How to build a better positive image of yourself to your Ph.D. advisor
If that's the first time you've copied a problem down incorrectly, I'd bet you're the type of student whose conscientious work is obvious to your advisor and other professors in your department (whom you should also be developing relationships with, even at this early stage, btw). Also, keep in mind that admissions committees (especially in departments that can attract &quot;famous&quot; professors) don't casually accept students; if you weren't cut out for it, you probably wouldn't be there to begin with.
In any case, no reasonable advisor would expect a first-year student to get every fact right or never make mistakes. In the unlikely case your advisor does happen to have those expectations and/or does think, based on two inconsequential mistakes, that you're not cut out for the program, you have plenty of time to find a new advisor, which is not at all unusual. More importantly, it sounds like you may be assuming the advisor thinks these things, since you don't mention any specific reaction he had that indicates as much. But even if he did give some feedback or correction, keep in mind that giving feedback and critique is what they're there for, and receiving that critique and advice is what you're there for. It's crucial for survival in grad school that students not take criticism personally—after all, if you're planning to be an academic at least, proposing some idea and having other people critique it is pretty much what you'll be doing for the rest of your career. If your advisor did give some clear indication he's displeased with your progress, you might consider explaining the circumstances or somehow remediating the situation—but if there wasn't an explicit indication (and even if there was) he's probably forgotten about the whole thing by now. You're likely better off just keeping on with the hard work and reminding yourself as often as you can that you belong there and that all grad students who aren't egomaniacs are constantly wracked with self-doubt and that you're probably doing just fine.
But here's the most important thing, which I don't think most grad students (of which I am one) realize soon enough*: the professors are there because they want to teach students! If you showed up and knew every single thing perfectly before your first year was over, their lives would be pointless (slight exaggeration, but still). Being a professor is about teaching people things they didn't know before and guiding students in their acquisition of knowledge, so don't feel bad when that occasionally happens. A good advisor will genuinely enjoy feeling like they're actually contributing to your development from student to colleague, and your making mistakes early on is part of that process.

*I didn't figure it out on my own either; I was lucky that someone very wise told me this when I was distraught after my first teaching evaluation (which in retrospect was actually full of constructive feedback that I took as harsh criticism simply because it included room to improve).