Problems

A triangle is to be circumscribed around a given circle. Prove that, if it is to have the minimum area, it must be equilateral.

Expand in a power series in \(x\), as far as the term in \(x^3\), $$e \log \log(e + x) - x e^{-x/e},$$ where \(e\) is the base of natural logarithms.

Prove that the curve given by \(x^y = y^x\) in the region \(x > 0\), \(y > 0\) of the Cartesian plane has just two branches, and sketch them. What are the coordinates of the point where they cross?

A curve is specified by its Cartesian coordinates \(x(t)\), \(y(t)\). \(s(t)\) is the arc-length along the curve, \(\psi(t)\) the angle between the tangent to the curve at the point \(t\) and the \(x\)-axis, \(R(t)\) the radius of curvature and \(X(t)\), \(Y(t)\) are the coordinates of the centre of curvature. Find equations enabling \(s\), \(\psi\), \(R\), \(X\), \(Y\) to be calculated. Defining also \(S(t)\) as the arc-length along the curve \((X(t), Y(t))\), show that \(|dS| = |dR|\), where \(dS\), \(dR\) are the infinitesimal changes in \(S\) and \(R\) corresponding to a change \(dt\) in the parameter \(t\). Interpret this result geometrically.

A set of functions \(y_n(x)\), \((n = 0, 1, 2, \ldots)\) is defined by $$y_n(x) = \cos(n \cos^{-1} x).$$ Show that

1. [(a)] \(y_{n+1} - 2xy_n + y_{n-1} = 0\);
2. [(b)] \(y_n\) is a polynomial in \(x\) of degree \(n\);
3. [(c)] \(y_n\) satisfies the differential equation $$(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + n^2y = 0;$$
4. [(d)] the roots of the equation \(y_n(x) = 0\) all lie in the range \(-1 < x < 1\).

Prove that $$\int_1^n \log x \, dx < \sum_{r=2}^n \log r < \int_1^n \log x \, dx + \log n.$$ Hence, or otherwise, evaluate $$\lim_{n \to \infty} \frac{(n!)^{1/n}}{n}.$$

Find the indefinite integrals

1. [(i)] \(\int x^{2n+1} e^{-x^2} dx\), where \(n = 0, 1, 2, \ldots\)
2. [(ii)] \(\int \frac{(x^2 + a^2)^{\frac{1}{2}}}{x^2} dx\).

A loudspeaker-horn has the form of the surface of revolution obtained by rotating the portion \(0 \leq x \leq a\) of the curve \(y = \frac{b}{3}x^{\frac{1}{2}}\) about the line \(y = 0\). Calculate the area of metal sheet used in its construction.

1. [(i)] Using the substitution \(x = e^t\), or otherwise, solve the differential equation $$x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0.$$
2. [(ii)] Solve $$(x^2 - 1) \frac{dy}{dx} + 2(x + 3)y = 8x(x + 1)^6.$$

\(\psi\) is a given function of the three variables \(x\), \(y\), \(f\). Show that, if the equation \(\psi = 0\) is used to define a function \(f(x, y)\) implicitly, then $$\frac{\partial f}{\partial x} = -\frac{\partial \psi/\partial x}{\partial \psi/\partial f}$$ with a similar expression for \(\partial f/\partial y\). Show also that, if \(\psi(x, y, f)\) actually takes the special form \(\psi = \phi(g, h)\) where \(g = y\), and \(h = f - \log x\), and \(\phi\) is any function of its two arguments, then \(f(x, y)\) as just defined satisfies the differential equation $$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 1.$$ [All derivatives required throughout the question may be assumed to exist.]