Problems

Discuss the reasoning in the following statements:

1. [(i)] Statistics show that there are more road accidents at 30 m.p.h. than at 60 m.p.h. This proves that 60 m.p.h. is the safer speed.
2. [(ii)] Smith usually gets about 10\% in his examinations, the pass mark being 50\%. He may therefore expect to pass about one examination in five.
3. [(iii)] Most towns in England are large, but most towns in Australia are small. It follows that most townspeople in England live in large towns, whereas most townspeople in Australia live in small towns.
4. [(iv)] The odds against a complete week of rainy weather are 100:1. If therefore the first six days of your holiday are washed out you can be fairly certain that the seventh day will be dry.

Assuming that the length of the circumference of a circle lies between the total lengths of side of the inscribed and circumscribed regular polygons of \(n\) sides, derive upper and lower bounds for \(\pi\). Show that when \(n = 8\) these bounds reduce to \[4\sqrt{(2-\sqrt{2})} < \pi < 8(\sqrt{2}-1).\] How would you continue to evaluate the corresponding bounds for \(n = 16, 32, 64, \ldots\), using no operation more advanced than the taking of a square root?

Evaluate the integrals \[\int_0^u \tan^{-1}x\,dx; \quad \int_0^v \sqrt{x(a-x)}\,dx \quad (0 \leq v \leq a).\] If \[I_n = \int (ax^2 + 2bx + c)^n\,dx,\] find a relation between \(I_n\) and \(I_{n-1}\), and comment on special cases.

Consider a complex variable \(z = x + iy\), and show that in the \((x, y)\) plane the two sets of equations \[\text{Re}(z^2) = \text{const.}, \quad \text{Im}(z^2) = \text{const.}\] describe two families of mutually orthogonal hyperbolae, also that \[\text{Re}(z^{-1}) = \text{const.}, \quad \text{Im}(z^{-1}) = \text{const.}\] describe families of mutually orthogonal circles. (By \(\text{Re}(\omega)\) and \(\text{Im}(\omega)\) are meant the real and imaginary parts of the complex variable \(\omega\).)

Find all the real roots of the two following equations in \(x\): \[\cos(x\sin x) = \frac{1}{2};\] \[\cos 2x + 2\cos a\cos x - 2\cos 2a = 1.\]

A rectangle lies wholly in the region \[0 < x < a,\] \[0 \leq y \leq \frac{\beta}{x^\gamma} \quad (\alpha, \beta, \gamma > 0),\] and has two of its sides along the coordinate axes. Determine the rectangle of this type which has greatest area, paying attention to the relative values of \(a\), \(\beta\) and \(\gamma\).

Sketch the curves \[x^n + y^n = 1\] for \(n = -1, 1, 2, 3, 4\). Also, sketch the curves \(y = f(x)\), \(y = f'(x)\), for a function \(f(x)\) which obeys \[f'(0) < 0, \quad f''(x) > 0;\] \[\frac{f(x)}{x} \to 1 \text{ as } x \to +\infty\] in the range \(x > 0\).

Prove the expansion \[f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + \ldots + \frac{h^{n-1}}{(n-1)!}f^{(n-1)}(x) + \frac{1}{(n-1)!}\int_0^h f^{(n)}(x+y)(h-y)^{n-1}\,dy\] by integrating the remainder term by parts. Assuming that the remainder term can be written \(h^n f^{(n)}(x + \theta h)/n!\), show that if \(f^{(n+1)}(x+y)\) is continuous in \(y\) at \(y = 0\) then \(\theta \to 1/(n+1)\) as \(h \to 0\).

Find an equation satisfied by the values of \(\theta\) for which the function \[\frac{1}{2}\theta^2 - k\cos\theta \quad (k > 0)\] has a local minimum, indicating graphically the appropriate roots. How would you determine the largest value of \(k\) for which the only minimum is the one at \(\theta = 0\)? If \(k\) is large, show that the minima adjacent to that at \(\theta = 0\) are approximately located at \[\theta = \pm 2\pi(1-1/k).\]

Find a relation connecting \(\alpha\) and \(\beta\) such that the equations \[x_0 = \beta(x_1 + x_2 + \ldots + x_n) + c_0\] \[x_j = \alpha x_{j-1} + c_j \quad (j = 1, 2, \ldots, n)\] have no solution for the unknowns \(x_0, x_1, \ldots, x_n\) unless the \(c_j\) satisfy a certain linear relation. Show further that, if the \(c_j\) satisfy this relation, and \(x_j = \xi_j\) is a solution, then so is \(\xi_j + kx^j\), whatever the value of \(k\).