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