I don’t know…I can’t begin to know the rules of your college or this particular class. You do seem to have a case, however with your strong upward trend…however, I don’t think you should get something (a grade bump, even one point) for nothing. Also, what about the relative fairness to your classmates?
If you’re going to ask your professor, what about proposing to do an extra project related to the earlier coursework that you failed, such as a short research paper or a written review of the earlier material, etc. This might be fair if other students are also given this extra-credit chance to improve grades. Know, however, this creates some extra grading work for the prof, so you must accept it if he/she declines and be incredibly gracious if the opportunity is extended.