\noindent{\bf Ans}: 

\begin{enumerate}
\item
Direct fixed point iteration with 
$g(x) = g_0 (x) = 0.01-3\sin(x)$ does not converge.
Instead, a proper choice of $\beta$ and 
$g(x) = \beta x + (1 - \beta)g_0(x)$ will 
result in local convergence {\bf (2 pts)}. 
One could choose 
$$
\beta = \frac{g_0'(\xi)}{g_0'(\xi)-1}
$$ 
for some $\xi$ near 0.
Since $\xi \approx 0$, 
$g_0'(\xi) \approx -3$, 
we take $\beta = \frac{-3}{-3-1} = \frac{3}{4}$. {\bf (2 pts)}

First check $g([-{1\over 2}, {1 \over 2}]) \subset [-{1\over 2}, {1 \over 2}]$.
$$
-\frac{1}{2} < -0.012931... = g(-\frac{1}{2}) \leq g(x) \leq g(\frac{1}{2}) = 0.017931... < \frac{1}{2}\;\textrm{{\bf (3 pts)}}
$$

Second check $|g'(x)| \leq k$ for some $k \in (0,1)$, $\forall\;x \in (-{1\over 2}, {1 \over 2})$. 
$$
|g'(x)| = \bigg|\frac{3}{4}(1-\cos(x))\bigg| \leq \frac{3}{4} < 1\;\forall\;x \in (-{1\over 2}, {1 \over 2})\;\textrm{{\bf (3 pts)}}
$$
\item
Estimation {\bf (3 pts)} Result $N$ {\bf (2 pts)}\\
$[\textrm{Method 1}]$
$$
|x_n-x_*| \leq k^n\max\{x_0-a,\;b-x_0\} 
=\left(\frac{3}{4}\right)^n\max\left\{0-\left(-\frac{1}{2}\right),\;\frac{1}{2}-0\right\} 
=\left(\frac{3}{4}\right)^n\frac{1}{2} < 10^{-30} 
$$
$$
\Rightarrow n > \frac{\log_{10}2-30}{\log_{10}\frac{3}{4}} = 237.71... \Rightarrow N = 238.
$$

$[\textrm{Method 2}]$
$$
|x_n-x_*| \leq \frac{k^n}{1-k}|x_1-x_0|
=4\left(\frac{3}{4}\right)^n|g(0)-0| 
=\left(\frac{3}{4}\right)^n0.01 < 10^{-30} 
$$
$$
\Rightarrow n > \frac{28}{\log_{10}\frac{4}{3}} = 224.11... \Rightarrow N = 225.
$$