Prove that, if \(f(x) = e^{ax} \sin bx\), then \[ f'(x) = r e^{ax} \sin (bx + \phi), \] and specify the values of \(r, \phi\) in terms of \(a\) and \(b\). Prove that \(f(x)\) has a sequence of maximum values which form a geometric progression whose common ratio is \(e^{2\pi a/b}\).
Prove that, if \(k\) is real and \(|k| < 1\), the function \(\cot x + k \operatorname{cosec} x\) takes all values as \(x\) varies through real values. Prove that, if \(|k| > 1\), the function takes all values except those included in an interval of length \(2\sqrt{(k^2 - 1)}\). Give rough sketches of the graph of \[ y = \cot x + k \operatorname{cosec} x \] for \(-\pi < x < \pi\), in the cases (i) \(0 < k < 1\), (ii) \(k > 1\).
When \(v\) is eliminated between the equations \(y = f(x, v)\) and \(z = g(x, v)\), the equation \(z = \phi(x, y)\) is obtained. Prove that \[ \frac{\partial \phi}{\partial x} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial x} \frac{\partial g}{\partial v} - \frac{\partial f}{\partial v} \frac{\partial g}{\partial x}. \] Verify this result when \begin{align*} y &= x \sin v + a \cos v, \\ z &= x \cos v - a \sin v, \end{align*} \(a\) being a constant.
The length of the equal sides of an isosceles triangle is given. Prove that, when the radius of the inscribed circle is a maximum, the angle between the equal sides has a value between \(76^\circ\) and \(76^\circ 30'\).
Using the notation \begin{align*} f(x) \ll g(x) \quad &\text{if } f(x)/g(x) \to 0 \text{ as } x \to \infty, \\ f(x) \prec g(x) \quad &\text{if } f(x)/g(x) \to \text{a limit between 0 and 1}, \\ f(x) \sim g(x) \quad &\text{if } f(x)/g(x) \to 1, \end{align*} arrange the following functions in order by means of the symbols \(\ll, \prec, \sim\). \[ 3^x, \quad x e^x, \quad \int_1^x t e^t dt, \quad \int_1^x 3^t dt. \]
By use of the series for \(\log(1+z)\), or otherwise, prove for a range of values of \(r\) to be specified that \[ r \sin \theta + \frac{1}{2}r^2 \sin 2\theta + \frac{1}{3}r^3 \sin 3\theta \dots = \arctan\left(\frac{r \sin \theta}{1-r \cos \theta}\right), \] making clear how the many-valued function \(\arctan\) is to be interpreted.
Find the equation of a curve which passes through the origin and is such that the area included between the curve, any ordinate and the \(x\)-axis is \(k\) times the cube of that ordinate. For a given value of \(k\), is there more than one such curve?
A function \(f(x)\) is defined, for \(x \ge 0\), by \[ f(x) = \int_{-1}^1 \frac{dt}{\sqrt{\{1 - 2xt + x^2\}}}, \] where the positive value of the square root is to be taken. Prove that, if \(0 \le x \le 1\), \(f(x)=2\). What is the value of \(f(x)\) if \(x > 1\)? Has the function \(f(x)\) a differential coefficient for \(x=1\)?
If \[ I_{m,n} = \int \frac{\sec^m x}{\tan^n x} dx, \] where \(m\) and \(n\) are positive integers and \(n > 1\), prove that \[ (n-1) I_{m,n} = (m-2) I_{m-2, n-2} - \frac{\sec^{m-2} x}{\tan^{n-1} x}. \] Hence evaluate \[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\sec^5 x}{\tan^4 x} dx. \]
\(A, B\) are fixed points distant \(2c\) apart. Find the polar equation of the locus of points \(P\) in a plane through \(AB\) such that \(PA.PB = c^2\). Prove that the points \(P\) of space such that \(PA.PB = c^2\) lie on a surface whose area is \(4\pi c^2 (2-\sqrt{2})\).