I’ve been asked to tutor a soon-to-be college freshman who has taken AP Calculus and successfully earned college credit for first semester calculus. He has been admitted to an Engineering program, and has been advised that there is a “gap” between where single-variable calculus (e.g. AP calculus) leaves off and multivariable calculus begins.
And so, the request to tutor this freshman to make up for “missed material”/the “gap”.
My recommendation (while I’ll help), is that the student purchase the text to be used in his upcoming Calculus II, obtain a syllabus from the Calc I & II class at the University, and compare what was covered (Calc I) and where Calc II starts, comparing his coverage of calculus in the AP class he was successful in.
It’s been a long time since I took a 3-semester, 18-credit university-honors calculus sequence, and I don’t recall when multivariable calculus began in that sequence. Even though I successfully completed two semesters of calculus early in high school, I was on an “honor’s track” for my BS in math, for which the three semester sequence I referred to was required. (It tackled the typical 3-semester calculus sequence, plus undergraduate real analysis/advanced calculus.) So, though I believe we started quite early with multivariable calc, my experience wasn’t typical.
I’m assuming the student received credit for only first semester calc, but I may be wrong.
My questions are, since I do not have the syllabi or text reference for the courses to be taken, nor the text from which the student studied AP calc:
If so, what is it that AP students could be covering/might want to cover, prior to beginning (continuing) calculus at the university level.
Also, any AP-students who received college credit for their AP studies, please feel free to share what your experience has been/was in terms of adjusting to the courses following the calculus class(es) for which you earned AP-credit.
This question will hopefully help students in a similar situation: those who have earned college credit in Calculus by passing the requisite AP exam, who may be uncertain as to their readiness for, say, Calc II as a 1st semester college student. Certainly, the summer prior to beginning college would be an ideal time to study what they might find they are expected to know, if that can be ascertained.
Thanks for taking the time to read the above. Your input is welcome!
Let me preface my answer with some possibly relevant comments. Each semester in the last three years I’ve tutored high school AP-calculus students (overall, about the same number of AB and BC students) and, in the two decades before this, I’ve taught around 30 calculus I and II courses (at 5 different colleges/universities and 3 years at one high school).
Everywhere I’ve heard of (outside of MIT and Caltech, I suppose), Calculus I ends just after the fundamental theorem of calculus and some basic work with u-substitution and Calculus II consists of 1-variable integral calculus, sequences and series, and perhaps some work with polar coordinates and parametric curves. So this situation, where Calculus II (and not Calculus III) involves multivariable calculus, is rather rare (in the U.S.).
That said, given the fact that this is for an Engineering program, I strongly suspect that the gaps include the following material: volumes of revolution using cylinders (only problems that can be solved by disks appear on the AP-exams), hyperbolic functions, partial fractions with repeated linear factors and irreducible quadratic factors (only non-repeated linear factor types appear on the AP-exams), many kinds of “old school” integration methods (especially all the various types of trig. substitutions), and certain specific physics applications such as center of mass, moments of inertia, fluid pressure, work, etc.
Depending on the university (for instance, it’s especially true of the university near where I live), your student might find that a lot more skill in algebraic fluency and “by hand calculations” are expected in Engineering calculus than one typically picks up in an AP-calculus course (where the emphases is more on conceptual understanding than on mastery of classical paper and pencil techniques). If you find this to be the case for the university your student will be attending, then you’ll probably need to do some reviewing of things such as: rapid expansion of binomials using Pascal’s triangle, review of denominator and numerator rationalization techniques, work with trig. identities, curve sketching ideas (i.e. precalculus methods for analyzing the graphs of rational functions, including slant asymptotes), and the like.
I’m not sure what that particular “gap” is (normally we just start the multivariate calculus right after we finish one variable) but I guess the main thing is to make sure that the student is well familiar with the one variable theory. In view of what is normally covered in the multivariate calculus, I would suggest to put the emphasis on the first and second order Taylor formula (a.k.a. “linearization”), the definition of the derivative, the extremal problems, and the Riemann sum definition of the integral and the Newton-Leibniz formula. The limits and the convergence of series are less important and more often than not you can get away with the “intuitive feeling”, though, of course, if you have time, going over the basic epsilon-delta stuff will be beneficial.
What is a real nightmare when teaching the multivariate calculus is that Linear Algebra is often not a formal prerequisite. So, if you find out that your student knows everything that I mentioned well, do some basic staff with him (solving linear equations, understanding linear mappings, a bit of determinants and quadratic forms). That really helps in the multivariate setting.
Of course, this is a “blind suggestion” and once you see the materials you requested from him, you’ll, probably, want to make some adjustments. The only other advice I can give right away is “Don’t take it for granted that the student knows anything well”. If you waste 15 minutes on a problem the student knows well how to solve in the beginning, it won’t put you far behind the schedule but it will give you a firm ground to build on.
Perhaps I can help by relating my rather unorthodox situation. My high school only offered the AP Calculus AB test (typically covering through traditional Calc. 1) and not the more comprehensive AP Calculus BC test (that I believe typically covers through Calc. 2). However, it turns out that the particular undergraduate university that I attended awarded credit for both Calc. 1 and Calc. 2 to those who earned a 5 on the AP Calc. AB exam. As a result, I started college in Honors Calc. 3 despite never having taken Calc. 2.
This particular Calc. 3 course was probably atypical in that it began by covering sequences and series (standard Calc. 2 material). So that would have been a gap for me had the Calc. 3 course been more standard. Aside from that, there were several topics that I missed out on that I didn’t really discover until it came time to TA for Calc. 2 in grad school. The big ones that come to mind were a variety of integration techniques (in particular trigonometric integrals) and polar coordinates. Polar coordinates became relevant in Calc. 3 in the form of cylindrical coordinates.
In my experience, the most critical gap between Calculus I and II has to do with vector analysis. It starts with fairly simple stuff, vector addition, dot product, cross product, systems of equations, “displacements,” etc.
But all this is the foundation of the next step up Calculus; partial derivatives, line integrals, etc. Then gradient, divergence, and curl. Followed by Green and Stokes’ theorem.
In fact, vectors are so important (and easy) that they should be taught in high school, “post algebra,” and “pre calculus,” in my opinion. But they’re usually not, and form an important gap that if not filled, could lead to severe problems later on.