What is special about this Calculus course?

Calculus courses have been taught at universities around the world for hundreds of years. The teaching materials for calculus, from traditional textbooks to modern computer software, have been reinvented and refined over the years and have become classical and standard. Thus, the most challenging question for this project is: why do we need to develop a new calculus course? The straightforward answer is that although the basic concepts and techniques of calculus have not changed, many fields where mathematics is applied have developed and advanced, especially in the biological sciences, and most importantly the students have changed. All these changes have increased concerns over science, technology, engineering and mathematics (STEM) education [see Project Kaleidoscope (2006)]. The reforms in STEM education demand a redesign of foundation courses in mathematics, among which calculus is the key to quantitative analysis in sciences.

Although we can teach and learn calculus from the pure and abstract mathematical point of view, the general consensus is that the most efficient way to study/teach Calculus is connecting the mathematical concepts with their applications. Classical applications for teaching Calculus include: moving objects, free fall problems, optimization problems involving area or volume and interest rate problems. These examples have been proved to be very efficient for engineering students but not for the life science majors. We have developed a set of application examples for Calculus, which are more biology oriented. These include: growth/decay problems in any organism population, gene regulation and dynamical changes in biological events such as monitoring the change of patients’ temperature along with the medications. By using these examples, the students would feel the connection between mathematics and their major subjects. Consequently, they are more motivated to study Calculus.

Traditionally, the first Calculus course does not include exponential functions and logarithm functions. Because of the applications as mentioned above, it is essential for us to discuss these two functions in our first Calculus course. With careful planning, this is not difficult to do. In fact, this course could be more efficient than the traditional Calculus I.

The objective of the first semester calculus is to train the students in the basic concepts and techniques of calculus: limit, continuity, differentiation and integration. This course is important because it transitions from high school mathematics to higher mathematical thinking with analytical rigor. It is also important because of its wide applicability in many fields, from science and engineering to economics and social science, allowing students to broaden their horizons of investigation and career options. We believe that most of the students would learn calculus well if they were motivated by the prospective usefulness of calculus in their future studies and careers. They would also appreciate mathematics more if they felt that they were connected with the applications as well as the theories. However, the traditional first-semester calculus focuses on applications in mechanics and physics. Although calculus textbooks nowadays contain some problems in economics and business, chemistry and biology applications are rare and instructors usually do not mention them at all in class, being somewhat unfamiliar with those fields. We will design a new first-semester calculus course which would break this tradition and contain a balanced set of application examples in biology, chemistry, economics and physics.  This will then serve as a gateway course for students from all fields so that they can have a broader view about calculus.

 
Michaelis-Menten
Figure 1. Plot of a Michaelis-Menten function. This function is always increasing and concave down. It has a horizontal asymptote, y=4.

For this part, we will cover all the theories and techniques that are covered in the traditional calculus-I course. Unlike in the traditional calculus-I course where most  of application problems taught are physics problems, we will carefully choose a mixed set of examples and homework problems to demonstrate the importance of calculus in biology, chemistry and physics, but emphasizing the biology applications.

Example 1. Traditionally, the first application discussed in Calculus I is the distance/velocity/acceleration problem for moving objects including the free-fall problem. For our Bio-enriched Calculus I, we will consider the Michaelis-Menten kinetics function [4][9]:

Equation 1

 

This function has many applications in biological fields. For example, it can be used for modeling in enzyme reaction or population growth. Here n could be the nutrient concentration and f  be the growth rate function for bacteria; Kmax and Kn are positive constant parameters standing for maximum growth rate and the nutrient density at which the bacteria growth rate reaches Kmax /2. This example can be used to introduce the dependence on nutrient as the first derivative and the acceleration (deceleration) of it as the second derivative. In the later discussions of related rates, we can revisit this example for the relationship of two time dependent functions, u(t) and n(t):

Equation 2

where u(t) and n(t) are bacteria density and nutrient concentration as functions of time, t.
Graphing of the Michaelis-Menten kinetics function can be one stone for two birds: using graphing techniques with derivatives and showing the biological significance of the two parameters Kmax and Kn (Figure 1).

Example 2. (Example given in [2] adapted from [1]) Ichthyosaurs are a group of marine reptiles that were fish-shaped and comparable in size to dolphins. They became extinct during the Cretaceous. Based on a study of 20 fossil skeletons, it was found that the skull length (in cm) and backbone length (in cm) of an individual were related through the allometric equation:

Equation 3

where S(x) is the skull length and B(x) is the backbone length at age x. After differentiation on both sides of the equation and a couple of manipulation steps, we end up with the equation:

Equation 4

The first equation gives the relationship between S(x) and B(x). However, it is the second equation that clearly shows that the backbone grows faster than the skull. This example contains several basic calculus concepts and techniques, derivative, power chain rule, relative growth rates and related growth rates. Plus it stirs the students’ curiosity with questions like why babies always seem to have big heads.

Although all application examples of calculus are interesting in some way, examples from microbiology and paleontology as given above are certainly more fascinating to the students in life sciences. Throughout the course, we will carefully integrate the application examples with the calculus concepts and techniques. By the end of the semester, we have two missions to complete: a solid introduction to calculus with rigorous standards of understanding and mastery, and building a real bridge between mathematics and life sciences.

References:
[1] Benton, M. J. and Harper, D (1997), Basic Paleomtology. Addison Wesley and Longman.

[2] Neuhauser, C. (2004), Calculus for Biology and Medicine, 2nd edition, Pearson Education, Inc..

[3] Project Kaleidoscope (2006), Transforming America’s Scientific and Technological Infrastructure: Recommendations for Urgent Action. Project Kaleidoscope, Washington, DC. http://www.pkal.org/documents/ReportOnReportsII.cfm.