OCR FP2 (Further Pure Mathematics 2) 2010 January

It is given that \(f(x) = x^2 - \sin x\).

1. The iteration \(x_{n+1} = \sqrt{\sin x_n}\), with \(x_1 = 0.875\), is to be used to find a real root, \(\alpha\), of the equation \(f(x) = 0\). Find \(x_2, x_3\) and \(x_4\), giving the answers correct to 6 decimal places. [2]
2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 0.876726\), correct to 6 decimal places, find \(e_3\) and \(e_4\). Given that \(g(x) = \sqrt{\sin x}\), use \(e_3\) and \(e_4\) to estimate \(g'(\alpha)\). [3]

It is given that \(f(x) = \tan^{-1}(1 + x)\).

1. Find \(f(0)\) and \(f'(0)\), and show that \(f''(0) = -\frac{1}{2}\). [4]
2. Hence find the Maclaurin series for \(f(x)\) up to and including the term in \(x^2\). [2]

\includegraphics{figure_3} A curve with no stationary points has equation \(y = f(x)\). The equation \(f(x) = 0\) has one real root \(\alpha\), and the Newton-Raphson method is to be used to find \(\alpha\). The tangent to the curve at the point \((x_1, f(x_1))\) meets the \(x\)-axis where \(x = x_2\) (see diagram).

1. Show that \(x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\). [3]
2. Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation \(x = x_1\), gives a sequence of approximations approaching \(\alpha\). [2]
3. Use the Newton-Raphson method, with a first approximation of 1, to find a second approximation to the root of \(x^2 - 2\sinh x + 2 = 0\). [2]

The equation of a curve, in polar coordinates, is $$r = e^{-2\theta}, \quad \text{for } 0 \leq \theta \leq \pi.$$

1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value. [2]
2. The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) respectively, lie on the curve. Given that \(\theta_2 > \theta_1\), show that the area of the region enclosed by the curve and the lines \(OP\) and \(OQ\) can be expressed as \(k(r_1^2 - r_2^2)\), where \(k\) is a constant to be found. [5]

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$\cosh^2 x - \sinh^2 x \equiv 1.$$ Deduce that \(1 - \tanh^2 x \equiv \operatorname{sech}^2 x\). [4]
2. Solve the equation \(2\tanh^2 x - \operatorname{sech} x = 1\), giving your answer(s) in logarithmic form. [4]

1. Express \(\frac{4}{(1-x)(1+x)(1+x^2)}\) in partial fractions. [5]
2. Show that \(\int_0^{\frac{\sqrt{3}}{3}} \frac{4}{1-x^4} dx = \ln\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) + \frac{1}{3}\pi\). [4]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sqrt{x}\), together with a set of \(n\) rectangles of unit width.

1. By considering the areas of these rectangles, explain why $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} > \int_0^n \sqrt{x} dx.$$ [2]
2. By drawing another set of rectangles and considering their areas, show that $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} < \int_1^{n+1} \sqrt{x} dx.$$ [3]
3. Hence find an approximation to \(\sum_{n=1}^{100} \sqrt{n}\), giving your answer correct to 2 significant figures. [3]

The equation of a curve is $$y = \frac{kx}{(x-1)^2},$$ where \(k\) is a positive constant.

1. Write down the equations of the asymptotes of the curve. [2]
2. Show that \(y \geq -\frac{1}{4}k\). [4]
3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve. [4]

1. Given that \(y = \tanh^{-1} x\), for \(-1 < x < 1\), prove that \(y = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\). [3]
2. It is given that \(f(x) = a\cosh x - b\sinh x\), where \(a\) and \(b\) are positive constants.
   1. Given that \(b \geq a\), show that the curve with equation \(y = f(x)\) has no stationary points. [3]
   2. In the case where \(a > 1\) and \(b = 1\), show that \(f(x)\) has a minimum value of \(\sqrt{a^2 - 1}\). [6]