1.10 Numerical Methods (A-level only)

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Question 88

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

t_{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]

1.10 Numerical Methods (A-level only)

Question 88

Markscheme

1.10 Numerical Methods (A-level only)

Question 88

Markscheme

1.10 Numerical Methods (A-level only)

Question 88

1.10 Numerical Methods (A-level only) Questions

1.10 Numerical Methods (A-level only)

Question 88

1.10 Numerical Methods (A-level only) Questions