How Rigorous are High School Calculus classes?

[Intraverno]

Hello everyone, I'm a high school student and wondering about the rigor of the AP Calculus classes that U.S. high school students often take. When I say rigor here, I'm talking about mathematical rigor as is commonly talked about with regard to math classes, and also whether or not the class is truly at college level.

The impression that I have had so far from what I've heard about the classes and here on the forums is that the two classes (AP Calc AB and AP Calc BC) are near the level of a normal college level math class but are not mathematically rigorous. Could anyone with experience with these classes verify this, or correct me if I'm wrong?

Another thing is, that the school I go to tends to have slightly modified AP courses. For example, the program that I'm enrolled in requires AP Environmental Science, which we are required to take without any of the prerequisites, so the class is modified to accommodate that. From what I have heard, our Calc classes are also different in that whereas some schools have the option to jump right into Calculus BC, ours requires AB first, followed by BC. They are also referred to as Calculus 1 and Calculus 2. Does this indicate that our classes are different and possibly more in depth? I would assume that being a STEM specialty program with highly accelerated math, my school wouldn't require you take a class you could get by without.

Any and all help is appreciated, and thanks for any replies!

 

[Vanadium 50]

I don't think there is a good answer. High school classes vary greatly, as do college classes. My observation is that high schools are rushing students into calculus, leaving them with modest algebra skills, weak trig skills, and almost nonexistent analytic geometry skills. But there are exceptions, and surely at least some colleges have modified their expectations (you can see that in discussions of "Calc IV").

 

[Intraverno]

It seems you're right, since I just found an online syllabus for our Calculus BC class and in comparison to things I've heard from people at different schools, which vary from Calculus BC being Calculus AB with added topics to Calculus AB being a slower version of BC, it seems our version spends only 10-11 days on reviewing the entire Calculus AB course (each class period is every other day all year for an average of an hour and a half per period).

Has anyone taken a class where it is done this way? If so, what was the class like, and how hard was it? I realize every class is different, but stories of the various ways the class has gone for others can't hurt.

 

[PhotonSSBM]

The classes vary based on the teacher and school district. The AP exams though, which are standardized, are usually very easy compared to most college exams. So take that however you will.

 

[MidgetDwarf]

Not very, I assign the label "rigorous calculus," to classes that use books, such as Spivak, Apostol, and Courant, as the main textbook. The AP/BC Calculus curriculum leaves out important topics. I believe Epsilon/Delta, Sequence, and Series are left out. I would get the AP credit for both Calculus 1 and 2, then when you get to the university, retake both calculus that may use a book at the level of Spivak.

 

[WWGD]

Maybe you can answer the rigor issue yourself: what statements are proved, and which/how many are just assumed ? I think this is the main basis of rigor.

 

[romsofia]

Calculus is calculus at the end of the day. No matter what course you take, even in college, it'll be up to you to get what you want out of it. If you find the course isn't rigorous enough, find more problems to challenge yourself.

 

[WWGD]

Another way of challenging yourself is to go over the proofs and do your best to prove as many statements as possible. And/or pick a Real Analysis book to help you through it.

 

[Intraverno]

Thanks for the great advice, everyone! I think I'm going to go through Spivak's Calculus in addition to the regular Calculus classes, most likely not in the school year because of how busy I am but in the summer instead. Also, I found out the textbook for the AB class is Calculus: Graphical, Numerical, Algebraic, and the BC class uses Larson Calculus of a single variable. Out of curiosity, how are these textbooks?

 

[romsofia]

I've used Larson's and it's fine, however, people will try to say it is a bad book because they want to compare it to books with different audiences. Just remember, it is your first time taking a look at calculus.