(i) Prove that \[ \frac{d^3x}{dy^3} = -\frac{d^3y}{dx^3} \left(\frac{dx}{dy}\right)^4 + 3\left(\frac{d^2y}{dx^2}\right)^2 \left(\frac{dx}{dy}\right)^5. \] % Note: The OCR for this question was garbled. The above is a standard identity that fits the visible structure. (ii) If \(ax^2+2hxy+by^2+2gx+2fy+c=0\), show that \[ \frac{d^2y}{dx^2} = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} (hx+by+f)^{-3}. \]
Find an expression for \(\frac{d^n}{dx^n} \tan^{-1}x\). \newline Prove that when \(x=0\) its value is zero if \(n\) is even and \(\pm(n-1)!\) if \(n\) is of the form \(4p\pm 1\), where \(p\) is an integer. \newline Hence obtain the series expansion of \(\tan^{-1}x\).
Show that, if \(p>q>0\) and if \(x>0\), then \[ \frac{x^p-1}{p} > \frac{x^q-1}{q}. \] Prove further that, for \(n\) a positive integer and \(s>0\), \[ \frac{1}{p(p+s)^n} \left|x^p - \frac{1}{s^n}\right| > \frac{1}{q(q+s)^n} \left|x^q - \frac{1}{s^n}\right|. \]
If \(\frac{\sin \theta}{x} = \frac{\sinh \phi}{y} = \cos \theta + \cosh \phi\), prove that \[ \frac{\partial x}{\partial \theta} = \frac{\partial y}{\partial \phi}, \quad \frac{\partial x}{\partial \phi} = -\frac{\partial y}{\partial \theta}. \] The function \(U(x,y)\) transforms by means of the above relations into \(V(\theta, \phi)\). Prove that \[ \left(\frac{\partial U}{\partial x}\right)^2 + \left(\frac{\partial U}{\partial y}\right)^2 = \left\{\left(\frac{\partial V}{\partial \theta}\right)^2 + \left(\frac{\partial V}{\partial \phi}\right)^2\right\} / (\cos \theta + \cosh \phi)^2. \]
(i) Find the limit of \((\cos x)^{\cot^2 x}\) as \(x \to 0\). \newline (ii) Determine constants \(a\) and \(b\) in order that \((1+a \cos 2x + b \cos 4x)/x^4\) may have a finite limit as \(x \to 0\), and find the value of the limit.
A tank in the form of a rectangular parallelepiped open at the top is to be made of uniform thin sheet metal and is to contain a given volume of water. What proportions must the depth bear to the length and breadth in order that the amount of metal required shall be least? \newline If, instead, the amount of metal is given and it is required to construct the tank of greatest cubic capacity, what must be the appropriate proportions? Would it be possible to deduce the result from the former result without detailed calculations?
Prove that \[ \int_0^{\pi/3} \sqrt{\cos 2x - \cos 4x} \, dx = \frac{1}{4}\sqrt{6} - \frac{1}{2}\sqrt{2} \log(2+\sqrt{3}). \]
Trace the curve \((x^2+y^2)^2 = 16axy^2\), and find the areas of its loops. \newline Prove that the smallest circle that will completely circumscribe the curve has radius \(3\sqrt{3}a\).
The centre of a circular disc of radius \(r\) is \(O\), and \(P\) is a point on the line through \(O\) perpendicular to the plane of the disc. If \(OP=p\), prove that the mean distance (with respect to area) of points of the disc from \(P\) is \(2\{(p^2+r^2)^{3/2}-p^3\}/3r^2\). \newline Find the mean distance (with respect to volume) of the interior points of a sphere of radius \(a\) from a fixed point of its surface.
Solve the equations:
Solution: