Problems

Find the limit of

1. [(i)] \(\dfrac{\sqrt{1+x}-1}{1-\sqrt{1-x}}\) as \(x \to 0\),
2. [(ii)] \(\dfrac{3^n + (-2)^n}{3^n - 2^n}\) as \(n \to \infty\),
3. [(iii)] \(\dfrac{n^2}{2^n}\) as \(n \to \infty\),
4. [(iv)] \(\sqrt[n]{n}\) as \(n \to \infty\),

where \(x\) is a continuous real variable and \(n\) a positive integral variable.

Prove that, if \[ \theta = \cot^{-1} x \quad (0 < \theta < \pi), \] then \[ \frac{d^n\theta}{dx^n} = (-1)^n (n-1)! \sin^n \theta \sin n\theta, \] where \(n\) is any positive integer. Show that the absolute value of \(d^n\theta/dx^n\) never exceeds \((n-1)!\) if \(n\) is odd, or \[ (n-1)! \cos^{n+1} \frac{\pi}{2(n+1)} \] if \(n\) is even.

Prove by differentiation, or otherwise, that \[ xy \le e^{x-1} + y \log y \] for all real \(x\) and all positive \(y\). When does the sign of equality hold?

A point \(M\) is taken on the curve \(y = \sin x\) (where \(x\) is measured in radians) such that the area bounded by the portions of the curve and of the line \(y=0\) between \(x=0\) and \(x=\pi\) is divided into two equal parts by the chord \(OM\) joining \(M\) to the origin \(O\). Show that the abscissa \(x_1\) of \(M\) satisfies the equation \(\cot x_1 = \frac{1}{2} - \frac{1}{2}x_1\). Show from the tables that this equation has a root between \(x_1 = 2.44\) (\(139^\circ 46'\) approx.) and \(x_1 = 2.48\) (\(142^\circ 6'\) approx.). By writing \(x_1 = 2.44 + \xi\) and neglecting higher powers of \(\xi\) than the first, find a further approximation to \(x_1\).

Find \[ \int_0^\infty \frac{x\,dx}{x^5 + x^2 + x + 1}, \quad \int \frac{dx}{(x^3 - 1)^{\frac{1}{3}}}, \quad \int x^3 \sin x^2 \,dx. \]

Obtain a relation between \(I_{n-1}\) and \(I_{n+1}\) (\(n>0\)), where \[ I_n = \int_0^x \frac{t^n}{1+t^2} dt. \] Prove that, for any fixed \(x\) in the range \(-1 < x \le 1\), \(I_n \to 0\) as \(n \to \infty\). Deduce an expansion of (i) \(\tan^{-1} x\), (ii) \(\log (1+x^2)\), in ascending powers of \(x\), valid for \(-1 \le x \le 1\).

Sketch the curve \[ a(x^2 - y^2) = y^3 \quad (a > 0). \] Find (i) the position of the centre of curvature of either branch of the curve at the origin, and (ii) the area of the loop.

The area \(\Delta\) of a triangle \(ABC\) is calculated from measurements of the sides \(a, b, c\). If each measurement is liable to a small error of \(p\) per cent., find the percentage error to which the calculated value of \(\Delta\) is liable (i) when \(A, B, C\) are all acute, (ii) when \(A\) is obtuse. Express your answers in terms of \(A, B, C\) and \(p\), to the first order in \(p\).

Prove that through any point \((x,y)\) in the upper half-plane \(y > 0\) there pass two members of the family of confocal parabolas \[ y^2 = 4\alpha (x+\alpha), \] corresponding to values \(\alpha = \lambda\) and \(\alpha = \mu\) of the parameter \(\alpha\), where \(\lambda > 0 > \mu\). Calculate

1. [(i)] \(\begin{vmatrix} \frac{\partial\lambda}{\partial x} & \frac{\partial\lambda}{\partial y} \\ \frac{\partial\mu}{\partial x} & \frac{\partial\mu}{\partial y} \end{vmatrix}\) when \(x\) and \(y\) are taken as independent variables;
2. [(ii)] \(\begin{vmatrix} \frac{\partial x}{\partial \lambda} & \frac{\partial x}{\partial \mu} \\ \frac{\partial y}{\partial \lambda} & \frac{\partial y}{\partial \mu} \end{vmatrix}\) when \(\lambda\) and \(\mu\) are taken as independent variables.

Verify that these two determinants are reciprocals of one another. Show also that \[ \frac{\partial x}{\partial \lambda} \frac{\partial x}{\partial \mu} + \frac{\partial y}{\partial \lambda} \frac{\partial y}{\partial \mu} = 0. \]

(i) Solve the equation \[ \frac{dy}{dx} \cos^2 x + y = \tan x, \] with the condition that \(y=0\) when \(x=0\). (ii) Solve the equation \[ \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = e^{-x} \sin^2 x. \]