A circle \(S\) rolls once round an equal circle \(S'\). Determine the area contained within the closed curve traced out by the point \(P\) of \(S\) which was first in contact with \(S'\).
(i) Prove that if \(f(x)\) is an even function of \(x\) (i.e. \(f(-x) = f(x)\)) then its first derivative (assumed to exist) is an odd function of \(x\) (i.e. \(f'(-x) = -f'(x)\)). Determine whether the converse is true, and give some justification for your answer. (ii) Do what is indicated in (i), but with 'odd' and 'even' interchanged.
Explain briefly how complex numbers may be represented as points in a plane. How many squares are there that have the points \(3-i\), \(1+4i\) as two of their corners? In each case find the remaining two corners. If \(z_1\), \(z_2\) and \(z_3\) are the complex numbers representing the vertices of an equilateral triangle, prove that $$z_1^2 + z_2^2 + z_3^2 = z_2 z_3 + z_3 z_1 + z_1 z_2.$$ If this condition is satisfied, what can you deduce about the points represented by \(z_1\), \(z_2\) and \(z_3\), and why?
(i) Given that \(\sum_{n=1}^{\infty} n^{-2} = S_{\infty}\), find \(\sum_{n=1}^{\infty} n^{-2}(n+1)^{-2}\). (ii) Find (a) \(\sum_{n=1}^{\infty} nr^n\) and (b) \(\sum_{n=1}^{\infty} n^2 r^n\). What can be concluded about the 'sum to infinity' \(\sum_{n=1}^{\infty} n^2 r^n\)?
(i) Explain how definite integrals may be obtained as the limit of suitable sums. Illustrate by obtaining \(\int_0^3 x^2 dx\) without making use of the relation \(dx^3/dx = 3x^2\). [You may assume that \(\displaystyle \sum_{n=1}^{N} n^2 = \frac{1}{6}N(N+1)(2N+1)\)] (ii) Assuming that \(\log n = \int_1^n x^{-1} dx\), find $$\lim_{n \to \infty} \left(\frac{1}{2n} + \frac{1}{2n+1} + \ldots + \frac{1}{3n}\right).$$
(i) Find the solution of the differential equation \(x dy/dx = 3y\) that takes the value 2 when \(x = 1\). (ii) Find a differential equation satisfied by the function \(g(x) = e^{-xf(x)}\) whenever \(f(x)\) is a function that satisfies the differential equation \(d^2f/dx^2 + xdf/dx - f = 0\).
Explain the relation between the greatest and least values taken by a function in an interval, the maxima and minima of the function, and the points where the first derivative of the function is zero. Illustrate by considering the functions (i) \(\exp[-(x^2-1)^2]\), \quad (ii) \(\exp[-|x^2-1|]\), in the interval \(-2 \leq x \leq 2\). Draw a rough sketch of each function. [exp \(y\) means \(e^y\).]
Evaluate (i) \(\int_0^{\pi} x^2 e^{2x} dx\); \quad (ii) \(\int_{-\pi}^{\pi} |\sin x| e^{i \cos x} dx\); (iii) \(\int_0^{\infty} \frac{dx}{(x+1)(2x+3)}\); \quad (iv) \(\int_{-\infty}^{\infty} \frac{\sin x}{1+x^2} dx\).
Calculate the volume of the solid of revolution formed by rotating the cardioid \(r = a(1-\cos\theta)\) about the line \(\theta = 0\).
If \(m\) and \(n\) are positive integers, with \(m > n\), determine (by graphical considerations, or otherwise) how many roots of the equation \(x \sin x = 2n\pi\) are in the interval \(0 \leq x \leq 2m\pi\). Show that if \(N\) is large enough there is exactly one root in the interval $$(N-\frac{1}{2})\pi \leq x \leq (N+\frac{1}{2})\pi,$$ and that this root is approximately equal to \(N\pi + (-1)^N 2n/N\) when \(N\) is large. Can you find a better approximation?