Learn English – Choice between ”in the freshman year’ and ‘during the freshman year’

Each of the examples above is grammatically correct, but with a slightly different meaning.

First, the question of "past" or "last" is less important, and it may depend on context. To me, "past" means specifically that each of the three years in question has already ended, whereas "last" might include the current year. Context is more important than the choice of words here.

The phrasing is more important. This is somewhat subjective, but to me the different prepositions indicate these connotations:

• "... in the past/last three years": Cash flows steadily improved during some interval of these years
• "... for the past/last three years": Cash flows started improving three years ago and never stopped
• "... the past/last three years": Same as "for", but a bit awkward to my ears
• "... during the past/last three years": Same as "in"
• "... over the past/last three years": Probably the same as "for", but now there is some ambiguity. "Over" could be a comparison between the current improved cash flow and that of the previous three years.

The sentence with during is the correct one; the other one is incorrect.

The preposition during generally refers to a time interval in its entirety: in your example, the entire day (meaning "daytime") is a time interval in which sleep is forbidden.

The preposition in is sometimes useable for referring to a small portion of a time interval. It doesn't work in your example, and I apologize but I can't explain why. It may be that it is grammatically correct, but "in the day" just isn't generally used, and doesn't sound natural.

Some examples using in that are correct and natural:

I will bring you to the doctor's office in the morning.

The fireflies will come out in the summertime.

We'll be on vacation in November.

This last example is a good one for the in vs. during contrast. If you say:

We'll be on vacation during November.

then you mean you'll be on vacation for the entire month, not just for a few days.