Problems

Obtain the cube roots of unity and establish their principal properties. Express in terms of the exponential function the sums of the infinite series

1. [(i)] \(1 \cdot \dfrac{a^6}{5!} + 2 \cdot \dfrac{a^{12}}{11!} + 3 \cdot \dfrac{a^{18}}{17!} + \dots\).
2. [(ii)] \(\sum_{r=0}^\infty \dfrac{a_r x^r}{r!}\), where \(a_r=0\) when \(r\) is of the form \(3n-1\), \(a_r=-1\) when \(r\) is of the form \(6n+1\), \(a_r=+1\) for other values of \(r\).

If \(\phi(x)\) is a polynomial of degree not greater than that of a polynomial \(f(x)\), shew that \[ \frac{\phi(x)}{(x-a)f(x)} \equiv \frac{\phi(a)}{(x-a)f'(a)} + \frac{\text{a polynomial in }x}{f(x)}, \] provided \(f(a)\neq 0\). Discuss the case \(f(a)=0\). Expand \(\dfrac{2x^8-5x^4+2x^3+6x^2-2}{(x+2)(x^2-1)^3}\) in a series of ascending powers of \(x\), stating carefully the general term.

Prove Taylor's theorem for a function \(f(x)\), in the range \(a \le x \le b\), stating the necessary restrictions on the behaviour of \(f(x)\) and its derivatives. Find the first four terms in the expansion of \(\log_e(1+\sin^2 x)\) in increasing powers of \(x\), and also the first three terms of the expansion of \(x\) in increasing powers of this function.

Explain the application of the Calculus to the discussion of inequalities, giving simple illustrations. Hence or otherwise prove that if \(0<\theta<1\), \[ \theta + \frac{\theta^3}{3} < \tan\theta < \theta + \frac{2\theta^3}{3}. \]

Trace the curve \(r\cos\theta+a\cos 2\theta = 0\). Shew that the area of the loop is \(a^2(2-\frac{\pi}{2})\), and that the area enclosed between the curve and its asymptote is \(a^2(2+\frac{\pi}{2})\).

Prove that for odd values of \(n\), \[ \int_0^\pi \frac{\cos n\theta}{\cos\theta} d\theta = (-1)^{\frac{n-1}{2}}\pi. \] If, for odd values of \(n\), \(I_n = \int_0^\pi \frac{\cos^2 n\theta}{\cos^2\theta}d\theta\), shew that \begin{align*} I_n &= I_{n-2}+2\pi \\ &= n\pi. \end{align*}

Shew that the area of the surface of the prolate spheroid obtained by the rotation of an ellipse of eccentricity \(e\) about its major axis (\(2a\)) is \[ A = 2\pi a^2\left[ \sqrt{1-e^2} + \frac{\sin^{-1} e}{e} \right] \] and that the centroid of the half surface bounded by the central circular section is at a distance \(d\) from the plane of that section, where \[ Ad = \frac{4\pi a^3}{3} \frac{1}{e^2}\left[ \sqrt{1-e^2} - (1-e^2)^{\frac{3}{2}} \right]. \]

Find the maximum and minimum values for real values of \(x,y,z\), of the quantity \(x^2+y^2+z^2\), subject to the conditions that \begin{align*} lx+my+nz &= 0, \\ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} &= 1, \end{align*} where \(a,b,c\) are positive and \(l,m,n\) are real. Verify that the values determined are real and positive.