I’d try to get the teacher and GC on the record, if they aren’t already, with a plan for how your daughter should meet this challenge and improve on the final test. Specific actions. Then make sure to do those things (see below for why).
I do know a few kids who for some reason hit a wall in the latter half of either HS or college calc sequence. They were good at math, it just took awhile for things to click. Once they got it there weren’t additional issues. They had no similar issues with linear algebra, discrete math, real analysis, etc. It was something about calc.
So, make sure your daughter is aware that this does happen and there’s nothing wrong with her. (Although I agree with the suggestions to make sure there isn’t an issue with the teacher or class that’s causing a problem).
I agree with all the advice about better tutoring, online resources like khan academy, trying to pull the grade up.
However, Princeton isn’t looking to rescind admitted students. And certainly not over a speed bump like one semester of math. They care more about the reasons and responses. Namely, that she wasn’t slacking and that she’s taking steps to try to learn the material.
So I think I’d be ready with a simple explanation of the issue and how your daughter responded. The explanation you gave in the first post is good: she had problems grasping some of the subject matter, she went to office hours, did what the teacher suggested, got a tutor, did all the homework, etc.
The teacher and GC should be in position to corroborate some of this if it becomes necessary. The teacher can note that your daughter was diligent about taking steps to get through this, including doing everything the teacher suggested. If the teacher isn’t providing a helpful action plan then I think the GC needs to get involved now on why that’s the case, and should help out later in explaining that your daughter tried to get help.
Perhaps none of this will be necessary and I think it’s very, very unlikely that admission is pulled over something like this. But that’d be my plan, and I’d be proactive about initiating the conversation if the final grade ends up as a D.