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.
So, 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 a non-dissipative case.
If your ODE solver is injecting a little bit of air 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 you're working with.