Problems

Prove Taylor's Theorem, obtaining a form for the remainder after \(n\) terms. Apply the theorem to obtain an infinite series for \(\log(1+x)\) valid when \(-1

Explain the term 'point of inflexion' of a plane curve, and prove that if \(y=f(x)\) has a point of inflexion whose abscissa is \(x_0\), then \(f''(x_0)=0\). The graph of a polynomial of the fourth degree in \(x\) touches the \(x\)-axis at \((a,0)\) and has a point of inflexion at \((-a,0)\). Prove that the graph passes through \((-2a,0)\) and that it has a second point of inflexion whose abscissa is \(a/2\).

Prove that for a plane curve \(\displaystyle p=r\frac{dr}{dp}\). Prove that the radius of curvature of \(r^n=a^n\cos n\theta\) is \(a^n/(n+1)r^{n-1}\) and find the \((p,r)\) equation of the evolute of the curve.

Integrate: \[ \int\frac{dx}{(a^2+x^2)^{3/2}}, \quad \int\frac{dx}{x\sqrt{1+x+x^2}}, \quad \int\frac{dx}{a+b\cos x}. \] Find \(\displaystyle\int_0^\infty e^{-ax}\sin^nx\,dx\) by the use of a formula of reduction.

Show that the area of the surface of the spheroid formed by revolving the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) about the axis of \(y\) is \(2\pi a^2\left[1+\frac{1-e^2}{e}\tanh^{-1}e\right]\), where \(e\) is the eccentricity of the ellipse.

Prove that

1. [(i)] \(\cos36^\circ-\cos72^\circ=\frac{1}{2}\).
2. [(ii)] \(\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{8}=\frac{\pi}{4}\).

Express \[ 1-\cos^2x-\cos^2y-\cos^2z+2\cos x\cos y\cos z \] as the product of four sines.

If \(a,b,c,d\) are the sides (taken in order) of a quadrilateral inscribed in a circle, prove that the area of the quadrilateral is equal to \(\sqrt{(s-a)(s-b)(s-c)(s-d)}\), where \(2s=a+b+c+d\). Prove also that the diagonals are in the ratio of \(ab+cd\) to \(ad+bc\).

Sum the following series:

1. [(i)] \(\sin a+\sin3a+\sin5a+\dots\) to \(n\) terms.
2. [(ii)] \(\sin\theta+2\sin2\theta+3\sin3\theta+\dots\) to \(n\) terms.
3. [(iii)] \(x\sin a - \frac{1}{3}x^3\sin3a+\frac{1}{5}x^5\sin5a-\dots\) to infinity. (\(x<1\))

Prove that \(\displaystyle\frac{\sin n\theta}{\sin\theta}\) is divisible by \(\cos\theta-\cos\alpha\), where \(n\) is an integer, and shew that \[ \frac{\sin n\theta}{\sin\theta} = 2^{n-1}\left(\cos\theta-\cos\frac{\pi}{n}\right)\left(\cos\theta-\cos\frac{2\pi}{n}\right)\dots\left(\cos\theta-\cos\frac{n-1}{n}\pi\right). \] Prove that \[ 32\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{3\pi}{11}\cos\frac{4\pi}{11}\cos\frac{5\pi}{11}=1. \]

The diagonals of a parallelogram are the straight lines whose equation referred to rectangular coordinates is \[ ax^2+2hxy+by^2=0, \] and \((\alpha,\beta)\) are the coordinates of the middle point of one side of the parallelogram. Find the equation of that side and shew that one of the angles of the parallelogram is \[ \tan^{-1}\frac{a\alpha^2+2h\alpha\beta+b\beta^2}{h(\alpha^2-\beta^2)+\alpha\beta(b-a)}. \]