1. Using pencil and paper, start with the energy expression for the simple pendulum given in the small angle approximation $E_i = \frac{1}{2}mL\theta_i^2 + \frac{1}{2}mgL\theta_i^2$, and answer the following questions.
(a) Using the Euler method, calculate $E_{i+1}$ and show that this method fails for the simple pendulum. In particular show that in the the energy will increase with time. With what power of the time step does the energy increase?
(b) Using the Euler-Cromer method, calculate $E_{i+1}$ and argue that the method has terms that average to zero over one cycle.
