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The first question on this quiz asks various things about the local truncation error of the forward Euler method
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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
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If you recall, this is actually the error of the forward Euler method, so this question is true
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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
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And notice here, we have the step size, not the step size squared, so this is also false
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The local truncation error in the forward Euler method is dependent on the dynamical landscape
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We can see this by the fact that theres this f(x) term
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So this question is false
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And the local truncation error of the forward Euler method is not proportional to the step size
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Its proportional to the step size squared, as you can see right here, so this question is also false
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Question 2 is whether finite-precision arithmetic causes truncation error, and this is false
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Finite-precision arithmetic causes roundoff or cutoff error
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Truncation error comes about by how we approximate the solution using the Taylor series: at which term of the Taylor series we truncate
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Question 3 asks if finite-precision arithmetic causes roundoff error
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This is true: finite-precision arithmetic can either cause roundoff error or cutoff error, depending on the system being used
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Question 4 asks whether observational error can snowball over the course of a numerical solution of an ODE
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This is false: observational error is something that can be seen as dirt on your glasses
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Its something that youre getting at the observation level, but it is not being fed back into the system
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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
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Observational error is just something that youre going to see on your glasses, not something that is fed back into the system
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For Question 5, the trapezoidal had lower error than both forward and backward Euler
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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
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Question 6 asks, why is it a good idea to adapt the time step of an ODE solver on the fly?
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And this is to account for the different curvatures that can occur in different parts of the dynamical landscapes in a nonlinear system