Methods2024 Exam 1 · Q7Tech-free9 marks

Trapezium rule and derivative of x sin(x)

Part of the graph of $f:[-\pi,\pi]\to \R,\ f(x)=x\sin(x)$ is shown below.

Graph of y = x sin(x) for 0 ≤ x ≤ π. The curve starts at the origin, rises to a maximum of just under 2 near x = 2π/3, and falls to zero at x = π. Gridlines are at x = π/3, 2π/3, π and y = 1, 2, 3.

Use the trapezium rule with a step size of $\frac{\pi}{3}$ to determine an approximation of the total area between the graph of $y=f(x)$ and the $x$ -axis over the interval $x\in[0,\pi]$ .

3 marks

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b.i.

Find $f'(x)$ .

1 mark

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b.ii.

Determine the range of $f'(x)$ over the interval $\left[\frac{\pi}{2},\frac{2\pi}{3}\right]$ .

1 mark

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b.iii.

Hence, verify that $f(x)$ has a stationary point for $x\in\left[\frac{\pi}{2},\frac{2\pi}{3}\right]$ .

1 mark

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