CAIE P3 2003 June — Question 8

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2003
SessionJune
TopicFixed Point Iteration

8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).
  1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
  3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.