Problems

Define the partial derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\) of a function \(f(x,y)\) of the two variables \(x\) and \(y\). If \(f(kx, ky) = k^n f(x,y)\) for all values of \(k, x\) and \(y\), where \(n\) is a constant independent of \(k\), obtain the relation \[ x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = n f(x,y). \] Verify this result explicitly when \(f(x,y) = (x^3+y^3)^{\frac{1}{3}}\tan^{-1}\frac{y}{x}\) by finding \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).

(i) Express \(\displaystyle \frac{3x^2+12x+8}{(x+1)^5}\) in partial fractions. (ii) Evaluate \(\displaystyle \int_0^2 \frac{x(x+1)}{(x-4)(x^2+4)} \, dx\).

Find the conditions that the roots of the equation \[ x^3+3px^2+3qx+r=0 \] should be (i) in arithmetic progression, (ii) in geometric progression, (iii) in harmonic progression.

The sequence \(u_0, u_1, \dots, u_n, \dots\) is defined by \(u_0=2, u_1=1\), and the recurrence relation \(u_{n+2}=u_{n+1}+u_n\). Show that \(u_n=A\alpha^n+B\beta^n\), where \(A, B, \alpha, \beta\) are independent of \(n\), and find \(A, B, \alpha, \beta\). Prove that

1. [(i)] \(\displaystyle\lim_{n\to\infty} \frac{u_{n+1}}{u_n} = \frac{1+\sqrt{5}}{2}\);
2. [(ii)] \(\displaystyle\left(\frac{1+\sqrt{5}}{2}\right)^{100}\) differs from an integer by less than \(10^{-20}\).

By considering the inequalities \[ \frac{1}{r(r+1)} < \frac{1}{r^2} < \frac{1}{r^2-1}, \] prove that \[ \frac{m}{ (m+1)(2m+1)} < \sum_{r=m+1}^{2m} \frac{1}{r^2} < \frac{m}{(m+1)(2m+1)} + \frac{3m+1}{4m(m+1)(2m+1)}. \] Hence find the value of \(\sum_{r=101}^{200} \frac{1}{r^2}\) with an error of less than \(2.10^{-5}\).

Sum the series \[ \sum_{r=0}^{n-1} \sin^2(\alpha+r\beta). \] Deduce that, if \(0 < \beta < \frac{\pi}{2n}\), \[ n > \frac{\sin n\beta}{\sin\beta} > n \frac{\cos 2(n-1)\beta}{\cos(n-1)\beta}, \] and hence prove that \[ \lim_{\beta\to 0} \frac{\sin n\beta}{\sin \beta} = n. \]

Prove that, if \(n\) is a positive integer, \[ (\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta. \] Deduce that \(\sin(2n-1)\theta\) can be expressed as a polynomial \(P(\sin\theta)\) of degree \(2n-1\) in \(\sin\theta\). Prove that, if \(\cos(2n-1)\alpha \ne 0\), the roots \(\beta_1, \dots, \beta_{2n-1}\) of \[ P(x) - \sin(2n-1)\alpha = 0 \] are \[ \beta_r = \sin\left(\alpha + \frac{2r\pi}{2n-1}\right), \quad \text{where } r=1, \dots, 2n-1. \] Deduce that, if \(n>1\), both \[ \sum_{r=1}^{2n-1} \sin\left(\alpha + \frac{2r\pi}{2n-1}\right) \quad \text{and} \quad \sum_{r=1}^{2n-1} \sin^2\left(\alpha + \frac{2r\pi}{2n-1}\right) \] are independent of \(\alpha\), and find the value of the first of them.

State, without proof, the binomial theorem for arbitrary real index. Express \(f(x) = \frac{(4-x)^2}{4(2+x)^2(1-x)}\) in partial fractions and show that, for \(0 < x < 1\), \(f(x) > \frac{1}{x} \log_e(1+x)\). (The expansion \(\log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots\), for \(-1

Evaluate:

1. [(i)] \(\displaystyle\int_0^1 \frac{x(1-x^2)}{(1+x^2)^2} \, dx\);
2. [(ii)] \(\displaystyle\int_0^\infty e^{-x}\sin^2 x \, dx\);
3. [(iii)] \(\displaystyle\int_0^{\pi/2} \frac{dx}{4\cos x+3\sin x}\).

Show that the radius of curvature of a plane curve \(C\) at the point \(P\) is \(r \frac{dr}{dp}\), where \(r\) is the distance from \(P\) to a fixed point \(O\), and \(p\) is the perpendicular distance from \(O\) to the tangent to \(C\) at \(P\). Find the polar equation of the evolute (locus of centres of curvature) of the curve \(r=ae^{k\theta}\).