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.