Explain graphically why, if \(x_1\) and \(x_2\) are each approximations to the same root of the equation \(f(x) = 0\), the expression $$\frac{x_1f(x_2) - x_2f(x_1)}{f(x_2) - f(x_1)}$$ may be expected to be a better approximation to the root. Show that $$f(x) \equiv x^3 + 3x^2 - 5x - 1 = 0$$ has a root between 1 and 1.5, and hence find an approximation to the root by the above formula, taking \(x_1 = 1\) and \(x_2 = 1.5\). Find a better approximation, and comment on its accuracy.
The function \(f(z)\) possesses a derivative \(f'(z)\) for all real values of \(z\), and is such that $$f(x + y) = f(x)f(y)$$ for all real values of the independent variables \(x\) and \(y\). By differentiating the relation with respect to \(x\) and \(y\) in turn, show that $$\frac{f'(x)}{f(x)} = \frac{f'(y)}{f(y)},$$ and hence determine the form of \(f\). Determine similarly the form of the function \(g\) that satisfies $$g(x+y) = \frac{g(x) + g(y)}{1 + g(x)g(y)}.$$
The components \(f_i(t)\) (\(i = 1, 2, \ldots, n\)) of the \(n\)-dimensional vector \(\mathbf{F}\) are functions of time \(t\), and not all of them are constant. Show that the vectors \(\mathbf{F}\) and \(\dot{\mathbf{F}}\) (where \(\dot{\mathbf{F}}\) is the vector with components \(df_i/dt\)) are orthogonal for all \(t\) if and only if \(\mathbf{F}\) has constant length. Is it possible for \(\mathbf{F}\) and \(\dot{\mathbf{F}}\) to be orthogonal for all \(t\) if \(\mathbf{F}\) has constant length? Another vector \(\mathbf{G}\), with components \(g_i(t)\) (\(i = 1, 2, \ldots, n\)), is parallel to \(\mathbf{F}\) for all \(t\). \(\mathbf{G}\) has constant length, and \(\mathbf{F}\) has length proportional to \(e^{kt}\), where \(k\) is a constant. Show that \(\dot{\mathbf{F}}\) and \(k\mathbf{G}+\dot{\mathbf{G}}\) are parallel for all \(t\). [If the vectors \(\mathbf{A}, \mathbf{B}\) have components \(a_i, b_i\) (\(i = 1, 2, \ldots, n\)) respectively, \(\mathbf{A}\) and \(\mathbf{B}\) are said to be orthogonal if \(\sum_{i=1}^n a_i b_i = 0\), and are said to be parallel if there is a scalar \(\lambda\) such that \(a_i = \lambda b_i\) for \(i = 1, 2, \ldots, n\). The length of \(\mathbf{A}\) is \(\sqrt{a_1^2 + a_2^2 + \ldots + a_n^2}\).]
A mapping of the \((X, Y)\) plane onto the \((x, y)\) plane is given by $$x = \sin X \cosh Y,$$ $$y = \cos X \sinh Y.$$ Find and sketch the curves in the \((x, y)\) plane which correspond under this mapping to the lines \(X = \text{const.}\) and \(Y = \text{const.}\) To which curves in the \((X, Y)\) plane do the lines \(x = 0, y = 0\) and \(x = y\) correspond?
A running track is in the form of a convex circuit. The width of the track is \(d\). By how much does the length of the outer edge of the track exceed that of the inner edge? (You should explain carefully how you arrive at your answer.)