Given that \(a\) and \(b\) are positive constants and \(x\) is a real variable, prove that \[f(x) = a \cot x + b \csc x\] takes all real values provided \(a > b\), but takes all real values except for a certain range if \(a < b\). Prove that the curve \(y=f(x)\) has real points of inflexion at \(x=\cos^{-1}\frac{\sqrt{a^2-b^2}-a}{b}\) when \(a > b\), but none when \( a < b\). What happens when \(a=b\)? Sketch graphs of \(y=f(x)\) for the three cases.
Starting from some (stated) definition of \(\log x\), prove from first principles that \((\log x)/x \to 0\) as \(x \to \infty\). Investigate the limit of \(x^{1/x}\) when (i) \(x \to \infty\), (ii) \(x \to 0\) by positive values. Sketch the graph of \(y=x^{1/x}\) for \(x>0\).
Prove that the \(n\)th derivative of \[ \frac{1}{x^2+b^2} \quad (b \ne 0)\] is \[ \frac{(-)^n n!}{b^{n+2}} \sin (n+1)\theta \sin^{n+1}\theta, \] where \(\theta=\tan^{-1}(b/x)\). Find the \(n\)th derivative of \(\tan^{-1}\frac{x+a}{b}\).
Light emitted from the point \(A\) on the circumference of a circle of centre \(O\) and radius \(a\) is reflected at the circumference in the plane of the circle. Prove that the once reflected rays all touch the curve \begin{align*} x &= \tfrac{1}{4}a (2\cos t - \cos 2t), \\ y &= \tfrac{1}{4}a (2\sin t - \sin 2t), \end{align*} where axes have been taken at \(O\) such that \(A\) is the point \((-a,0)\). Identify this curve as a cardioid, and show its position in a sketch.
Evaluate
Obtain a reduction formula for \(\displaystyle\int \frac{dx}{x^{2k}\sqrt{x^2-a^2}}\). Hence or otherwise evaluate \(\displaystyle\int_a^\infty \frac{dx}{x^6\sqrt{x^2-a^2}} \quad (a>0)\).
If \(y=u\) is known to be a solution of the differential equation \[ py''+qy'+ry=0, \] where \(p, q\) and \(r\) are given functions of \(x\), show that the solution of \[ py''+qy'+ry=s, \] where \(s\) is a given function of \(x\), can be reduced by means of the substitution \(y=uv\) to the solution of a first order linear differential equation for \(v'\). Given that \(y=(1+x^2)^{-1/2}\) is a solution of \[ (1+x^2)y''+2xy'+y/(1+x^2)=0, \] obtain the solution of \[ (1+x^2)y''+2xy'+y/(1+x^2)=x/(1+x^2) \] which vanishes at \(x=0\) and \(x=1\).
Given \(z=F(x,y)\) and \(y=f(x)\), explain the difference between \(dz/dx\) and \(\partial z/\partial x\), and find an expression for \(d^2z/dx^2\) in terms of derivatives of \(f(x)\) and \(F(x,y)\). If \(z=x+y^2\) and \(xy=\sin y\), show that \(d^2z/dx^2\) vanishes when \(\tan y = \frac{3y}{3-y^2}\).
Prove that a single loop of the curve \(r=2a \cos n\theta\) (\(n>1\)) has the same area and perimeter as an ellipse with semi-axes \(a\) and \(a/n\). If \(n=1+\epsilon\), where \(\epsilon\) is small, obtain an approximate expression for the perimeter of the loop as a series of powers of \(\epsilon\) up to the term containing \(\epsilon^2\).
Sketch the curve \[ (y^2-1)^2 - x^2(2x+3) = 0. \]