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1.12 I: Numerical methods (A-level only)

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Question 84

The height of a submerged probe, hhh metres, relative to a fixed monitoring station at time ttt seconds is modeled by the equation

h(t)=2cos⁡3t−3t+4,t>0 h(t) = 2 \cos 3t - 3t + 4, \quad t > 0 h(t)=2cos3t−3t+4,t>0

where the argument of the cosine function is in radians. The probe first reaches a height of zero at time t=αt = \alphat=α, denoted by point PPP in its path.

a.

Show that α\alphaα lies between 0.8 and 0.9.

[2]
b.

The Newton-Raphson method can be used to find an approximate value for α\alphaα. The iteration formula for this method is given by

tn+1=tn+2cos⁡3tn−3tn+46sin⁡3tn+3 t_{n+1} = t_n + \frac{2 \cos 3t_n - 3t_n + 4}{6 \sin 3t_n + 3} tn+1​=tn​+6sin3tn​+32cos3tn​−3tn​+4​

Using this iteration formula with t1=0.8t_1 = 0.8t1​=0.8, find, to 4 decimal places, the value of: (i) t2t_2t2​ (ii) t5t_5t5​

[3]
c.

The points QQQ and RRR are the first and second local minimum points on the probe's depth-time curve respectively for t>0t > 0t>0.

Using calculus, find the exact value of the ttt-coordinate of QQQ and the exact value of the ttt-coordinate of RRR.

[4]

1.12 I: Numerical methods (A-level only) Questions

  1. A Level
  2. /Maths
  3. /1.12 I: Numerical methods (A-level only)

Practise AQA A Level Maths 1.12 I: Numerical methods (A-level only) with exam-style questions for A Level Maths. 107 questions covering 1.12.1 Locating roots by change of sign (A-level only), 1.12.2 Iterative methods and Newton-Raphson (A-level only), 1.12.3 Numerical integration (A-level only), and 1.12.4 Numerical methods in context (A-level only), matched to the AQA A Level Maths (7357) 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.

Question bank