From which point to which point is latent heat measured? I think I understand the idea of latent heat, but the wording (or lack thereof) of practice problems instantly makes me think that more information should be given. Either that, or I'm not understanding it correctly.
Textbooks often will ask a question like "A 100 watt heater is used to melt ice. After 60 seconds, 0.02 kg of ice has melted. Find the latent heat of fusion for ice."
When I imagine ice, I imagine something that is below zero degrees Celsius. Look at this nice graph that I made showing the temperature change of a material (H2O, for example) over time:

I want to understand that latent heat of fusion is the energy required to get from point B to point C on my graph. Is that correct?
The wording of most questions/explanations I see make me think that it is the energy required to get from point A (an arbitrary point below the melting point) to point C. For example, in the practice problem I provided earlier, it doesn't mention the original starting temperature of the ice.
In order to do experiments like this to determine the latent heat of fusion for a material (ice, for example), shouldn't the ice be just a tiny fraction of a degree off of the melting point? Or, ideally, just at the melting point?
 A: The latent heat of fusion is the energy required to go from point B to point C. (or vice versa)
Generally in these questions if no temperature is given, you assume it's known the ice is perfectly at it's melting point (point B).
The point of these questions is to understand what the latent heat of fusion is and what it does.  It is an energy barrier between the phase changes.
In reality; your ice probably wont be perfectly at it's melting point (and phase changes are actually more complicated than that anyways; especially going from liquid to solid).  So yes, there would also be some required heat to get it to the melting temperature.
As long as you knew the properties of the ice and it's temperature, you could determine that energy as well.  Since you are not given that information; it seems the problems are to illustrate that fusion itself requires energy, or "latent heat" (because the temperature does not change during this heating).
If you are just being introduced to all this, I think it's great you have this question.  You've recognized that the textbook is making an assumption and is not clearly stating it; but I think it's safe to assume they are not trying to perfectly model reality, but are just trying to illustrate an example.
