By David Bressoud @dbressoud
As of 2024, new Launchings columns appear on the third Tuesday of the month.
FDWK refers to Ross Finney, Frank Demana, Bert Waits, and Dan Kennedy, the team of authors who for many years and multiple editions updated Cal culus: Graphical, Numerical, Algebraic, one of the most popular textbooks for the College Board’s Advanced Placement Calculus.
Single variable calculus has to contend with the problem that there is too much material for a single semester and too little to fill up two semesters. The common solution is to add a treatment of infinite series in the second semester. I am not convinced that students really learn anything useful about infinite series at the end of their year of calculus, but the College Board bows to what has become standard practice.
While I question the usefulness of an extended study of infinite series within the first year of calculus, Taylor polynomials do constitute a topic that belongs in the first year and is extremely important. After all, these are simply the natural extension of the central idea that the first derivative enables one to find a local linear approximation. The second derivative enables the construction of a local quadratic approximation, the third derivative a local cubic approximation, and so on. In fact, if students could come away with a solid understanding and working knowledge of Taylor polynomials and the Lagrange error bound, I would be very happy.
Our book follows the College Board Course and Exam Description (CED) as it starts the chapter on infinite series. We explain what is meant by a convergent as opposed to a divergent series, and then explore geometric series, a topic that may well have been covered in precalculus. What I do not like about the CED is that it then heads into eight lessons on the various tests of convergence. Only after they have thoroughly intimidated and confused students with these eight tests do they introduce Taylor polynomials, the Lagrange error bound, the radius of convergence, and general power series. I suspect this was done in the mistaken belief that power series, which are series of functions, must be more complicated than series of constants.
There is another way in which we chose to diverge from the CED. The last two units for College Board BC Calculus consist of Parametric Equations, Polar Coordinates, and Vector-Valued Functions as Unit 9 followed by Unit 10 on Infinite Sequences and Series. We chose to put infinite series before parametric, vector-valued, and polar functions. Both of these units are coming at the end of a long and packed year. Students know that there will be few questions on the exam based on these units, and there are a lot of unfamiliar ideas to be mastered. It would be nice if students would master calculus as applied to these multi-dimensional functions, but they are less likely to be emphasized on the test than Taylor polynomials and power series.
Regarding the Lagrange remainder, both the CED and our book simply state it as something to be memorized. I wish we could spend more time on it, emphasizing that this is simply an extension of the mean value theorem. I did write up a discussion of how this theorem can be approached for my Launchings column of January, 2025, but that write-up seems to have been lost. The interested reader can access it among my personal files at https://tinyurl.com/LagrangeError.
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org