FP1 January 2013 Q3
3. \[\mathrm{f}(x) = 2x^{\frac{1}{2}} + x^{-\frac{1}{2}} - 5, \qquad x > 0\]
(a) Find \(\mathrm{f}'(x)\). (2)
The equation \(\mathrm{f}(x) = 0\) has a root \(\alpha\) in the interval \([4.5,\ 5.5]\).
(b) Using \(x_0 = 5\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm{f}(x)\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures. (4)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}'(x) = x^{-\frac{1}{2}} - \tfrac{1}{2}x^{-\frac{3}{2}}\) | M1 A1 |
| (2) |
Notes
(a) M for at least one of \(\pm ax^{-\frac{1}{2}}\) or \(\pm bx^{-\frac{3}{2}}\), A for correct (equivalent) answer only
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(5) = -0.0807\) | B1 |
| \(\mathrm{f}'(5) = 0.4025\) | M1 |
| \(x_1 = x_0 - \dfrac{\mathrm{f}(x_0)}{\mathrm{f}'(x_0)} = 5 - \dfrac{-0.0807}{0.4025}\) | M1 |
| \(= 5.2(0)\) | A1 |
| (4) | |
| [6] |
Notes
The B and M marks are implied by a correct answer only with no working or by \(\tfrac{5}{9}(10\sqrt{5} - 13)\)
(b) B for awrt \(-0.0807\), first M for attempting their \(\mathrm{f}'(5)\), M for correct formula and attempt to substitute, A for awrt 5.20, but accept 5.2