Solution
= Solution
For $\theta=1/2$,
$$
R(z)=\frac{1+z/2}{1-z/2},
$$
so $|R(z)|\le1$ exactly when $\operatorname{Re}z\le0$. Thus the first method is A-stable with stability domain the closed left half-plane. The $\theta=3/4$ method is not A-stable: along the negative real axis its quadratic numerator makes $|R(z)|$ grow without bound as $z\to-\infty$.