C3 January 2007 Q4
4.
(i) The curve \(C\) has equation\[y = \frac{x}{9 + x^2}.\]Use calculus to find the coordinates of the turning points of \(C\). (6)
(ii) Given that\[y = (1 + \mathrm{e}^{2x})^{\frac{3}{2}},\]find the value of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) at \(x = \dfrac{1}{2}\ln 3\). (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{(9 + x^2) - x(2x)}{(9 + x^2)^2}\) \(\left(= \dfrac{9 - x^2}{(9 + x^2)^2}\right)\) | M1 A1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0 \Rightarrow 9 - x^2 = 0 \Rightarrow x = \pm 3\) | M1 A1 |
| \(\left(3, \dfrac{1}{6}\right), \left(-3, -\dfrac{1}{6}\right)\) Final two A marks depend on second M only | A1, A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3}{2}(1 + \mathrm{e}^{2x})^{\frac{1}{2}} \times 2\mathrm{e}^{2x}\) | M1 A1 A1 |
| \(x = \dfrac{1}{2}\ln 3 \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3}{2}(1 + \mathrm{e}^{\ln 3})^{\frac{1}{2}} \times 2\mathrm{e}^{\ln 3} = 3 \times 4^{\frac{1}{2}} \times 3 = 18\) | M1 A1 |
| (5) | |
| (11 marks) |