10.3 The Newton-Raphson Method

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The vertical height, $h\,$ metres, of a buoyant sensor bobbing in a turbulent drainage pipe relative to a fixed datum is modelled by the equation $\sin 2t - 4t + 10, \quad t > 0$ where $t\,$ is the time in minutes and $t\,$ is measured in radians.

The sensor first hits a submerged blockage at the point $P$ , where $h = 0$ . Given that the $t$ -coordinate of $P\,$ is $α\alpha$ :

Show that $\alpha\,$ lies between 1.8 and 1.9.

[2]

The Newton-Raphson iteration formula $tn+1=4tncos⁡2tn−2sin⁡2tn−54cos⁡2tn−2t_{n+1} = \frac{4t_n \cos 2t_n - 2 \sin 2t_n - 5}{4 \cos 2t_n - 2}$ can be used to find an approximate value for $α\alpha$ .

Using this iteration formula with $t_1 = 1.7$ , find, to 4 decimal places, the value of: (i) $t_2$ (ii) $t_5$

[3]

The points $Q\,$ and $R\,$ represent the times at which the sensor reaches its first and second local minimum heights respectively for $t > 0$ .

Using calculus, find the exact value of the $t$ -coordinate of $Q\,$ and the exact value of the $t$ -coordinate of $R$ .

[5]

10.3 The Newton-Raphson Method Questions

Practise Edexcel A Level Maths 10.3 The Newton-Raphson Method with exam-style questions for A Level Maths. 23 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.