Show that, for \(0 < \lambda < 1\), the least positive root of the equation $$\sin x = \lambda x \qquad (1)$$ is a decreasing function of \(\lambda\). How many real positive roots of (1) are there when $$\lambda = \frac{2}{(4n+1)\pi},$$ with \(n\) an integer?
Let \(y_0(x) = x\), \(y_n(x) = 1 - \cos y_{n-1}(x)\) (\(n \geq 1\)). For fixed \(n\), find the limit of \(x^{-2^n}y_n(x)\) as \(x\) tends to zero.
Sketch the curves described by the following equations:
Show that, if \(y = \tanh^{-1} x\), then $$(1-x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} = 0.$$ Hence determine the value of the \(n\)th derivative of \(y\) at \(x = 0\). [You may use the theorem of Leibnitz.]
For positive \(Q\), evaluate the integrals $$I(Q) = \int_0^{\pi/2} \frac{\sin^3 \theta \, d\theta}{1 + Q^2 \cos^2 \theta}, \quad J(Q) = \int_0^{\pi/2} \frac{\cos^2 \theta \sin \theta \, d\theta}{1 + Q^2 \cos^2 \theta},$$ and show that \(I(Q) > J(Q)\) when \(Q\) is sufficiently large.
A container in the form of a right circular cone with semi-vertical angle \(\alpha\) is held with its axis vertical and vertex downwards. Water is supplied to the container at a constant volume-rate \(Q\), and it escapes through a leak at the vertex at a rate \(ky\), where \(y\) is the depth of water in the cone, and \(k\) is a constant. Show that $$\pi \tan^2 \alpha \, y^2 \frac{dy}{dt} = Q - ky,$$ and find how long it takes for the water level to rise from zero to \(Q/2k\).
Show that, if $$f(x) = \sum_{n=1}^{\infty} a_n \sin nx, \qquad (1)$$ then $$a_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin nx \, dx. \qquad (2)$$ Assuming that the function $$f(x) = x(\pi - x) \qquad (0 \leq x \leq \pi)$$ can be expressed as an infinite series $$\sum_{n=1}^{\infty} a_n \sin nx,$$ and that the coefficients are still given by the formula (2), show that in this case $$a_{2m} = 0, \quad a_{2m+1} = \frac{8}{\pi(2m+1)^3},$$ and hence sum the series $$1 - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \cdots$$
Let \(f(x)\) be a continuous decreasing function of \(x\) for \(x > 0\), and \(m\) and \(n\) be positive integers with \(m < n\). Show that $$\int_m^{n+1} f(x) \, dx < \sum_{r=m}^n f(r) < \int_{m-1}^n f(x) \, dx,$$ and hence that $$1.19 < \sum_{r=1}^{\infty} \frac{1}{r^3} < 1.22.$$
State Simpson's rule for the numerical evaluation of \(\int_0^a f(x) \, dx\), and show that it is exact when \(f(x)\) is a cubic polynomial. By applications of this rule using three ordinates to $$\int_0^{1/3} \frac{dx}{\sqrt{1-x^2}} \quad \text{and} \quad \int_0^1 \frac{dx}{\sqrt{1-x^2}}$$ find expressions approximating to \(\frac{1}{4}\pi\) and \(\frac{1}{2}\pi\). Which result would you expect to yield the closer approximation, and why?
The real pairs \((x,y)\) and \((u,v)\) are related by $$x + iy = k(u + iv)^2 \qquad (k \text{ real}).$$ Identify the curves in the \((x,y)\) plane which correspond to \(u = \text{constant}\) and \(v = \text{constant}\), and show that they intersect at right angles.