If \(f(x) = e^{-ax}\sin(bx+c)\), \(a > 0\), and \(b > 0\), show that the values of \(x\) for which \(f(x)\) has either a maximum or a minimum form an arithmetic progression with difference \(\pi/b\). Show further that the values of \(f(x)\) at successive maxima form a geometric progression with ratio \(e^{-\pi a/b}\). Find the points of inflexion of \(f(x)\). Describe a physical problem for which \(f(x)\) might be a solution.
Sketch the curve whose equation, in polar coordinates, is \begin{equation*} \frac{l}{r} = 1+e\cos\theta, \end{equation*} \(e\) being a positive constant; distinguish between the cases \(e < 1\), \(e = 1\), \(e > 1\). By using the substitution \begin{equation*} \cos\phi = \frac{\cos\theta+e}{1+e\cos\theta} \quad (0 \leq \theta \leq \pi), \end{equation*} or otherwise, find the area enclosed by the curve when it is closed.
If \(f(x) = \sin(a\sin^{-1}x)\), \(-1 \leq x \leq 1\), show that \begin{equation*} (1-x^2)f''(x) - xf'(x) + a^2f(x) = 0. \end{equation*} Use Leibnitz' theorem to show that \begin{equation*} f^{(n+2)}(0) = (n^2-a^2)f^{(n)}(0), \end{equation*} and hence find \(\sin(5\sin^{-1}x)\) as a polynomial in \(x\). (Assume that there is such a polynomial.)
Prove the formulae \begin{align*} \sin\frac{y}{2} \sum_{m=0}^{N} \sin my &= \sin\frac{(N+1)y}{2}\sin\frac{Ny}{2},\\ \sin\frac{y}{2} \sum_{m=0}^{N} \cos my &= \sin\frac{(N+1)y}{2}\cos\frac{Ny}{2}. \end{align*} A numerical integration formula is \begin{equation*} \int_{0}^{2\pi} f(x)dx \simeq \frac{2\pi}{M}\sum_{m=0}^{M-1} f(x_m), \quad \text{where } x_m = \frac{2\pi m}{M}. \end{equation*} For what values of \(M\) will all functions of the form \begin{equation*} f(x) = \sum_{r=0}^{R} a_r\cos rx + \sum_{s=0}^{S} b_s\sin sx \end{equation*} be integrated exactly by this formula? (Here \(R\) and \(S\) are fixed integers, but \(a_r\) and \(b_s\) have any values.)
Verify that \begin{equation*} \frac{1}{2}\{f(n)+f(n+1)\} - \int_{n}^{n+1} f(x)dx = \frac{1}{2}\int_{0}^{1} t(1-t)f''(t+n)dt. \end{equation*} Using the inequality \begin{equation*} 0 \leq t(1-t) \leq \frac{1}{4} \quad \text{if } 0 \leq t \leq 1, \end{equation*} show that \begin{equation*} \frac{1}{2}\{\log n + \log(n+1)\} = \int_{n}^{n+1}\log x dx - r_n \quad (n > 0), \end{equation*} where \begin{equation*} 0 \leq r_n \leq \frac{1}{8}\left(\frac{1}{n} - \frac{1}{n+1}\right). \end{equation*} Deduce that, for all positive integers \(N\), \begin{equation*} \log N! = \left(N+\frac{1}{2}\right)\log N - N + 1 - R_N, \end{equation*} where \begin{equation*} 0 \leq R_N \leq \frac{1}{8}\left(1-\frac{1}{N}\right). \end{equation*}
A hospital buys batches of a certain tablet from a pharmaceutical company. A tablet is considered unsatisfactory if it contains more than 1 microgram of arsenic. It is known that within any batch of tablets the arsenic content is normally distributed with standard deviation 0.05 micrograms about a mean which depends on the batch. From every batch the hospital randomly selects \(n\) tablets for analysis, and rejects the batch if the mean arsenic content of the \(n\) tablets is greater than \(C\). What values should be chosen for \(n\) and \(C\) if the desired chances of rejecting batches with 0.1\% and 1\% of defective tablets, respectively, are 20\% and 90\%?
Tests are to be carried out to discover which of a large number of people have a particular disease. To keep the number of tests low, samples of blood from 40 people are mixed and tested together. If the test indicates that the disease is absent, all 40 people are free from it, but if the test shows that the disease is present, all 40 people are retested individually. Assuming that there is a constant and independent chance \(p\) that a person has the disease, determine the mean number of tests that have to be carried out. The following modified procedure is proposed with the aim of reducing the number of tests: whenever the group test shows that the disease is present, samples from 20 of the group are mixed and tested, and samples from the other 20 are then tested individually. Either or both sets of people are then tested individually if necessary. Show that this procedure does result in a smaller mean number of tests if \(p\) is small enough. Can you suggest any way of improving the procedure further?
Two opponents play a series of games in each of which they have an equal chance of winning. The loser of each game pays the winner one unit of capital. The first player begins with \(k\) units of capital and the second player has all \(\alpha\) units of capital. Let \(p_k\) be the probability that the first player wins the series. Write down a relation between \(p_{k-1}, p_k\) and \(p_{k+1}\); and hence show that \(p_k = k/\alpha\).
A shopkeeper has to meet a continuous demand of \(r\) units per unit of time from his customers. At intervals of \(T\) units of time, he buys a quantity of \(Q\) units from a wholesaler, where \(Q \geq rT\). The cost of placing the order is \(a\) pounds and its cost per unit is \(b\) pounds. If he runs out of stock at any unit time, his customers go elsewhere (at no cost to him per unit of time); but as soon as his shop is set again (through loss of customers, or other business) for the period during which the capital tied up against these losses he makes a net profit on this line of business if \(p^2 > 2ab/r\), where \(p\) is the amount of money per unit sold. Show that he can make a maximum profit per unit time of \(X\) which will maximise his profit per unit time.
Two smooth planes meet at right angles in a horizontal line. A rod, whose density is not necessarily uniform, is placed above this line and perpendicular to it, and rests on the planes. If the steeper plane is inclined at an angle \(\theta\) to the horizontal, find the equilibrium positions of the rod. Discuss explicitly the following special cases: