(a) Find the limit, as \(x\) tends to zero, of (i) \((b^x - a^x)/x\) where \(a\) and \(b\) are positive; (ii) \(x \sin x / \log \cos x\). (b) If \(n\) is a positive integer, show that \[ \left(1+\frac{1}{n}\right)^n \le 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!} < 3. \]
Prove that the equation \(x^5+5x+3=0\) has only one real root. Calculate this root correct to 3 decimal places.
Find the indefinite integrals
The co-ordinates \((x, y)\) of a point on a simple closed plane curve are expressed in terms of a parameter \(t\). Show that the area enclosed by the curve is given by \[ \frac{1}{2} \int \left(x \frac{dy}{dt} - y \frac{dx}{dt}\right) dt \] taken between suitable limits. Sketch the curve \(x=a(\cos^3 t + \sin^3 t)\); \(y=a(\sin^3 t - \cos^3 t)\), and find its area.
A family of curves is given by the equation \(y = \cos x + \lambda \cos 3x\), where \(\lambda\) is a positive parameter. Examine the maximum and minimum points of these curves, showing that they fall into two classes, (a) those whose \(x\)-coordinate depends on \(\lambda\), and (b) those whose \(x\)-coordinate is independent of \(\lambda\). Show that the points in class (a) lie on the curve \[ 3(3-4\sin^2 x)(\cos x - y) = \cos 3x \] for all relevant values of \(\lambda\).
If \(F(m, n) = \int_1^\infty (x-1)^m x^{-n} dx\), where \(m\) and \(n\) are positive integers satisfying \(n > m+2\), find relations between \(F(m, n-1)\) and \(F(m,n)\), and between \(F(m-1, n)\) and \(F(m, n)\). Hence find the value of \(F(m, n)\).
Sketch the curve whose equation in polar coordinates is \(r=1+\cos 2\theta\). Prove that the length of the curve corresponding to \(0 \le \theta \le 2\pi\) is \[ 8 + \frac{4}{\sqrt{3}} \log(2+\sqrt{3}). \]
A point \(P\) is situated at a distance \(f\) from the centre of a thin spherical shell of radius \(a\), and \(R\) is the distance of \(P\) from any point of the shell. Show that the mean value of \(R\), averaged with respect to elements of area of the shell, is \(f+(a^2/3f)\) or \(a+(f^2/3a)\) according as \(f\) is greater or less than \(a\).
If \(y=\psi_n(x)\) is a solution of the equation \[ \frac{d^2y}{dx^2} + \frac{2(n+1)}{x} \frac{dy}{dx} + y = 0, \] show that \(Y=\frac{1}{x}\frac{d}{dx}\{\psi_n(x)\}\) satisfies the equation \[ \frac{d^2Y}{dx^2} + \frac{2(n+2)}{x} \frac{dY}{dx} + Y=0. \] Hence show that, if \(n\) is a positive integer, \(\psi_n(x) = \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}\) satisfies the former differential equation.
If \(y\) is defined as a function of \(x\) by the equation \(f(x,y)=0\), and subscripts denote partial differentiation, show that \[ \frac{d^2y}{dx^2} = -\frac{f_{xx}f_y^2 - 2f_{xy}f_x f_y + f_{yy} f_x^2}{f_y^3}. \] Illustrate the use of this formula by finding the radius of curvature of the ellipse \(x^2+xy+y^2=3\) at the point \((1,1)\).