Problems

What is the equation of the chord of the parabola \(y^2 = 4a(x - k)\) joining the points \((at^2+k, 2at)\) and \((as^2+k, 2as)\)? What is the equation of the tangent at $(at^2+k, 2at)$? Show that the chords joining \((at^2 + k, 2at)\) to \((as^2 + k, 2as)\), where \(s = t + \lambda\) (\(\lambda\) fixed, \(t\) varying), all touch another parabola, and find its equation.

Show that \(f(t) = t - \sin t\) is an increasing function of \(t\), and deduce that the curve (a cycloid) given by the parametric equations \[x = a (t - \sin t), \quad y = a (1 - \cos t)\] has one value of \(y\) for each value of \(x\). Sketch the curve. The segment of the curve between \(x = 0\) and \(x = 2\pi a\) is now rotated about the \(x\)-axis. Find the surface area swept out.

Show that \[\int_{n-1}^{n} \log x dx \leq \log n \leq \int_{n}^{n+1} \log x dx \text{ for all integers } n \geq 2.\] Deduce that \[\int_{1}^{n} \log x dx \leq \log n! \leq \int_{2}^{n+1} \log x dx.\] Hence, or otherwise, show that \(e \leq n! (e/n)^n \leq \frac{1}{4}en(1 + 1/n)^{n+1}\).

The great grey green greasy Limpopo river is 1 kilometre wide and flows with negligible speed between parallel banks. A young elephant wishes to reach a fever tree \(h > 0\) kilometres upstream on the other side as quickly as possible. He gallops a distance \(x \geq 0\) kilometres (upstream) and then plunges in and swims directly to the tree. If he gallops at a speed of \(v\) kilometres an hour and swims at a speed of \(u\) kilometres an hour \((v \geq u > 0)\) what value of \(x\) should he choose? Explain briefly why the character of the solution is different for large and small \(h\).

Show that \(x \geq \sin x\) for \(x \geq 0\). Show further that for each \(\pi/2 \geq \delta > 0\) we can find a \(\lambda\) (depending on \(\delta\)) with \(1 > \lambda > 0\) such that \(\lambda x \geq \sin x\) for all \(x\) with \(\pi/2 \geq x \geq \delta\). Deduce that, if \(\pi/2 \geq x_0 \geq 0\) and \(x_{n+1} = \sin x_n\) (\(n \geq 0\)), then \(x_n \to 0\) as \(n \to \infty\).

Let \(n\) be a positive integer. What is the largest number \(M\) of maxima that the polynomial \[f(x) = x^n+a_1x^{n-1}+ \ldots +a_{n-1}x+a_n\] can have? For each \(n\), give an example of a polynomial which has \(M\) maxima, and justify your answer.

A curve is given parametrically in plane polar coordinates by \((r, \theta) = (e^t, 2\pi t)\) \((0 \leq t < \infty)\). Sketch the section of the curve for \(n \leq t \leq n + 1\), where \(n\) is an integer. Calculate the length of this section, and the area enclosed by it and the line \(\theta = 0\), \(r^n \leq r \leq r^{n+1}\).

A second order linear differential equation for \(y\) is given by \[\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0.\] Make the substitution \(y = uv\), where \(u\) and \(v\) are functions of \(x\), and obtain a differential equation for \(u\) in terms of \(P\), \(Q\) and \(v\). What first order differential equation must \(v\) satisfy in order to eliminate the term in \(du/dx\)? Hence, or otherwise, solve \[\frac{d^2y}{dx^2} - \frac{1}{x}\frac{dy}{dx} + \left(1+\frac{3}{4x^2}\right) y = 0\] for \(x > 0\) when \(y\) satisfies the conditions \[y = (\frac{1}{2}\pi)^{\frac{1}{2}} \text{ at } x = \frac{1}{2}\pi,\] \[y=0 \text{ at } x = \pi.\]

1. [(i)] Let \(\displaystyle I_n = \int_0^{\infty} x^n e^{-x^2}dx\). Obtain formulae for \(I_n\) in terms of \(I_0\) or \(I_1\), according to whether \(n\) is even or odd, and evaluate \(I_1\).
2. [(ii)] Evaluate \[\int_1^{\infty} \frac{dx}{x[1+(\log x)^2]}.\]

Let \(\displaystyle L(x) = \int_1^x \frac{ds}{s}\) for \(x > 0\).

1. [(i)] Prove that \(L(xy) = L(x) + L(y)\).
2. [(ii)]Show that, for \(t \neq 1\), \[\frac{1}{1-t} = 1 + t + t^2 + \ldots + t^{n-1} + \frac{t^n}{1-t},\] and deduce that \[L(1-x) = -x - \frac{x^2}{2} - \ldots - \frac{x^n}{n} - \int_0^x \frac{t^n}{1-t}dt.\]
3. [(iii)] For \(|x| < 1\), show that \[\left|\int_0^x \frac{t^n}{1-t}dt\right| \leq \frac{1}{n+1} \frac{|x|^{n+1}}{1-|x|},\] and deduce that \[L(1-x) = -x - \frac{x^2}{2} - \ldots - \frac{x^n}{n} - \ldots\]