Let \(f\) be a positive function of \(x\) with a negative first derivative for \(x \geq 1\). Show that \[\sum_{m=2}^{n} f(m) \leq \int_{1}^{n} f(x)dx \leq \sum_{m=1}^{n-1} f(m)\] where \(n > 2\) is an integer. Hence, or otherwise, show that (i) \(\sum_{m=1}^{n} \frac{1}{m^2} \leq 2\), for all \(n \geq 1\), (ii) \(\sum_{m=1}^{n} \frac{1}{m}\) is unbounded as \(n \to \infty\), (iii) \(0 \leq \left(\sum_{m=1}^{n} \frac{1}{m}\right) - \ln n \leq 1\), for all \(n \geq 1\). Show also that \[\sum_{m=4}^{n} \frac{1}{m\ln^2 m}\] is bounded as \(n \to \infty\).
The function \(f(x)\) has first and second derivatives for all values of \(x\) and satisfies the equation \[xf''(x) + f'(x) + xf(x) = 0,\] together with the condition \(f(a) = 0\) for some \(a > 0\). By considering the derivates with respect to \(x\) of \((xf(x)f'(x))\) and \((x^2f'(x)^2)\), or otherwise, show that \[\int_{0}^{a} xf(x)^2 dx = \int_{0}^{a} xf'(x)^2 dx = \frac{1}{2}a^2[f'(a)]^2.\]
A hole of circular cross-section is drilled through a spherical ball of radius \(a\), so that the axis of the hole goes through the centre of the sphere. The diameter of the hole is such that its length is \(b(<2a)\). What is the volume and total surface area of that part of the sphere that remains?
A canny Cambridge student attempts to build a rapid fuelless transport system which operates by dropping vehicles into straight frictionless tunnels that connect the major cities of the world. Assuming the density of the Earth to be uniform, show that the anticipated duration of each journey is \(T = \pi\sqrt{(R/g)}\), irrespective of the destination, where \(R\) is the radius of the Earth and \(g\) is the gravitational acceleration on the Earth's surface. An inevitable small frictional force of magnitude \(kv\), where \(k\) is constant, prevents the prototype vehicle from reaching its destination. Small motors fitted to subsequent models are adjusted to propel the vehicles such that they always travel at the speed \(v\) they would have attained had friction been absent. Show that the work done by the motors never exceeds \(\frac{1}{2}kgTR\) per journey. [You may assume the gravitational acceleration inside a uniform spherical body at a distance \(r\) from its centre is proportional to \(r\).]
The gravitational attraction between two pointlike bodies of masses \(m_1\) and \(m_2\) is \(\frac{Gm_1m_2}{r^2}\), where \(G\) is a constant and \(r\) is the distance between the bodies. The bodies are initially at rest a distance \(a\) apart. Because \(m_2 \gg m_1\), the body of mass \(m_2\) can be taken to remain at rest in the subsequent motion. Show that the bodies will collide after a time \(\frac{\pi a^{3/2}}{2(2Gm_2)^{1/2}}\). [You may find that \[\int_{0}^{1}\left(\frac{1}{x} - 1\right)^{-1} dx = \frac{\pi}{2}\] is useful.] By considering the motion of the earth round the sun as circular, show that if the earth were suddenly stopped in its orbit around the sun it would take about 65 days to reach the sun under the gravitational attraction between the earth and the sun.
A particle of unit mass is projected vertically upwards with initial speed \(V\). There is a resisting force \(kgv\), where \(v\) is the speed of the particle. Show that the particle reaches a maximum height above the point of projection \[\frac{1}{kg}\left\{V - \frac{1}{k}\ln(1 + kV)\right\}.\] Find the total time taken, and the distance below the point of projection, for the particle to attain speed \(V\) again. Can it always do this?