Hint
The quotient formula requires \(\displaystyle \frac{dx}{dt}\ne 0\). Divide the two coordinate derivatives.
Worked solution
(a)
\(\displaystyle \frac{dy}{dx}=\frac{3 t^{2} + 2}{2 t}\); substitute \(\displaystyle t=2\).
Answer: \(\displaystyle \frac{7}{2}\).
Answer: \(\displaystyle \frac{7}{2}\)
(b)
\(\displaystyle \frac{d^2y}{dx^2}=\frac{3 - \frac{3 t^{2} + 2}{2 t^{2}}}{2 t}\); substitute \(\displaystyle t=2\).
Answer: \(\displaystyle \frac{5}{16}\).
Answer: \(\displaystyle \frac{5}{16}\)
(c)
\(\displaystyle c=y-mx=- \frac{81}{2}\)
Answer: \(\displaystyle - \frac{81}{2}\).
Answer: \(\displaystyle - \frac{81}{2}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Divide the two coordinate derivatives.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not differentiate \(\displaystyle \frac{dy}{dx}\) with respect to t and stop there.
(b) · 2 marks
- 1 mark: Differentiate the gradient by t, then divide by \(\displaystyle \frac{dx}{dt}\).
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The parameter value is not automatically the Cartesian x-coordinate.
(c) · 2 marks
- 1 mark: Evaluate the point before constructing the tangent.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not differentiate \(\displaystyle \frac{dy}{dx}\) with respect to t and stop there.
Total: 6 marks.