1.
$$f ( x ) = \sec x + 3 x - 2 , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
- Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ 0.2,0.4 ]\)
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form
$$x = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x }$$
The solution of \(\mathrm { f } ( x ) = 0\) is \(\alpha\), where \(\alpha = 0.3\) to 1 decimal place.
- Starting with \(x _ { 0 } = 0.3\), use the iterative formula
$$x _ { n + 1 } = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x _ { n } }$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
- State the value of \(\alpha\) correct to 3 decimal places.