FP1 January 2009 Q5
5. \[\mathrm{f}(x) = 3\sqrt{x} + \frac{18}{\sqrt{x}} - 20\]
(a) Show that the equation \(\mathrm{f}(x) = 0\) has a root \(\alpha\) in the interval \([1.1,\ 1.2]\). (2)
(b) Find \(\mathrm{f}'(x)\). (3)
(c) Using \(x_0 = 1.1\) 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 |
|---|---|
| attempt evaluation of f(1.1) and f(1.2) (– looking for sign change) | M1 |
| \(\mathrm{f}(1.1) = 0.30875\), \(\mathrm{f}(1.2) = -0.28199\) Change of sign in \(\mathrm{f}(x) \Rightarrow\) root in the interval | A1 |
| (2) |
Notes
(a) awrt 0.3 and \(-0.3\) and indication of sign change for first A1
| Scheme | Marks |
|---|---|
| \(\mathrm{f}'(x) = \dfrac{3}{2}x^{-\frac{1}{2}} - 9x^{-1\frac{1}{2}}\) | M1 A1 A1 |
| (3) |
Notes
(b) Multiply by power and subtract 1 from power for evidence of differentiation and award of first M1
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(1.1) = 0.30875\ldots \qquad \mathrm{f}'(1.1) = -6.37086\ldots\) | B1 B1 |
| \(x_1 = 1.1 - \dfrac{0.30875\ldots}{-6.37086\ldots}\) | M1 |
| \(= 1.15\) (to 3 sig.figs.) | A1 |
| (4) | |
| [9] |
Notes
(c) awrt 0.309 B1 and awrt \(-6.37\) B1 if answer incorrect
Evidence of Newton-Raphson for M1
Evidence of Newton-Raphson and awrt 1.15 award 4/4