Justify Newton's method for approximating to a root of the equation \(f(x)=0\), namely, that if \(a\) is a first approximation, \(a_1 = a - \frac{f(a)}{f'(a)}\) is in general a better approximation. Illustrate as simply as you can, graphically or otherwise, the general circumstances in which (i) \(a_1\) is nearer to the actual root than \(a\), (ii) the actual root lies between \(a\) and \(a_1\). Consider the positive root between 1 and 2 of the equation \(3\sin x = 2x\) by taking \(a = \frac{\pi}{2}\). Find the next approximation and state which of the two cases mentioned above it illustrates.
State Leibnitz's theorem for the \(n\)th differential coefficient of the product of two functions. If \[ y = (1-x^2)^{\frac{1}{2}m} \frac{d^m u}{dx^m}, \] where \(m\) is a positive integer, and \(u\) satisfies the equation \[ (1-x^2) \frac{d^2u}{dx^2} - 2x \frac{du}{dx} + n(n+1)u=0, \] prove that \(y\) satisfies the equation \[ (1-x^2) \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \left\{ (n-m)(n+m+1) - \frac{m^2}{1-x^2} \right\} y = 0. \]
Give sufficient conditions for the truth of Rolle's theorem, that if \(f(x)\) has equal values for \(x=a\) and \(x=b\), then \(f'(x)\) will be zero for some value of \(x\) between \(a\) and \(b\). By considering the function \[ f(x) = \begin{vmatrix} 1 & \phi(x) & \psi(x) \\ 1 & \phi(a) & \psi(a) \\ 1 & \phi(b) & \psi(b) \end{vmatrix}, \] prove that if \(\phi'(x)\) and \(\psi'(x)\) exist for \(a \le x \le b\), and \(\psi'(x) \ne 0\), then there is a value \(c\) such that \(a < c < b\), and \[ \frac{\phi(b)-\phi(a)}{\psi(b)-\psi(a)} = \frac{\phi'(c)}{\psi'(c)}. \] Evaluate \[ \lim_{x \to 0} \frac{e^x-1-\log(1+x)}{1-\cos x}. \]
Prove that the equation \(x^3-3px^2+4q=0\) will have three real roots if \(p\) and \(q\) are the same sign and \(p^6>q^2\). Show that two roots will be positive or negative as the sign of \(p\) and \(q\) is positive or negative. (It may be assumed that neither \(p\) nor \(q\) vanish.) Find out by these results as much as possible about the roots of the equation \[ x^3 - 6x^2 + 16 = 0. \]
If \[ I(p,q) = \int_0^{\log(1+\sqrt{2})} \sinh^p x \cosh^q x \, dx, \] where \(p>1\), prove that \[ (p+q)I(p,q) = 2^{\frac{q-1}{2}} + (q-1)I(p,q-2) = 2^{\frac{q+1}{2}} - (p-1)I(p-2,q). \]
Give a rough sketch of the curve whose coordinates are given by \[ \begin{cases} x = a\phi+b\sin\phi, \\ y = b(1-\cos\phi), \end{cases} \] where \(\phi\) is a parameter, and \(a>b>0\). Find the equation of the normal at any point and show that the coordinates of the centre of curvature are \begin{align*} x &= a\phi - b\sin\phi - (a^2-b^2)\frac{\sin\phi}{b+a\cos\phi}, \\ y &= 3a+b-\frac{b^2}{a}+b\cos\phi + \frac{(a^2-b^2)^2}{ab}\frac{1}{b+a\cos\phi}. \end{align*}
\(ABC\) is a triangular lamina, and \(D, E, F\) are points in the sides \(BC, CA, AB\) respectively such that \(BD=\frac{1}{3}BC\), \(CE=\frac{1}{3}CA\), \(AF=\frac{1}{3}AB\). Forces of magnitudes \(kAD, kBE, kCF\) act along \(AD, BE, CF\) respectively. Show that these forces are together equivalent to a couple of moment \(k\Delta\), where \(\Delta\) is the area of the triangle \(ABC\).
A straight rod \(ABC\) of weight \(3W\) rests horizontally on a nearly flat surface, making contact only at the points \(A, B\) and \(C\), where \(B\) is the mid-point of \(AC\). The normal reaction at each of the points of contact is \(W\) and the coefficient of friction is \(\mu\). A gradually increasing horizontal force at right angles to the rod is applied to the rod at \(A\). Find the greatest magnitude of this force for which equilibrium is possible, and describe how the equilibrium is broken.
A rough circular cylinder of radius \(r\) is fixed with its axis horizontal. A uniform cubical block of weight \(W\) with edges of length \(a\) is placed symmetrically upon the cylinder with four edges vertical and four edges parallel to the axis of the cylinder. The block is then rolled upon the cylinder, without slipping, until it has turned through an angle \(\theta\). Calculate (i) the work \(V\) done against gravity in this rotation, (ii) the moment \(M\) of the couple needed to hold the block in equilibrium in the position \(\theta\), indicating the sense of this moment. Hence determine the condition that the position of equilibrium \(\theta=0\) should be stable.
To a motorist driving due West along a level road with constant speed \(V\) the wind appears to be blowing in a direction \(\alpha\) East of North. When he is driving with the same speed \(V\) due East, the apparent direction of the wind is \(\beta\) East of North. Show that, when he is driving at a speed \(2V\) due East, the apparent direction of the wind is \[ \tan^{-1} \left(\frac{1}{3} \tan\beta - \tan\alpha\right) \] East of North. Find also the true speed of the wind.