Find from first principles the differential coefficient of \(\tan x\). If \(\tan y = \{(e^x+1)/(e^x-1)\}^{1/2}\), prove that \[ \frac{d^2y}{dx^2} = 1 + 12\left(\frac{dy}{dx}\right)^2\left\{1+4\left(\frac{dy}{dx}\right)^2\right\}. \]
Trace the curve \(y=e^{1/x}\). Find the inflexions and the asymptotes.
Prove the following formulae for the radius of curvature at any point of a plane curve \[ \text{(i) } r\frac{dr}{dp}, \quad \text{(ii) } \left\{u^2+\left(\frac{du}{d\theta}\right)^2\right\}^{3/2} / \left\{u^3\left(u+\frac{d^2u}{d\theta^2}\right)\right\} \quad (u=1/r). \] Prove that the distance between the origin and the centre of curvature at any point of \(r^n=a^n\cos n\theta\) is \[ \{a^{2n}+(n^2-1)r^{2n}\}^{1/2} / \{(n+1)r^{n-1}\}. \]
Prove that the area of one loop of the curve \(x^4-2xy a^2+a^2y^2=0\) is \(\frac{1}{6}a^2\).
Show that \((y-c)^2+\frac{1}{2}(x-c)^3=0\) is a family of solutions of \[ y+\frac{1}{2}p^3-(x+p^2)=0, \quad \text{where } p=\frac{dy}{dx}. \] Find the envelope of the family, and show that \(y=x\) is a cusp locus.