The first question on this quiz asks
some 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,

and then we can base all the other
questions off of this.

If we 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's
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 of the
forward Euler method

is dependent on the dynamical
landscape.

We can see this by the fact that
there's this f prime prime term (f''),

so this question is false.

And the local truncation error of the
forward Euler method

is not proportional to the step size:

it's proportional to the
step size squared,

as we can see right here,
so this question is also false.

Question 2 is whether the 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 a Taylor series, and 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 to an ODE.

This is false. Observational error is
something that can be seen as

dirt on your glasses.

It's something that you're 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 errors are something
that you're going to see on your glasses,

not something you fed
back into the system.

For question 5, the trapezoidal
ODE solver has lower error

than both forward and backward
Euler.

The trapezoidal method has local
truncation error

on the order of h cubed (h^3),

versus forward and backward
Euler, which both have error

on the order of h squared (h^2).

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
landscape in a nonlinear system.