The first question on this quiz asks various things about the local truncation error of the forward Euler method

I think the easiest way to start is to know what the actual error is, then we can base all the other questions of this

If you recall, this is actually the error of the forward Euler method, so this question is true

Notice that, in this answer, we only have a first derivative here, which is not the case  we actually use the first derivative term in the Taylor expansion, so this is false

And notice here, we have the step size, not the step size squared, so this is also false

The local truncation error in the forward Euler method is dependent on the dynamical landscape

We can see this by the fact that theres this f(x) term

So this question is false

And the local truncation error of the forward Euler method is not proportional to the step size

Its proportional to the step size squared, as you can see right here, so this question is also false

Question 2 is whether finite-precision arithmetic causes truncation error, and this is false

Finite-precision arithmetic causes roundoff or cutoff error

Truncation error comes about by how we approximate the solution using the Taylor series: at which term of the Taylor series we truncate

Question 3 asks if finite-precision arithmetic causes roundoff error

This is true: finite-precision arithmetic can either cause roundoff error or cutoff error, depending on the system being used

Question 4 asks whether observational error can snowball over the course of a numerical solution of an ODE

This is false: observational error is something that can be seen as dirt on your glasses

Its something that youre getting at the observation level, but it is not being fed back into the system

The snowballing effect occurs whenever you have some kind of dynamical error, or numerical error, that is fed back into the system at every time step

Observational error is just something that youre going to see on your glasses, not something that is fed back into the system

For Question 5, the trapezoidal had lower error than both forward and backward Euler

The trapezoidal method has local truncation error on the order of h cubed, versus forward and backward Euler, which both have error on the order of h squared

Question 6 asks, why is it a good idea to adapt the time step of an ODE solver on the fly?

And this is to account for the different curvatures that can occur in different parts of the dynamical landscapes in a nonlinear system