dy}{dx}\).
(b) · 2 marks
- 1 mark: Evaluate both coordinates and the gradient at the same parameter.
- 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 second derivative needs another division by \(\displaystyle \frac{dx}{dt}\).
(c) · 2 marks
- 1 mark: Differentiate \(\displaystyle \frac{dy}{dx}\) in t and divide once more by \(\displaystyle \frac{dx}{dt}\) before evaluation.
- 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 confuse \(\displaystyle \frac{dy}{dt}\) with \(\displaystyle \frac{dy}{dx}\).
Total: 6 marks.