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.