How to request a professor to let you into their class
Lizzie, After reading the details which I asked you to add earlier, I can only come up to this solution: Go and talk to her in person. Being a teacher myself, I find it difficult to ignore the requests of an 'eager learner'. Which in your case is "you". Tell her how important it is for you to take that course and specifically from her and how you have performed in your first quiz. Ask her very politely if it is possible for her to allow you in that course (Politeness matters A LOT) and that you will continue to work harder. Try this and I hope it'll work. Good luck :)