Show that \[\cos 3\theta = 4\cos^3\theta - 3\cos\theta\] (thus expressing \(\cos 3\theta\) as a cubic in \(\cos\theta\)). Show that if we can express \(\cos m\theta\) and \(\sin\theta\sin(m-1)\theta\) as polynomials of degree at most \(m\) in \(\cos\theta\) for all \(m\) with \(1 \leq m \leq n\), then we can express \(\cos(n+1)\theta\) and \(\sin\theta\sin n\theta\) as polynomials of degree at most \(n+1\) in \(\cos\theta\). Deduce that \[\cos n\theta = \sum_{r=0}^{n} a_{nr}(\cos\theta)^r\] for suitable real numbers \(a_{n0}, a_{n1}, \ldots, a_{nn}\). If we write \[T_n(x) = \sum_{r=0}^{n} a_{nr}x^r,\] show, using the fact that \(T_n(x) = \cos(n\cos^{-1}x)\) for \(|x| \leq 1\), or otherwise, that \begin{align} \text{(i)} \quad & |T_n(x)| \leq 1 \text{ for } |x| \leq 1,\\ \text{yet (ii)} \quad & |T_n'(1)| = n^2. \end{align} [Hint for (ii): If \(f\) is continuous then, automatically, \(f(1) = \lim_{x \to 1}f(x)\).]
Find a solution to the differential equation \[\frac{dy}{dt} = 2(2y^{\frac{1}{2}} - y^2)^{\frac{1}{2}}\] for which \(y = 1\) when \(t = 0\).
Prove that the curves \(y = \frac{3x}{2}\) and \(y = \sin^{-1}x\) intersect precisely once in the range \(0 < x \leq 1\); \(\sin^{-1}x\) is to be interpreted as the value of \(\theta\) between 0 and \(\frac{1}{2}\pi\) for which \(\sin\theta = x\). Sketch, on the same axes, these two functions for this range of \(x\). Use this sketch to illustrate graphically the sequence of numbers \(q_n\) governed by \[q_{n+1} = \sin\left(\frac{3q_n}{2}\right), \quad q_0 = \frac{1}{2},\] and deduce from the picture that the sequence converges as \(n \to \infty\) to a number less than 1.
Evaluate the indefinite integral \[\int \frac{d\theta}{a + \cos\theta},\] where \(a > 1\), using the substitution \(t = \tan\frac{1}{2}\theta\) or otherwise. What is the value of \[\int_0^{2\pi}\frac{d\theta}{a+\cos\theta}?\] What happens to the latter integral as \(a \to 1\) from above?
Let \((a,b)\) be a fixed point, and \((x,y)\) a variable point, on the curve \(y = f(x)\) (where \(z > a\), \(f'(x) \geq 0\)). The curve divides the rectangle with vertices \((a,b)\), \((a,y)\), \((x,y)\) and \((x,b)\) into two portions, the lower of which has always half the area of the upper. Show that the curve is a parabola with its vertex at \((a,b)\).
By using diagrams or otherwise, explain why \[\sum_{r=n}^{\infty} r^{-2} > \int_{n}^{\infty} x^{-2}dx > \sum_{r=n+1}^{\infty} r^{-2}.\] If we write \(A = \sum_{r=1}^{\infty} r^{-2}\), show that \[n^{-1} > A - \sum_{r=1}^{n} r^{-2} > (n+1)^{-1}.\] How large must we take \(n\) to ensure that \(\sum_{r=1}^{n} r^{-2}\) approximates \(A\) with an error of less than \(10^{-4}\)? Show that, for the same \(n\), \[(n+1)^{-1} + \sum_{r=1}^{n} r^{-2}\] approximates \(A\) with an error of less than about \(10^{-8}\).