Sketch the graph of the function \[\phi_n(x) = e^{-x} \left(1+x+\frac{x^2}{2!}+ \ldots +\frac{x^n}{n!}\right)-k,\] where \(k\) is a constant, \(0 < k < 1\); distinguish as you think fit between different values of \(n\). Show that there is just one positive value of \(x\) for which \[1+x+\frac{x^2}{2!}+ \ldots +\frac{x^n}{n!} = ke^x.\] Denoting this by \(x_n\), show that \(x_n < x_{n+1}\). [It may be assumed that, for any \(m\), \(x^me^{-x} \to 0\) as \(x \to \infty\).]
(i) Sketch, in the same diagram, the graphs of \[y = \tan x, \quad y = \tan^{-1} x,\] where \(\tan^{-1} x\) denotes the principal value. Show that the equation \[\tan x = \tan^{-1} x\] has just one root between \((n - \frac{1}{2})\pi\) and \((n + \frac{1}{2})\pi\) for \(n \geq 1\); how many such roots are there when \(n = 0\)? Give an estimate for the root when \(n\) is large. (ii) The continuous curve \(y = f(x)\) is such that \[f'(x) > 0, \quad = 0, \quad < 0\] according as \[f(x) > 0, \quad = 0, \quad < 0 \quad \text{respectively}.\] By considering the function \(\{f(x)\}^2\), or otherwise, show that if \(f(x_0) = 0\), then \(f(x) = 0\) for all \(x < x_0\). Is there any corresponding result if (4) is replaced by \[f'(x) < 0, \quad = 0, \quad > 0\] according as \[f(x) > 0, \quad = 0, \quad < 0 \quad \text{respectively}?\]
A boiling fluid, which is initially a mixture of equal amounts of fluids \(A\) and \(B\), evaporates at a constant rate, and evaporates completely in ten seconds. At any time, the ratio of the rate of evaporation of fluid \(A\) to the rate of evaporation of fluid \(B\) is twice the ratio of the amount of fluid \(A\) to the amount of fluid \(B\). How long elapses before the two fluids are evaporating at exactly the same rate?
Let \begin{equation*} L_n = \int_{0}^{\pi} \sin^n \theta\, d\theta. \end{equation*} Show that \(L_{2m-1} > L_{2m} > L_{2m+1}\). Establish a recurrence relation between \(L_{n+2}\) and \(L_n\), and by solving this (for a value \(p_m\) and for \(n\) odd) show that \begin{equation*} \frac{2m+1}{2m}p_m > \frac{\pi}{2} > p_m, \end{equation*} where \begin{equation*} p_m = \frac{(2m)^2(2m-2)^2\ldots 2^2}{(2m+1)(2m-1)^2\ldots 3^2\cdot 1^2}. \end{equation*}
The curve \(x^2+(y-a)^2 = a^2\) \((-a \leq x \leq a, 0 \leq y \leq a)\) is rotated about the \(x\)-axis. Find the volume contained between the resulting surface and the planes \(x = -a\) and \(x = a\). Find also the centre of gravity of the plane area bounded by the curve, the lines \(x = -a\) and \(x = a\), and the \(x\)-axis.
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*}