(a) Differentiate with respect to \(x\): (i) \(x^{x^{\cosh^{-1}x}}\); (ii) \(\tan^{-1}\left[\tan x \frac{1+\cos 2x}{1-\cos 2x}\right]\). (b) Expand \(y=\tan^{-1}x\) as a series in increasing powers of \(x\), stating any theorems you may use, or conditions you may apply to \(x\).
Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve at a point \(P\) where the length of the radius vector is \(r\) and the length of the perpendicular from the origin on to the tangent at \(P\) is \(p\). Sketch roughly the curve for which \(\frac{p}{r}\) is a constant and shew that \(\rho\) also is in constant ratio to \(r\) and \(p\). Prove that the radius of curvature can never be less than the corresponding radius vector.
Shew that if \(f(x)\) and \(\phi(x)\) are functions of \(x\) having derivatives \(f'(x), \phi'(x)\) in the range \((a,b)\), then \(\frac{f'(\xi)}{\phi'(\xi)} = \frac{f(b)-f(a)}{\phi(b)-\phi(a)}\) for some value \(\xi\) of \(x\) between \(a\) and \(b\). What restrictions (if any) are to be placed on the behaviour in the interval \((a,b)\) of \(f'(x)\) and \(\phi'(x)\)? Hence or otherwise prove that \(x^\lambda \log_e x \to 0\) as \(x \to 0\), if \(\lambda > 0\).
Find a reduction formula for \(f(m,n) = \int_0^1 x^{n-1}(1-x)^{m-1}dx\) and shew that \[ f(m,n) = \frac{(m-1)!}{n(n+1)\dots(n+m-1)}, \] \(m\) and \(n\) being positive integers. Shew further that if \(I(n) = \int_0^\infty e^{-x} x^{n-1}dx\), then (i) \(I(n+1) = n.I(n)\), (ii) \(f(m,n) = \frac{I(n).I(m)}{I(m+n)}\).
If \(x,y,z\) are three variables each of which may be regarded as a function of the other two, shew that (i) \(\left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{(\frac{\partial y}{\partial x})_z}\), where \(\left(\frac{\partial x}{\partial y}\right)_z\) denotes the partial differential coefficient of \(x\) with respect to \(y\), \(z\) being constant. (ii) \(\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y + 1 = 0\). By changing the independent variables to \(\xi=x+y, \eta=x-y\) prove that the solution of \[ \frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial y^2} \] is given by \(z=F_1(x+y)+F_2(x-y)\), where \(F_1(t), F_2(t)\) denote arbitrary functions of \(t\).
Prove that the volume enclosed by rotating a closed plane curve about a non-intersecting coplanar axis is given by the product of the area enclosed by the curve, and the length of the path traced by the centroid of the area. Shew that the volume of the surface formed by rotating the larger part of an ellipse about the latus rectum which is its boundary, is given by \[ 2\pi\frac{l^3}{(1-e^2)^3}\left\{ e\cos^{-1}(-e) + \sqrt{1-e^2}\left(\frac{2+e^2}{3}\right) \right\}, \] where \(e\) is the eccentricity and \(l\) the semi-latus rectum of the ellipse.