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The diagram shows the part of the curve \(y = 3 x \sin 2 x\) for which \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
The maximum point on the curve is denoted by \(P\).
- Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2 x + 2 x = 0\).
- Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration.
- The trapezium rule, with four strips of equal width, is used to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\).
Show that the result can be expressed as \(k \pi ^ { 2 } ( \sqrt { 2 } + 1 )\), where \(k\) is a rational number to be determined.
- Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\).
- Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3 x \sin 2 x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
- Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case.