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I've taught calculus in college for five years, and it's alwways interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even sem like they needed the course. Others struggle.

I took Calculus in high school, and I don't remember doing Newton's method or his law of heating and cooling. In fact, even though my high school class was a 'double class' (counting for two credits), I don't remember covering as much material as we did in college; but I didn't feel behind in my next classes.

In the United States, what are the main differences between the way high school Calculus and college Calculus are taught?

I'm interested in amount of time spent in class, average homework loads, topics covered, and any other differences you feel are important.

Edit: I'm specifically talking about AP Calculus BC in highschool compared to a one year sequence of Calculus at a four-year university. Both as currently taught.

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I doubt that it is possible to give an accurate characterization of a typical US AP calculus course. There's an excellent science magnet school in my town where I'm sure AP calc is far more rigorous than at the community college. But within the last decade or so, there has also been a vast proliferation of AP courses and vastly increased numbers of students taking the courses. It's not uncommon for a certain school to offer an AP course in a certain subject, and have 10% of their students or less get a passing grade on the AP exam. –  Ben Crowell May 8 at 14:55
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I took AP calculus BC in high school, and since TA's and taught calculus at two universities since (I'm a grad student, but I've lectured as well as TA'd). These experiences have amalgamated together into this answer.

I had 170 days of calculus instruction in high school, each one hour long. We used a standard college text of Larson and Edwards (whatever edition it was at the time). There were weekly homeworks, a test roughly each month, and a final each semester (and an AP test afterwards, I suppose). We did not prove things, and we did not need to use the $\epsilon- \delta$ formulation of a limit. In practice, most functions were continuous and differentiable. In fact, most functions were elementary and smooth. We did learn Newton's law of cooling, some numerical integration bits like the the trapezoid rule and Simpson's rule, Taylor series with remainder, u-substitution, integration by parts, trig substitution (to mention topics of various difficulty that I feel might be omitted in some high school classes). We did not talk about many differential equations (we learned initial value problems, and Euler's method, but I simply cannot remember if we did much else).

A very important component that was missing from my high school education was understanding why anything was true. We might heuristic our way through things and prove a (small) subset of the things we came across, but in general there were few proofs. Math was a gift, set down whole before us like a swaddled babe from a stork.

With respect to the homework, I never did any calculus homework at home - many of us were able to finish it all during school hours. In this regard, we couldn't have done too much homework timewise. But the homework was graded completely and returned with notes, and we did an incredibly large amount number-of-problems-wise.

Now, university calculus. At my current university, calculus meets for almost 4 hours per week: roughly 2.5 hours for lecture and roughly 1.5 hour for recitation. There are roughly 15 weeks weeks, so that there are approximately 60 hours of instruction per semester. My high school calculus BC course is the equivalent of two semesters of calculus at my university (roughly corresponding to differentiation and then integration), so we should really be thinking of 170 hours vs. 120 hours. (At my undergraduate university, calculus courses met for 6 hours per week and BC calculus was roughly the equivalent of the first semester and the first fourth of the second semester of calculus - the other three-fourths were actually linear algebra).

We teach Newton's law of cooling, but we do almost no numerical integration. In fact, we barely cover Riemann sums at all. The lectures will talk about Riemann sums insofar as to define integration, and they aren't really brought up again. (At my undergrad, we similarly skimmed over Riemann sums, though there was more variability: some professors care and others don't). Thus we don't talk about any numerical integration techniques. We don't talk about "differentials," by which I mean approximating $f(x+\delta)$ by using some sort of $f(x) + f'(x)\delta$ style argument, whereas in high school we did, including some with multiple steps (like calculating $\arctan{1.5}$ using three steps of length $.5$ from $0$, or something). We similarly skipped Euler's method for approximating solutions to differential equations. This is a recurring theme: we skimp out on numericals.

I partially attribute this to calculator usage. In high school, we used calculators as part of the AP test uses calculators. At my university, we use no calculators ever - so no numerics and usually the problems themselves are easier. I don't know whether this is a good thing or a bad thing.

At my university, we teach $\epsilon-\delta$ but do not test it (a conceptual hurdle that the powers that be have decided isn't worth the struggle, but which is not skipped at my undergrad). We teach Taylor remainder estimation but do not test it (another numerical problem skipped). This was a large and challenging part of high school calculus entirely missed. (This was not skipped at my undergrad).

On the flipside, we do teach first and (some) second order differential equations during the second semester - students learn integrating factor and separable differential equations in particular. Sometimes, depending on the professor/speed of the class, we teach power series solutions to differential equations. The homeworks that we assign were very long (at my undergrad) and are moderately long at my current university. But the graders at both were hired undergrads, for whatever that is worth.

And every lecturer I know proves everything from the ground up, even though the students are not held directly accountable to the proof material.

In general, I find high school calculus to be a much better course, with more contact hours, required attendance, more and more completely graded homeworks, and most topics (except $\epsilon-\delta$ and some differential equations bits that are covered in some university classes). In my university classes, we cover fewer subjects, but we cover them deeper (i.e. with proof), but there are fewer (though perhaps harder) homeworks that are graded by undergrads. And we omit most numerics.

I would suspect a student who got a 5 on the AP calc BC test to get an A or B at the end of the second semester of calculus at my university. But before one concludes that I think high school better, I find that most of my students (including those who took calculus in high school) have almost no conceptual understanding of calculus coming in. Some do when they pass the course.

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I've noted that trigonometric substitution is not taught in most BC calc classes(it's not tested on the AP test so teachers probably figure why should they). Looking at the calculus courses at my university I would expect someone who did well on BC Calculus to do good in second semester, but not know the majority of it to start. –  ruler501 Mar 27 at 6:50
    
I mean this constructively: I think this answer is longer than it needed to be. It seems like the main differences you found could have been put into a short bulleted list, and then your own interpretations would appear more front-and-center as the meat of the answer. That said, I especially liked the insight about calculator usage being a driving force behind many of the differences. Thanks for the good read! –  Chris Cunningham Mar 27 at 21:15
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