<p>I’m not a Caltech student and I can’t comment on your chances, but there are a few comments I would like to make. </p>
<p>In my sophomore year of high school, I also started to learn Calculus on my own and proceeded to take the AP Calc AB test my sophomore year. If it’s avoidable I would try to take Calculus earlier and as someone mentioned above I recommend taking AP Calculus BC instead of or in addition to Calculus AB. Other than the fact that it will probably look better, if you’re already starting to learn calculus now then taking the class two years later will be extremely boring as most AP Calculus classes don’t go in depth. What most people don’t realize is that you don’t necessarily have to take an AP class in order to take the corresponding AP test, and so if you’re up for it I suggest contacting the AP coordinator at your school about signing up for the test on your own. My senior year I self-studied AP Bio, AP Physics C, and AP US History and while no one at my school took the AP Physics C examinations, they still ordered the exam and provided me with a proctor. If you do choose this route, I suggest just having the discipline to study on your own and try to get your hands on past exams to get a feel for the test. As for studying calculus, a couple excellent books are by Spivak and Apostol and I recommend that you take a look at these books in the future. They are fairly rigorous and difficult as far as Calculus books go, but if you really spend time to work through them, you’ll gain a thorough understanding of Single Variable Calculus. (I’m actually working through Apostol’s Calculus Vol. 2 right now and I love it.) I also second the earlier notions of looking into Art of Problem solving and getting interested in competition mathematics, something that I missed out on when I was in high school. Personally I think competitions like ARML, AMC, and just various math competitions are a lot of fun and can teach you a lot about mathematics. They help show that you have passion for mathematics and in general just help you connect with other people who love math. And I don’t know your situation, but don’t throw the idea of taking college classes completely out the door. I don’t necessarily recommend this, but if you keep learning math at the pace you are currently going then there is the option of taking math classes like Linear Algebra, Multi-variable calculus, or Differential equations your senior year at a local college. </p>
<p>I myself wanted to go to Caltech, but I never really spent the time in high school to be a competitive candidate. Anyways, good luck with your future endeavors and continue enjoying math!</p>