A point moves in a plane in such a way that its least distances from two fixed non-intersecting circles in that plane are equal. Describe the locus of the point in each of the various cases which may arise and justify your answers.
Show that, if \(f(x)\) is an increasing positive function for \(0 \leq x \leq 1\), then \[\frac{1}{n} \sum_{r=0}^{n-1} f\left(\frac{r}{n}\right) \leq \int_0^1 f(x) dx \leq \frac{1}{n} \sum_{r=1}^n f\left(\frac{r}{n}\right).\] Deduce that, for \(k \geq 0\), \[\left|n^{-k-1} \sum_{r=0}^n r^k - \frac{1}{k+1}\right| \leq \frac{1}{n}.\] Use similar arguments to show that \[\left|\sum_{r=n}^{2n} \frac{1}{r} - \log 2\right|\] can be made as small as we like by taking \(n\) large enough.
British Rail have found that their income from a route is given by \(I(v) = hv\), where \(v\) is the average speed (in appropriate units) of trains over the route, and \(h\) is constant. The capital cost \(C\) of improvements to track and signalling to obtain a speed \(v\) is estimated as follows:
The expenditures \(x(t)\) and \(y(t)\) on armaments at time \(t\) of two countries are governed by the equations \[\frac{dx}{dt} = -ax+by+k_1,\] \[\frac{dy}{dt} = -ay+bx+k_2,\] where \(a > 0\), \(b > 0\), \(k_1 > 0\), \(k_2 > 0\); also \(x(0) = y(0) = 0\). Show that the total expenditure \(x(t) + y(t)\) on armaments will increase without bound as \(t \to \infty\) if \(b \geq a\), but will tend to a limit if \(b < a\). Find \(x(t)\) and \(y(t)\) when \(b \neq a\).
Let \(f(x) = 2\cos x^2 - \frac{1}{x^2}\sin x^2\) for \(x \geq 1\). Find \(\int_1^t f(x)dx\) for \(t \geq 1\), and show that it tends to a limit as \(t \to \infty\) but that \(f(x)\) does not tend to zero as \(x \to \infty\). Give an example of a function \(h(x) \geq 0\), defined for \(x \geq 1\) and such that \(\int_1^t h(x)dx \to \infty\) as \(t \to \infty\) but \(h(x) \to 0\) as \(x \to \infty\). Give an example of a function \(l(x) \geq 0\), defined for \(x \geq 1\), and such that \(dl/dx > 0\) for all \(x \geq 1\) but \(l(x) \leq 1\) for all \(x \geq 1\). [It may be helpful to sketch such an \(l(x)\); but an explicit formula should be given.]
Show by using the binomial expansion or otherwise that \((1 + x)^n \geq nx\) whenever \(x \geq 0\) and \(n\) is a positive integer. Deduce that if \(y > 1\) then, given any number \(K\), we can find an \(N\) such that \(y^n \geq K\) for all integers \(n \geq N\). Show similarly that if \(y > 1\) then, given any \(K\), we can find an \(N\) such that \(\frac{y^n}{n} \geq K\) for all integers \(n \geq N\).