Sketch and describe the three curves given in polar coordinates by \begin{align*} (i)&~ r = \sin\theta \quad (0 < \theta < \pi);\\ (ii)&~ r^{-1} = \sin\theta \quad (0 < \theta < \pi);\\ (iii)&~ r^{-2} = \sin 2\theta \quad (0 < \theta < \pi/2). \end{align*} [No credit will be given for solutions obtained by numerical methods alone.]
A sequence of numbers \(u_1, u_2, u_3 \ldots\) is defined by the relations \begin{align*} u_1 &= a+b\\ u_n &= a+b-\frac{ab}{u_{n-1}}, \end{align*} where \(a+b \neq 0\). Show that if \(a \neq b\) then \[u_n = \frac{a^{n+1}-b^{n+1}}{a^n-b^n},\] and when \(a > b > 0\) determine the limit to which \(u_n\) tends as \(n\) tends to infinity. Find a formula for \(u_n\) when \(a = b\), and determine the limit to which \(u_n\) tends as \(n\) tends to infinity.
Express the function \[f(x) = \frac{x^3-x}{(x^2-4)^2}\] in partial fractions with constant numerators. Find the \(n\)th derivative of \(f(x)\) at \(x = 0\).
Show that if \[e^x\sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \ldots + \frac{a_n}{n!}x^n + \ldots\] then \(a_0 = 0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).
Evaluate the integrals \[\int_0^\infty e^{-t}\cos xt \,dt \quad \text{and} \quad \int_0^\infty e^{-t}\sin xt\,dt.\] Hence or otherwise evaluate \[\int_0^\infty \int_0^\infty e^{-(s+t)}\cos x(s+t)\,ds\,dt.\]
A chemist wishes to conduct an experiment in which a process takes place for a fixed interval of time \(t_0\) in a pressurised vessel. Initially the pressure in the vessel is \(p_0\). Theoretical considerations show that there are positive parameters \(A\) and \(\alpha\) (with \(\alpha = 1\) or 2) such that the pressure \(p\) satisfies \[\frac{dp}{dt} \leq Ap^\alpha.\] The chemist asks you how strong his vessel should be. Advise him.
(i) If \(z_1\) and \(z_2\) are two complex numbers, prove algebraically that \[|z_1|-|z_2| \leq |z_1-z_2| \leq |z_1|+|z_2|.\] Interpret these inequalities on the Argand diagram. (ii) Obtain \(\sqrt{(1+i)}\) in a form \(a+ib\) (for \(a\) and \(b\) real) and show that it is a root of the equation \(z^4 = 2i\). What are the other roots of this equation?
In the differential equations \begin{align*} \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= 0, \quad (A)\\ \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= f(x), \quad (B) \end{align*} \(p\) and \(q\) are constants. Prove that
Evaluate \(\int_1^x (\log_e t)^2\,dt\), for \(x > 0\). Let \(J_n = \log_e(1+\frac{1}{n})\), where \(n\) is a positive integer. By considering an upper bound for \(\int_1^{1+1/n} (\log_e t)^2\,dt\), or otherwise, show that \[J_n^2 - 2(1+\frac{1}{n})J_n + \frac{2}{n} \leq 0.\]
Evaluate \begin{align*} (i)&\int\frac{dx}{x+\sqrt{(2x-1)}} \quad (x > \frac{1}{2});\\ (ii)&\int\frac{dx}{(a^2-x^2)^{3/2}} \quad (|x| < |a|). \end{align*}