2015 NYJC P2 Q4

1. Prove that \(\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\tan }^{-1}}x \right)=\frac{1}{1+{{x}^{2}}}\).[2]

The diagram shows a sketch of the graph of \(y=\frac{1}{1+{{x}^{2}}}\).

2. By considering the shaded rectangle and the area of the region between the graph and the \(x\)-axis for \(n-1\le x\le n\), where \(n\ge 1\), show that
   \(\frac{1}{1+{{n}^{2}}}<{{\tan }^{-1}}\left( n \right)-{{\tan }^{-1}}\left( n-1 \right)\).[2]
3. Deduce that \(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\ldots +\frac{1}{1+{{n}^{2}}}<{{\tan }^{-1}}\left( n \right)\).[2]

The region \(R\) is bounded by the curve \(y=\frac{1}{1+{{x}^{2}}}\) and the lines \(x=3\) and \(y=\frac{1}{2}\). Write down the equation of the curve obtained when the curve \(y=\frac{1}{1+{{x}^{2}}}\) is translated by \(-3\) units in the direction of the \(x\)-axis. Hence, or otherwise, find the volume of the solid formed when \(R\) is rotated completely about the line \(x=3\).[4]