Problems

Find the number of stationary values of the function \(y=x^2+6\cos x\), distinguishing between maxima and minima, and find the number of points of inflexion.

1. If \(f_n(x) = \dfrac{d^n}{dx^n} \dfrac{\log x}{x}\) for \(x>0\), \(n=0, 1, 2, \dots\), show that \[ f_{n+1}(x) + (n+1)x^{-1}f_n(x) = (-1)^n n! x^{-n-2}. \]
2. If \(g_n(x) = \dfrac{d^n}{dx^n} \dfrac{3x^2+2x+1}{(x^2+1)(x+1)}\), show that for \(n=0, 1, 2, \dots\) \[ g_{4n+1}(1) = -4^{-(2n+1)}(4n+1)! \]

Evaluate \[ \int_0^\infty \frac{x^2\,dx}{(1+x^2)^{5/2}} \] and \[ \int_{-\infty}^\infty \frac{dx}{(e^{x/2}+1)(e^{-x/2}+1)}. \]

Find the volume of the body defined by \(z^2 \le e^{-(x^2+y^2)}\) and \(x^2+y^2 \le a^2\).

Prove that \(\displaystyle\int_0^x \frac{\sin y}{y}\,dy\) is positive when \(x\) is positive.

Show that, if the variables \(x, y\) and \(r, \theta\) are connected by the relations \[ x=r\cos\theta, \quad y=r\sin\theta \] and if \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0, \] then \[ \frac{\partial^2 f}{\partial r^2} + \frac{1}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} = 0. \]

On a plane inclined at an angle \(\alpha\) to the horizontal a uniform circular cylinder of radius \(a\), length \(2a\), and weight \(W\) rests with its axis horizontal and with one of its generators completely in contact with a uniform cube of side \(2a\) and weight \(w\) which rests on the inclined plane below the cylinder. The cube is about to begin to overturn. Prove that \[ \tan\alpha < \frac{w+3W}{w+W}, \] \[ 2\sin(\alpha-\lambda) + \cos(\alpha+\lambda) < \frac{W}{w}\sin\alpha(\sin\lambda - 2\cos\lambda), \] where \(\lambda\) is the angle of friction between the cube and the plane.

Explain the use of the force and funicular polygons in finding the resultant of a system of coplanar forces, and apply the method to find the magnitude and line of action of the resultant of forces 1, 2, 3, 4 acting respectively in the sides of a square taken in order.

Prove that, in general, a system of coplanar forces may be reduced to a force acting through an arbitrary point of the plane together with a couple. A piece of uniform thin wire is bent into the form of the three sides \(AB, BC, CD\) of a square and the wire is freely suspended at \(A\). Express the magnitudes of the couples acting at \(B\) and \(C\) in terms of the weight and length of the wire.

\(AB, BC\) are two similar uniform rods each of length \(a\), smoothly jointed at \(B\), and freely suspended at \(A\). \(C\) is tied by a light inextensible string of length \(2a\) to a point \(D\) distant \(3a\) from \(A\) and such that \(AD\) is horizontal. If \(AB, BC\) are inclined to \(AD\) at angles \(\theta, \phi\) respectively when the system is in equilibrium, shew that the reaction at \(A\) is inclined to \(AD\) at an angle \[ \tan^{-1} \frac{1}{2} \left(\frac{12-3\cos\theta-\cos\phi}{3\cos\theta+\cos\phi}\right)(\tan\theta-3\tan\phi). \]