Here is a public service announcement from your friendly internet college professor. AP Calc BC is not really equivalent to the first year of calculus at a really rigorous college.
At a lot of colleges, yes, AP Calc BC is good enough and you can test out of the first year of college. If you’re going to Caltech or MIT or Harvard or U Michigan or Cornell, it’s not a replacement. Even at the University of Minnesota, where I’ve taught a lot, most freshmen who come in with a 3 on the AP Calc exam aren’t exactly prepared for the next semester. We adjust the curriculum for second semester calc knowing that students who start there due to a 3 on the AP are going to have a little bit of a slower start.
What’s the difference between calc at a tough college and AP Calc? Depth. At a tough college, all of your instructors are going to be working on or already have a PhD. Whatever their personal qualities or teaching ability, they’ve been through a lot of math and they have a lot of depth of knowledge. High school teachers, on the other hand, have been through school for a teaching certificate. All of them have a lot more education in education than college instructors. Their math background, though, varies from “I have a PhD or master’s and wanted to teach high school!” to “I got a math major and it was great!” to “I was supposed to be the French teacher but they couldn’t find anyone to cover this class.” I am not kidding: check out the stats on the percentage of high school math teachers in the US who don’t have a math major or a certificate in math. The most important factor in your success besides your own effort is having a good teacher. Either you have one already, or you need to find one (and I’m volunteering!).
That’s why a lot of you get, “Because I said so,” or “because that’s how they want it written,” as answers to your questions in class in high school. That’s why your high school teacher might care about whether you write 1 1/2 versus 3/2. (FYI we don’t use mixed fractions in college math. They’re useless for computation. Fractions or decimals. That’s it.) College math teachers have their own problems, notably a complete lack of education in how to teach unless they went out and got this training themselves — but they usually know a lot of math.
If you really want to be ready for college math, it might help to get some insight from someone who’s already there. Talk to your friends who have gone on to college already — no one can give quite the same insight as someone who went to your high school and is currently taking college math. But also look further along the road: someone like me, who has taught a lot of college math, can give you a 30,000 foot view of high school math, college math, and math out in the real world.
If you’re planning on taking the AP test this spring, sign up for emails with study help and links to videos on the free response section of the AP calculus exam! And good luck!