10.3 The Newton-Raphson Method

List view

The height of a submerged probe, $h$ metres, relative to a fixed monitoring station at time $t$ seconds is modeled by the equation $\cos 3t - 3t + 4, \quad t > 0$ where the argument of the cosine function is in radians. The probe first reaches a height of zero at time $\alpha$ , denoted by point $P$ in its path.

Show that $α\alpha$ lies between 0.8 and 0.9.

[2]

The Newton-Raphson method can be used to find an approximate value for $α\alpha$ . The iteration formula for this method is given by $tn+1=tn+2cos⁡3tn−3tn+46sin⁡3tn+3t_{n+1} = t_n + \frac{2 \cos 3t_n - 3t_n + 4}{6 \sin 3t_n + 3}$

Using this iteration formula with $t_1 = 0.8$ , find, to 4 decimal places, the value of: (i) $t_2$ (ii) $t_5$

[3]

The points $Q$ and $R$ are the first and second local minimum points on the probe's depth-time curve 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 $R$ .

[4]

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.