Problems

Show that it is not always possible to inscribe a circle within a convex quadrilateral with sides (taken consecutively) of given lengths, but that if it is possible then a circle may be inscribed in any convex quadrilateral with sides (taken consecutively) of these lengths.

Evaluate the following.

1. [(i)] \(\displaystyle \int_{-\pi}^{\pi} |\sin x + \cos x| dx,\)
2. [(ii)] \(\displaystyle \int_{-\pi}^{\pi} x^5 \cos x \, dx.\)

Show that if \(e(x)\) is a differentiable function with \(e'(x) = e(x)\) and \(e(0) = 1\) then, if \(a\) is any fixed real number, \[\frac{d}{dx}[e(a-x)e(x)] = 0.\] Deduce that \(e(x)e(y) = e(x+y)\) for all \(x\) and \(y\). Let \(c(x)\), \(s(x)\) be differentiable functions such that \(c'(x) = -s(x)\), \(s'(x) = c(x)\), \(s(0) = 0\) and \(c(0) = 1\). Show that \(c(x+y) = c(x)c(y) - s(x)s(y)\).

1. [(i)] Given that \(e^x = \sum_{r=0}^{\infty} \frac{x^r}{r!}\) \([0! = 1]\) prove that \(|e^{1.9} - 1\cdot 405| < 10^{-3}/2\).
2. [(ii)] The following is a table of the kinematic viscosity \(\nu\) of dry air at a pressure of one atmosphere at a given temperature \(T\), correct to three decimal places:
   \begin{tabular}{c|ccccc} \(T\) & 0 & 10 & 20 & 30 & 40 \\ \hline \(\nu\) & 0.132 & 0.141 & 0.150 & 0.160 & 0.159 \end{tabular}
   Would you be prepared to guess the value of \(\nu\) when \(T = 18\), \(T = 25\), \(T = 42\), \(T = 50\), \(T = 100\) or \(T = 500\)? What guess would you make if you did guess? How good would you expect your guesses to be? Give your reasons briefly but clearly.

Are the following statements true or false? If they are true give an example of a function \(f(x)\) defined for all real \(x\) with the stated behaviour. If they are false prove that no such function exists.

1. [(i)] Given \(g(x) > 0\) we can find \(f(x) > 0\) such that \(g(x)f(x) \to \infty\) as \(x \to \infty\).
2. [(ii)] There exists a function \(f(x)\) with an infinite number of maxima.
3. [(iii)] There exists a function \(f(x)\) with \(1 \geq f(x) \geq 0\) for all \(x\) such that \[\int_{-1}^{1} f(x) dx = \frac{1}{2} \quad \text{and} \quad \int_{-1}^{1} [f(x)]^2 dx = 1.\]
4. [(iv)] There exists a strictly increasing function \(f(x)\) [i.e. a function such that \(f(y) > f(z)\) whenever \(y > z\)] with no zeros [i.e. the equation \(f(x) = 0\) has no solution].

1. Show that \(\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}\) for \(|x| < 1\).
2. Deduce that \(\sum_{n=0}^{\infty} (n+r)(n+r-1)\ldots(n+1)x^n = \frac{r!}{(1-x)^{r+1}}\).
3. \(K\) brothers inherit \(\mathcal{L}n\). Show that the number \(N\) of ways of sharing this sum between them so that each receives a whole number of pounds (possibly zero) is equal to the coefficient of \(x^n\) in \((1+x+x^2+\ldots)^K\).
4. What is the value of \(N\) in terms of \(K\) and \(n\)? [Except in part (a) you need not justify your manipulations of power series.]

Let \(C\) be the arc of the parabola \(y = \frac{1}{2}x^2\) between \(x = 0\) and \(x = a\). Calculate the length of \(C\) and the area swept out when \(C\) is rotated about the \(x\)-axis.

(i) Show that if \(|x| < 1\) then \[(1+x)(1+x^2)(1+x^4)(1+x^8)\ldots(1+x^{2^n}) \to \frac{1}{1-x}\] as \(n \to \infty\). (ii) Show that \(\sum_{r=1}^{\infty} \frac{x^{2^r-1}}{1-x^{2^r}}\) converges to \(\frac{x}{1-x}\) if \(|x| < 1\) and to \(\frac{1}{1-x^{-1}}\) if \(|x| > 1\).

Prove that if \(|x| \leq \frac{1}{2}\) then \(x \geq \log (1+x) \geq x-x^2\). By taking logarithms, or otherwise, show that for any positive integer \(k\) \[\left(1-\frac{1}{n^2}\right)\left(1-\frac{2}{n^2}\right)\ldots\left(1-\frac{kn}{n^2}\right) \to e^{-k^2/2}\] as \(n \to \infty\).

Find the straight line which gives the best fit to \(x \cos x\) for \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\); i.e., find constants \(a\), \(b\) such that \[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x\cos x - ax - b)^2 dx\] is as small as possible.