C3 June 2008 Q1
1. The point \(P\) lies on the curve with equation\[y = 4\mathrm{e}^{2x+1}.\]The \(y\)-coordinate of \(P\) is 8.
(a) Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\). (2)
(b) Find the equation of the tangent to the curve at the point \(P\) in the form \(y = ax + b\), where \(a\) and \(b\) are exact constants to be found. (4)
| Scheme | Marks |
|---|---|
| \(\mathrm{e}^{2x+1} = 2\) \(2x + 1 = \ln 2\) | M1 |
| \(x = \dfrac{1}{2}(\ln 2 - 1)\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 8\mathrm{e}^{2x+1}\) | B1 |
| \(x = \dfrac{1}{2}(\ln 2 - 1) \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = 16\) | B1 |
| \(y - 8 = 16\left(x - \dfrac{1}{2}(\ln 2 - 1)\right)\) | M1 |
| \(y = 16x + 16 - 8\ln 2\) | A1 |
| (4) | |
| (6 marks) |