Wolfram Computation Meets Knowledge

College Calculus with Mathematica

Calculus has occupied a central position in scientific thought ever since its discovery by Newton and Leibniz more than 300 years ago. The combination of elegance, utility, and rigor that characterize this subject have led to its extensive use in theoretical approaches to diverse fields such as economics, finance, and biology. Indeed, calculus is one of the greatest intellectual achievements of humankind, which explains its important role in the training of students all over the world.

It has been my privilege to present a “College Calculus with Mathematica talk as part of a series of free online seminars organized by the Wolfram Education Group. Today, I would like to give you a glimpse of the seminar’s topics and write about the advantages of online instruction.

As a Mathematica kernel developer for calculus functions, I have collected many fascinating examples that illustrate the world-class functionality for computing integrals, solving differential equations, and other operations available in Mathematica. I have always been deeply interested in education, and taught high school and college mathematics for several years. It was, therefore, natural for me to spend some time in comparing the Mathematica algorithms for calculus functions with the methods for hand computation that are taught to students in the classroom.

When Tom Wickham-Jones, Director of Kernel Technology, suggested that I present the online seminar, I wrote down a list of objectives that would help in conveying some of the beauty and utility of calculus, as seen through the eyes of Mathematica, to a wide audience of students, educators, and others.

The first objective of the seminar is to show how to use functions such as Integrate, DSolve, and Sum in an effective manner. In order to do this, a three-step routine is repeated several times during the talk. The first step is to examine the structure of the input carefully. For instance, the following input deals with the integration of a rational function (a fraction in which the numerator and the denominator are polynomials).

In[1]:= RationalIntegral=\[Integral](28x^4+70x^3+83x^2+49x+11)/((7x+4)(2x^2+3x+1)^2) \[DifferentialD]x

The next step is to ask the question, “Does the answer returned by Mathematica seem plausible?” Well, the result of the above integral has two parts, a rational part (like the input) and a logarithmic part, as one would expect from the theory of these integrals.

The final step is to verify the answer—usually very easy to do in Mathematica. As seen below, after differentiation using D we do get back the rational function initially input in this example.

In[2]:= Together[D[RationalIntegral,x]]

Another recurring theme during the seminar is to combine symbolic computation with Mathematica‘s spectacular capabilities for graphics and dynamic interactivity to gain further insight into calculus. For example, the following system of differential equations can be solved in terms of Fresnel integrals.

In[3]:= sol=DSolve[{x^\[Prime](s)\[LongEqual]sin(t(s)),y^\[Prime](s)\[LongEqual]cos(t(s)),t^\[Prime](s)\[LongEqual]s,x(0)\[LongEqual]0,y(0)\[LongEqual]0,t(0)\[LongEqual]0},{x,y,t},s]

These differential equations define a Cornu spiral, and a plot of the solution clearly reveals that the spiral has infinite curvature as one approaches infinity in both the positive and the negative directions.

In[4]:= ParametricPlot[Evaluate[{x(s),y(s)}/.sol[[1]]],{s,-20,20},PlotStyle->Red,Background->Yellow]

The second objective of the seminar is to illustrate how calculus has been used to solve problems arising from the real world. The application examples were chosen based on their importance in the historical evolution of the subject and the fundamental simplicity of the relevant mathematical ideas. Above all, I wanted to show how students with a basic knowledge of calculus can understand some of the greatest advances in scientific knowledge.

Thus, the seminar recalls the discovery of solitary waves by John Scott Russell, the work of Karl Weierstrass on minimal surfaces, the ideas of Paul Painlevé on nonlinear differential equations, and the pioneering research of the female mathematician Sofia Kovaleskaya. This portion of the seminar, too, includes animations and graphics, such as the plot of the Costa surface shown below.

Plot of the Costa surface

There is an impressive balance between theoretical concepts and computational techniques in the present curriculum for college calculus. The standard sequence of calculus courses provides excellent training for applying the subject in practice. However, it appears to me that, due to the large number of topics that need to be covered, learning often proceeds at a breathless pace, leaving little time for students to reflect on calculus as a human endeavor. I feel that including one or two applications of the type discussed in the seminar, perhaps as Mathematica laboratory assignments, will give students a deeper appreciation of calculus and make it seem even more worthwhile for them to have spent time in mastering the technical aspects of the subject.

A third major seminar objective is to compare the evaluation of integrals, differential equations, and infinite series in Mathematica with the methods used in hand calculation. In my opinion, there is a great deal of similarity at the conceptual level between these seemingly different approaches to computation. For example, the evaluation of the following exponential integral can be understood using the standard college-calculus tricks of variable substitution and partial fractions. In turn, some of these ideas also underlie the Risch algorithm, which is used in Mathematica for indefinite integration.

\[Integral](E^x (E^x x-8 E^(2 x) x-15 E^(3 x) x-3 E^(4 x) x+3 E^(5 x) x+4 x-5 E^x-E^(2 x)+E^(3 x)-3))/((-3+E^x) (1+E^x)^2 x) \[DifferentialD]x

Two other important objectives for the seminar are to give attendees a tour of Mathematica‘s documentation and to emphasize the importance of discrete notions in calculus. Indeed, Mathematica 7 includes state-of-the-art tools for discrete calculus operations such as symbolic summation, sequence recognition, and solving difference equations. For example, one can use the new RecurrenceTable function to study the fractal dynamics of Douady’s rabbit, pictured below.

Douady's Rabbit Fractal

I have been pleasantly surprised by the popularity of the calculus seminar, and presenting it has been a source of great happiness and inspiration for me. Overall, the audiences seem to enjoy the combination of graphics, symbolics, and historical snippets, and their constructive feedback has given us future directions for online calculus seminars. I have learned a lot from discussions with my senior colleagues Paul Wellin and Harry Calkins, who have a profound understanding of how Mathematica relates to education.

The seminar series has taught us about the immense benefits of online instruction. It is clearly advantageous to deliver talks without any travel, and we have been able to keep the show going even under severe winter weather conditions here in Champaign, Illinois. It is a thrilling experience to interact with people from around the world and respond to their questions about Mathematica. The seminar series has now been attended by people from about 100 countries—colored blue in the following map, prepared using CountryData (darker shades of blue indicate countries with more attendees).

Seminar series attendance

Of course, whether it is online or otherwise, teaching at any level requires a substantial amount of preparation. For me, this means four or five quick rehearsals using the seminar notebooks in my office during the week before each seminar, followed by a complete practice run-through in our multimedia room on the day before the seminar, and an audio check around four minutes before the seminar finally begins with “Hello! Welcome to College Calculus with Mathematica!”

The free online seminars are offered on a regular basis, at different hours to facilitate attendance by people living in different time zones. The seminar calendar gives details about upcoming sessions, and it is easy to register for one of these events from that page. I invite you to attend a College Calculus seminar soon, if you have not done so already.