A circular cylinder rolls on a horizontal plane with uniform angular velocity; within it rolls a smaller cylinder also with uniform angular velocity. If the smaller rolls round the circumference of the larger in the time taken by the latter to make one revolution, compare their angular velocities.
A smooth sphere suspended by a string is struck directly by an equal sphere moving downwards, the line of impact being inclined to the vertical. Prove that if there is perfect restitution no kinetic energy is destroyed by the impact.
Find the \(n\)th differential coefficients with respect to \(x\) of \[ x\log x, \quad \sin^3 x, \quad 1/(x^2+1). \]
Determine the stationary values of the function \(e^{ax}\sin bx\), where \(a\) and \(b\) are positive, and illustrate the results by a figure.
The relation between the variables being \(f(x,y)=0\), find \(\dfrac{d^2y}{dx^2}\) in terms of the partial differential coefficients of \(f(x,y)\) with respect to \(x\) and \(y\).
The perpendicular from the origin on the tangent to a curve being denoted by \(p\), and the angle this perpendicular makes with a fixed line by \(\phi\), find an expression for the projection of the radius vector on the tangent. Obtain the relation between \(p\) and \(\phi\) for the curve given by \[ x = a\cos t, \quad y = a\sin^3 t. \]
Shew how to determine the asymptotes of an algebraic curve, including the cases in which the curve has (i) asymptotes parallel to an axis, (ii) a pair of parallel asymptotes. Find the asymptotes of the curve \[ 2x(x+y)^2 - (x+y)^2(x-2y)+x=0. \]
Define the curvature at any point of a curve, and obtain that of the curve \(x=f(t), y=F(t)\) at the point \(t\). Find the radius of curvature at any point of the curve \[ x^2y = x^3+y^3. \]
Integrate with respect to \(x\) \[ \frac{1}{1+x+x^2}, \quad \frac{1}{(x+1)\sqrt{x^2+x+2}}, \quad x\sin 2x\sin 3x. \] By means of the substitution \[ \tan\tfrac{1}{2}\theta = \sqrt{\frac{1+e}{1-e}}\tan\tfrac{1}{2}\phi, \] evaluate \[ \int_0^\pi \frac{d\theta}{(1+e\cos\theta)^2}. \]
Explain the method of integration by parts, and shew that if \(\int \phi(x)\,dx\) is known then \(\int \phi^{-1}(x)\,dx\) can be found, where \(\phi^{-1}(x)\) is the inverse function corresponding to \(\phi(x)\).