Newton-Raphson with trigonometric or exponential functions

Questions where the function involves trigonometric functions (including inverse trig) or exponential functions, requiring differentiation of these transcendental functions before applying Newton-Raphson.

2 questions

OCR FP2 2007 January Q2
2 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x\).
  1. Show by calculation that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval \(0.8 < x < 0.9\).
  2. Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation to this root. Give your answer correct to 3 decimal places.
OCR FP2 Q2
2 Use the Newton-Raphson method to find the root of the equation \(\mathrm { e } ^ { - x } = x\) which is close to \(x = 0.5\). Give the root correct to 3 decimal places.