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The diagram shows the curve \(y = \tan ^ { - 1 } x\) and its asymptotes \(y = \pm a\).
- State the exact value of \(a\).
- Find the value of \(x\) for which \(\tan ^ { - 1 } x = \frac { 1 } { 2 } a\).
The equation of another curve is \(y = 2 \tan ^ { - 1 } ( x - 1 )\).
- Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of \(a\).
- Verify by calculation that the value of \(x\) at the point of intersection of the two curves is 1.54 , correct to 2 decimal places.
Another curve (which you are not asked to sketch) has equation \(y = \left( \tan ^ { - 1 } x \right) ^ { 2 }\).
- Use Simpson's rule, with 4 strips, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x\).