[MOSAIC-news] Friday's Project MOSAIC M-CAST (AntiD operator)

Nicholas Horton nhorton at smith.edu
Wed Dec 8 09:42:06 CST 2010

Friday's Project MOSAIC M-CAST
Dec. 10, 12:30 CST (1:30 EST or 10:30 PST)

Presenter: Danny Kaplan (Macalester College)

The AntiD Operator

Abstract: One of the most beautiful notations in mathematics is the sweeping
S-curve of the integral sign.  But carrying out the operation itself is
challenging.  There are substantially different procedures for
similar-looking functions, the results are often elaborate and complicated,
and many basic functions don't have any integral that can be written with
elementary functions.

Integration is an important technique in many branches of science and
technology.  To make it accessible to students, particularly when
introducing it in a first calculus course, it would be nice to be able
side-step the algebraic difficulties of integration, presenting it as a
unified technique that can be explored and applied.

I'll present a way to do this, a computer operator that I call antiD.  It
carries out integrals quantitatively, returning an anti-derivative function.
It involves hardly any algebra.  Since the output of antiD is a function,
you can differentiate it, you can integrate it, you can evaluate it.  It
works with just about any function you like, not just the standard forms of
traditional calculus, but splines and other piecewise-defined functions,
sigmoidals, the Gaussian, ... whatever.

I'll show some ways to use antiD, both to illustrate concepts such as the
fundamental theorem of calculus and to help students see what integration is
about and how it connects to modeling.

At Macalester, we use R to teach the introductory calculus course. For that
reason, I've implemented antiD in the R language, but it could be just as
easily implemented in Matlab or any of the computer algebra systems such as
Mathematica, Maple or Sage.

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MOSAIC M-Casts are held the 2nd and 4th Friday of each month.  See the
upcoming M-Casts and recordings of past M-Casts on our schedule


Project MOSAIC is a community of educators working to develop a new way   to
introduce mathematics, statistics, computation and modeling to students in
colleges and universities.
Our goal: Provide a broader approach to quantitative studies that provides
better support for work in science and technology.  The focus of the project
is to tie together better diverse aspects of quantitative work that students
in science, technology, and engineering will need in their professional
lives, but which are today usually taught in isolation, if at all.
* Modeling. The ability to create, manipulate and investigate  useful and
informative mathematical representations of a real-world situations.
* Statistics. The analysis of variability that draws on our ability to
quantify uncertainty and to draw logical inferences from observations and
* Computation.  The capacity to  think algorithmically, to manage data on
large scales, to visualize and interact with models, and to automate tasks
for  efficiency, accuracy, and reproducibility.
* Calculus. The traditional mathematical entry point for college and
university students and a subject that still has the potential to provide
important insights to today's students.
The name MOSAIC reflects the first letters --- M, S, C, C --- of these
important components of a quantitative education. Project MOSAIC is
motivated by a vision of quantitative education as a mosaic where the basic
materials come together to form a complete and compelling picture.

More information can be found at: www.mosaic-web.org

Nicholas Horton 
Department of Mathematics and Statistics, Smith College
Clark Science Center, Northampton, MA 01063-0001

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