You must have hit her instead of the sign on the floor.
You'll need to give it a few days for her to calm down.
Suggestion: If you've done at least the beginning of the main questline, you'll have a couple of shouts available. She'll also let you in if you tell her your the Dragonborn and demonstrate a shout (I suggest whirlwind or ethereal as Fus Do Rah is kind of agressive).
I still find it odd that she would ask for a firebolt spell. I have never had her ask for an offensive spell from me. The closest I've come is having her ask for me to conjure a flame antronach.
BTW: Why do you need to get in for Forbidden Legend? I don't remember the college even being a part of that questline. I could be wrong, but, isn't that the one with the 3 brothers? Oh! I see, you need to go with Tolfidir. Now it makes sense.
Do you have a prior save you can revert to before you pissed Faralda off? She's a pretty bada** sorceress so I wouldn't mess with her.