Find the limit, as \(x\) tends to zero, of \[ \frac{x\cos x - \sin x}{x^3}. \] Sketch the curve \[ y = \frac{\sin x}{x} \] and discuss briefly its form for small values of \(x\). Show that, for large values of \(x\), there are maxima and minima near the points \(x=(n+\frac{1}{2})\pi\), where \(n\) is an integer. Obtain a closer approximation to the values of \(x\) at the maxima and minima, with an error of order \(\dfrac{1}{n^3}\).
Show that the equation \[ x^4 + 3x^2 - 3 = 0 \] has one positive root. Find to three decimal places an approximation to this root.
A man standing on the edge of a circular pond wishes to reach the diametrically opposite point by running or swimming or a combination of both. Assuming that he can run \(k\) times as fast as he can swim, find what is his quickest method, for any given \(k>1\).
From the equations \(y=f(x)\), \(x=\xi\cos\alpha - \eta\sin\alpha\) and \(y=\xi\sin\alpha+\eta\cos\alpha\) (where \(\alpha\) is constant) it is deduced that \(\eta=\phi(\xi)\). Prove that \[ \frac{\frac{d^2y}{dx^2}}{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}} = \frac{\frac{d^2\eta}{d\xi^2}}{\left[1+\left(\frac{d\eta}{d\xi}\right)^2\right]^{\frac{3}{2}}}, \] and interpret this result geometrically.
If \(y\) is defined as a function of \(x\) by the equation \(y\sqrt{1+x^2}=\log[x+\sqrt{1+x^2}]\), prove that \[ (1+x^2)y' + xy = 1 \] and express \(y\) as a series in ascending powers of \(x\). Hence show that the sum of the series \[ 1 - \frac{1}{3!} + \frac{(2!)^2}{5!} - \frac{(3!)^2}{7!} + \dots \] is \(\dfrac{4}{\sqrt{5}}\log\tfrac{1}{2}(1+\sqrt{5})\).
Evaluate the integrals
A curve is given by the parametric equations \[ x=f(t), \quad y=g(t). \] Explain the significance of the expression \[ \frac{1}{2} \int_{t_0}^{t_1} (fg' - f'g) dt. \] Sketch the curve whose parametric equations are \[ x = \frac{1-t^2}{1+t^2}, \quad y = \frac{t(1-t^2)}{1+t^2}, \] and calculate the area of the loop.
Obtain a recurrence relation between integrals of the type \[ I_n = \int x^n e^{ax} \cosh bx \, dx. \]
\(Q, R, S\) are the points \((\alpha, \beta)\), \((-l, 0)\) and \((l, 0)\), and \(P\) is a variable point \((x, 0)\) on \(RS\). Evaluate \[ \int_{-l}^l \frac{dx}{PQ}, \] and show that it can be expressed as a function of \(RS\) and \(QR+QS\).
Show that the solution of the equation \[ y'' + n^2 y = a \sin pt \] (where \(n\neq 0\) and \(p^2 \neq n^2\)), such that \(y=0\) and \(y'=0\) when \(t=0\), is \[ y = \frac{a}{n^2-p^2}\left(\sin pt - \frac{p}{n}\sin nt\right). \] Show also that, as \(p\) tends to \(n\), \(y\) tends to \[ \frac{a}{2n}\left(\frac{1}{n}\sin nt - t\cos nt\right), \] and verify that this is the solution when \(p\) is equal to \(n\).