Investigate the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). Prove that the expression \(\frac{x+a}{x^2+bx+c^2}\) will be capable of all real values if \(a^2+c^2 < ab\).
Find the sum of the squares, and the sum of the cubes of the first \(n\) natural numbers. Sum the series to \(n\) terms
Prove the law of formation of successive convergents of the continued fraction \[ a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \frac{1}{a_4+} \dots. \] Prove that the difference between the continued fraction and \(p_n/q_n\) its \(n\)th convergent is less than \(1/q_n q_{n+1}\) and greater than \(a_{n+2}/q_n q_{n+2}\).
Find the differential coefficients of \(f(x)/\phi(x)\) and of \(f\{\phi(x)\}\). Find the \(n\)th differential coefficients of \(\frac{x}{x^2-3x+2}\) and of \(\tan^{-1}x\).
Prove the method of determining and discriminating between maximum and minimum values of a function of a single variable by means of its differential coefficients. If \(O\) be the centre of an ellipse whose semi-axes are \(a\) and \(b\), \(ON\) the perpendicular to the tangent at \(P\), shew that the maximum area of the triangle \(OPN\) is \(\frac{1}{4}(a^2-b^2)\).
Investigate the equation of the tangent at any point of the curve \(f(x,y)=0\). Write down the equation of the normal at the point of the curve \(ay^2=x^3\) where \(x=\frac{1}{2}a\), and prove that it touches the curve.
Prove the expressions for the radius of curvature of a curve \[ \text{(i) } \rho = r\frac{dr}{dp}, \quad \text{(ii) } \rho = p + \frac{d^2p}{d\psi^2}. \] Find \(\rho\) and \(p\) at a point of \(r^n=a^n\sin n\theta\).
Evaluate the integrals \[ \int \frac{dx}{x^2+x+2}, \quad \int \frac{d\theta}{5-3\cos\theta}, \quad \int a^x\cos x dx. \] Find a formula of reduction for \[ \int_0^{\pi/2} \cos^m x \sin^n x dx. \]
Shew how to find the area of a curve given in polar coordinates. Trace the curve \(r=a(2\cos\theta+\sqrt{3})\) and find the area between the two loops.
If \(\alpha, \beta\) are the values of \(x\) which satisfy \[ x^2y^2+1+a(x^2+y^2)+bxy=0 \] for any given value of \(y\), show that \[ \alpha^2\beta^2+1+A(\alpha^2+\beta^2)+B\alpha\beta=0, \] where \[ A = \frac{(1-a^2)^2}{ab^2}, \quad B = \frac{2(1-a^2)^2-b^2(1+a^2)}{ab^2}. \]