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## Main Question or Discussion Point

Hello I am looking at Stat Mech problem 2 from here (page 8) with solution here. I am confused about their approximations. They are all valid, but they are different. For example in part a) they use $$\frac{\partial P}{\partial x}(x+\frac{1}{2}\Delta x,t)=\frac{P(x+\Delta x,t)-P(x,t)}{\Delta x}$$ and a bit lower they use $$\frac{\partial P}{\partial x}(x,t)=\frac{P(x,t+\Delta t)-P(x,t)}{\Delta t}$$ Why would I use one over the other? If you know what answer you need, you might figure it out, but in general I think you would get a different answer if using different approximations (by a factor of 2 or something?). Also in part b) they use a Taylor series, which could have been used equally well in part a) and I am not sure why they used it here but not there.