Problems

Evaluate the following limits: \[ \frac{\sqrt[3]{x} - \sqrt[3]{a}}{\sqrt[4]{x} - \sqrt[4]{a}} \quad \text{as } x \to a \quad (a>0), \] \[ (\pi - 2x)\tan x \quad \text{as } x \to \tfrac{1}{2}\pi \quad (x < \tfrac{1}{2}\pi), \] \[ \frac{n}{n^2} + \frac{n+1}{n^2} + \frac{n+2}{n^2} + \dots + \frac{2n}{n^2} \quad \text{as } n \to \infty. \]

A slide rule consists of a fixed scale and a sliding scale, each 10 in. long. On each scale the numbers from 1 to 10 are marked in such a way that the distance between the marks 1 and \(x\) is proportional to \(\log x\). In order to multiply together two numbers \(x, y\) between 1 and 10 whose product is less than 10, the mark 1 on the slide is brought into coincidence with the mark \(x\) on the fixed scale. The mark \(z\) on the fixed scale which then coincides with the mark \(y\) on the slide gives the product \(xy\). If marks \(\frac{1}{100}\) in. apart are liable to be judged coincident, find to two significant figures the percentage error to which the reading \(z\) is liable. If an increase of temperature causes the fixed scale to increase in length by one part in 2000, and the slide, owing to a difference in construction, to increase by one part in 1000, what is the percentage error in \(z\), assuming that coincidences of marks are judged accurately?

Justify the statement that, if \(n\) is a positive integer or positive fraction, \[ (\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta. \] Prove that, if \(y = 2\cos\theta\), \[ 2\cos 7\theta = y^7 - 7y^5 + 14y^3 - 7y. \] Hence find the cubic with the roots \(4\cos^2\pi/14\), \(4\cos^2 3\pi/14\) and \(4\cos^2 5\pi/14\).

1. [(i)] Prove that, for positive values of \(x\), \[ \log(1+x) < \frac{x(2+x)}{2(1+x)}. \]
2. [(ii)] Find whether \(e^{-x^2}\sec^2 x\) has a maximum or a minimum value for \(x=0\).

1. [(i)] If \(x=f(y)\) determines \(y\) as a function of \(x\), calculate \(\frac{dy}{dx}, \frac{d^2y}{dx^2}\) and \(\frac{d^3y}{dx^3}\) in terms of \(f'(y), f''(y)\) and \(f'''(y)\). Verify your results by taking \(x=y^3\).
2. [(ii)] If \(x\) and \(y\) are defined as functions of \(t\) by the equations \(f(x, y, t) = 0\) and \(\phi(x, y, t) = 0\), calculate \(\frac{dy}{dt}\) in terms of the partial derivatives of \(f(x, y, t)\) and \(\phi(x, y, t)\).

Obtain the coordinates of the centre of curvature at any point of the curve \(x=f(t), y=g(t)\). Sketch the curve \(x=at\cos t, y=at\sin t\), and prove that the centre of curvature at any point lies inside the circle \(x^2+y^2=a^2\). Mark on your sketch the approximate position of the centre of curvature of a point given by a large value of \(t\).

Evaluate \(\int_1^\infty \frac{dx}{(1+x)\sqrt[3]{x}}, \quad \int_0^{2\pi} |1+2\cos x| \, dx, \quad \int_2^5 \frac{x\,dx}{\sqrt{\{(5-x)(x-2)\}}}\).

Show that \(\int_0^{\log 2} \cosh^5 x \, dx = 1.079\) approximately.

Solve

1. [(i)] \(\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = x+2\).
2. [(ii)] \(y \sin x - \cos^3 x + \cos x \frac{dy}{dx} = 0\).

State, without proof, the conditions that the expression \(A\lambda^2 + 2H\lambda + B\) should be positive for all real values of \(\lambda\). If \(f(t)\) and \(g(t)\) are real continuous functions, show, by expressing \(\int_a^b [\lambda f(t)+g(t)]^2 dt\) in the form \(A\lambda^2 + 2H\lambda + B\), that \[ \left[ \int_a^b f(t)g(t)\,dt \right]^2 \le \int_a^b [f(t)]^2\,dt \cdot \int_a^b [g(t)]^2\,dt. \] State under what conditions the equality holds. Prove that, if \(x > 0\), then \[ e^x - 1 < \int_0^x \sqrt{(e^{2t} + e^{-t})}\,dt < \sqrt{\{\tfrac{1}{2}(e^x-1)(e^{2x}-\tfrac{1}{2})\}}. \]