Sketch the curve \[ x=t(t^2-1), \quad y=t^3(t^2-1), \] and find the coordinates of the points at which the tangent to the curve is parallel to either the \(x\)-axis or the \(y\)-axis.
Using the equation \[ \tan^{-1}x = \int_0^x \frac{dt}{1+t^2} \] show that, if \(x>0\), \(\tan^{-1}x\) lies between \(x-\displaystyle\frac{x^3}{3}\) and \(x-\displaystyle\frac{x^3}{3}+\frac{x^5}{5}\). Use this result to evaluate \(\tan^{-1}\frac{1}{11}\) correct to five places of decimals.