Outline and Goals of a One-Year Calculus Sequence

Question

Our department is considering restructuring our traditional three semester calculus sequence so that the calculus requirement for our majors is satisfied in two semesters.

Does your department offer such a two semester sequence and, if so, could you provide a rough outline of topics covered and/or textbooks used? What are your department's learning goals for this sequence? If your department does not offer such a sequence, has it been considered?

We are particularly interested in responses from faculty at small liberal arts colleges.

Addendum: It has been suggested that we consider, for the first semester, accelerating the Calculus I (differentiation) part of the course by reviewing for a week and then proceeding with Calculus II (integration) at the usual pace. The second semester is then dedicated to multivariable calculus. If your institution has tried this, feedback on this would be helpful.

Note: We intend to offer the abridged sequence only for mathematics majors (or perhaps mathematics and physics majors).

Answer 1

Mathematics department at Koç University used to offer two-semester Calculus courses.

The first one used to cover limits and continuity; derivative and properties of differentiable functions; mean value theorems, Taylor's formula, extreme values; indefinite integral and integral rules; Riemann integral and fundamental theorem of calculus; L'Hospital's rule; improper integrals, sequence and series of numbers; power series and their properties; Taylor and Maclaurin series.

And the second one used to cover functions of several variables; partial differentiation; directional derivatives; exact differentials; multiple integrals and their applications; vector analysis; line and surface integrals; Green's, Divergence and Stoke's theorems in the 2/3 of the class and vector spaces; linear operators; algebra of matrices; systems of linear equations; eigenvalue problems in the last 1/3.

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These courses were the required courses for Science and Engineering students. They used Stewart Calculus as a textbook. For the Linear Algebra, they used Prof. Attila Aşkar's notes on Linear Algebra.

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While I was taking these courses, I thought that the second one was intense, actually. Anyway, the faculty has decided to make it three instead of two, which is better I think.

Answer 2

We are slightly similar in that we are trying to put lots of content into a smaller space with our Calc Sequence. We have 6 credits to do Calc 1, 2 and anything else vitally important that needs more than Calc 2, i.e. difference equations, partial derivatives and intro to differential equations. I contextualize most everything in the biological, environmental and social science as opposed to the strictly physics approach of many texts.

I am using a yet unpublished book by Gross, Bodine and Lenhart, so that won't help you, but another good one I've seen is MAA's Online calculus book by Smith.

I did a survey of faculty who had Calc as a program requisite or a course prereq and listed all different topics, followed by some explanation as to applications of the content. What it came down to was a preference for our case concept over computation, and an understanding of equilibria and modeling. We eliminated several proofs of derivative formulas as well as did not discuss several integration techniques. Skipped L'Hospital and convergence of series. What we DID get to that many don't, especially in a 3-3 sequence do was some intro programming, discrete and continuous population growth models, and SIR models.

I have taught Calc I and II in one 3 credit semester before using Cullen's Mathematics for Bioscience and it was rather accelerated. I highly recommend the use of a CAS for an accelerated sequence with less emphasis on hand calculation techniques.