A wedge of given total surface area \(S\) has the form of a right cylindrical figure whose base is the sector of a circle with given sectorial angle \(\alpha\). Show that the volume of the wedge does not exceed \(\dfrac{\sqrt{\alpha}\,S\,\sqrt{S}}{3\sqrt{3}(2+\alpha)}\). If only the total surface area \(S\) is given, show that the volume cannot exceed \(\dfrac{S\sqrt{S}}{6\sqrt{6}}\).
The equation \(f(x)=0\), where \(f(x)\) is a polynomial, has a root \(\xi\) such that \(f'(\xi) \neq 0\). Show that, if \(\xi_1\) is a sufficiently good approximation to \(\xi\), then \[ \xi_2 = \xi_1 - \frac{f(\xi_1)}{f'(\xi_1)} \] is a better approximation to \(\xi\). Use the above formula to evaluate \(\sqrt[3]{3}\) to three decimal places.
Prove that the three (distinct) complex numbers \(z_1, z_2, z_3\) represent the vertices of an equilateral triangle in the Argand diagram if and only if \[ z_1^2+z_2^2+z_3^2-z_2z_3-z_3z_1-z_1z_2 = 0. \]
If \(\epsilon\) is small in magnitude compared with unity, show that the perimeter of the curve \[ r = 1 + \epsilon \cos\theta \] is approximately \(\frac{1}{2}\pi(4+\epsilon^2)\).
Obtain the equation and perimeter of the evolute (locus of centres of curvature) of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
Sketch the curve \[ (x+y)(x^2+y^2) = 2xy, \] and obtain the area of its loop.
Solution: Let \(X = x+y\) and \(Y = x-y\), notice that \(4xy = X^2-Y^2\) so our equation becomes \begin{align*} && \frac12 (X^2-Y^2) &= X(X^2-\frac12(X^2-Y^2)) \\ &&&= \frac12X(X^2+Y^2) \\ \Rightarrow && Y^2(X+1) &= X^2 - X^3 \\ \Rightarrow && Y^2 &= \frac{X^2-X^3}{X+1} \end{align*}
Obtain a recurrence relation between integrals of the type \[ \int x \sec^n x \,dx. \] Evaluate \[ \int_0^{\pi/4} x \sec^4 x \,dx. \]
Prove that \[ \int_1^x \frac{dt}{t+\alpha} \le \log x \le \int_1^x \frac{dt}{t-\alpha}, \] where \(x>0\) and \(\alpha>0\). Hence show that \[ \log x = \lim_{n\to\infty} n(\sqrt[n]{x}-1). \] Use the above expression for \(\log x\) to prove that \[ \log(x^m) = m \log x \] for positive integral values of \(m\).
It is given that \(u=f(x,y)\) satisfies the relation \[ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = nu, \] where \(n\) is a constant. Prove that \[ k\frac{\partial}{\partial k}f(kx, ky) = n f(kx, ky), \] and deduce that \(f(x,y)\) is a homogeneous function of degree \(n\) (i.e. \(f(kx, ky)=k^n f(x,y)\) for all positive \(k\)).
Solve the differential equation \[ \frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0 \] with the conditions \(y=2\) and \(\dfrac{dy}{dx}=-5\) at \(x=0\). Hence, or otherwise, find \(u_n\), given that \[ u_{n+2}+6u_{n+1}+9u_n = 0 \] for \(n\ge 0\), and \(u_0=2, u_1=-5\).