Question 1 asks if forward Euler is a single step method
If you remember from the lecture, this is true
In fact, forward Euler is the most basic of all single-step methods
It takes exactly one step, and no interpolation is done
Question 2 asks whether forward Euler and trapezoidal are members of the family of Runge-Kutta methods
This is true
Forward Euler is a first order Runge-Kutta method, and trapezoidal is a second order Runge-Kutta method
So this is true
Question 3 asks what symplectic ODE solvers are good for
Symplectic ODE solvers are good for systems that are conservative
That is, systems where you want to conserve energy, or area in the more abstract case
With these systems, failure to use a symplectic ODE solver can sometimes inject friction, essentially
You can imagine, with a pendulum for example, in the non-dissipative case, if your ODE solver is injecting a little bit of error every step size, every time you take a step, this could effectively act as a numerical friction term
This loss of energy fails to conserve energy, and symplectic ODE solvers are developed specifically to keep the energy in the system
Or the area, depending on the system youre working with