Question#$4(20$ points$)$: Secant Method

Determine the lowest positive root of

$f(x)=8sin(x)e^{-x}-1$:

a$.$ Using the secant method $($two iterations, $x_{-1}=0\mathrm{.}5$ and $x_0=0\mathrm{.}4)$

b$.$ Using the modified secant method$($ two iterations, $x_0=0\mathrm{.}3,=0\mathrm{.}01)$

iteratio$\frac{n}{\text{\#}}1$

$f(0\mathrm{.}4)=8sin(0\mathrm{.}4)e^{-0\mathrm{.}4}-1=8(0\mathrm{.}3894)(0\mathrm{.}6703)-1=2\mathrm{.}0878-1=1\mathrm{.}088$

$f(0\mathrm{.}5)=8sin(0\mathrm{.}5)e^{-0\mathrm{.}5}-1=8(0\mathrm{.}4794)(0\mathrm{.}6065)-1=2\mathrm{.}3251-1=1\mathrm{.}326$