Math 233-234: A Modeling Approach to Calculus

Experimental Course Overview

At James Madison University, the traditional calculus track suitable for STEM majors has two pathways. Math 235 is the one-semester, traditional approach to a first-semester university course in calculus. Over 10 years ago, JMU implemented a second track, Math 231-232, that is a two-semester sequence that integrates the content of first-semester calculus with much of the prerequisite knowledge students need to succeed in calculus. Rather than take a semester of precalculus followed by calculus, this course sequence was designed to review the pre-calculus just in time for the calculus where that material would be needed.

The proposed sequence, Math 233-234, aims to accomplish a similar objective to Math 231-232, namely learn the necessary content of first-semester calculus over the course of two semesters, but with a completely different philosophy. Rather than focus on mathematics for the sake of learning mathematics, we focus on the use of mathematics to model dynamics in our world to motivate the development and use of mathematics. Applications, interpretation and communication of mathematical ideas are emphasized, connecting the mathematics with other scientific fields.

Dr. Brian Walton previously experimented with components of this approach in several past sections of Math 231 and Math 232. This will be the first time the approach can be pursued for the full course. Multiple sections of Math 233 will be offered in Fall 2015 and multiple sections of Math 234 will be offered in Spring 2016 to complete the sequence. For additional information, please contact Dr. Walton.

Course Description/Invitation to Learn

In 1623, Galileo wrote that the book of nature — the universe — is written in the language of mathematics. Physics and chemistry each have a long history with mathematics. Increasingly, the earth and life sciences also require strong mathematical thinking. In 2003, Dr. Rita Colwell, Director of the National Science Foundation, stated:

Where mathematics and computing meet biology, they serve as biology’s new microscope, illuminating otherwise invisible entities. Math and computing shed light on bioscience problems that are too big (the biosphere), too slow (evolution), too remote in time (early extinctions), too complex (the brain), or exceed our capabilities in other ways.
( https://www.nsf.gov/news/speeches/colwell/rc030627gwubiomedical.htm, accessed June 2014)

Why is mathematics the universal language of science?

I invite you to begin an examination of the world through the lens of modeling and computation. I hope that your experiences will prepare you to perceive measurements as the source of discovering predictable patterns, to appreciate the power of dynamic models (i.e., models of change) in prediction, to employ productive problem-solving strategies, and to think critically about assumptions underlying mathematical predictions relevant to you individually or as an informed member of society.

(Proposed) Catalog Description

MATH 233*-234. Modeling Approach to Calculus A/B. (3 credits)
Two-semester sequence applying static and dynamic models in the context of scientific applications. Includes development and application of differential and integral calculus. Discrete and continuous models involving functions, difference equations and differential equations. Prerequisite: MATH 155, MATH 156 or sufficient score on the Mathematics Placement Exam. MATH 233-234 together are equivalent to MATH 235 for all prerequisites. Not open to students who have already earned credit in MATH 232 or MATH 235.

Course Objectives

Students will be able to …

• Translate observations and assumptions about the relations between physical variables into static or dynamic mathematical models.
• Select and apply the appropriate tools of algebra and calculus (i.e., limits, derivatives and integrals) in the analysis and interpretation of models.
• Integrate communication about mathematical models and analysis in the context of real-world situations.
• State the basic definitions and theorems of calculus and characterize the deductive relationships between them.

At a more refined level, students will:

• Create and interpret graphical representations of data and relationships between vari- ables.
• Interpret variables in the context of measurements, including units and dimensions.
• Understand and apply the relationship between an equation and a graph.
• Apply basic functional models for classic relationships: linear, exponential, power laws, and periodic (trigonometry) functions
• Apply properties of logarithms to explore multiplicative (exponential and power law) relationships, including log-log and semi-log plots
• Use functions as a model for interpolation and extrapolation.
• Consider sequences as a discrete time model of a dynamical system.
• Develop models of population growth using an additive model of contributions and multiplicative models of interactions and density dependence.
• Interpret limiting behaviors of sequences in the context of equilibrium or unbounded growth.
• Properties of limits of sequences
• Determine and interpret the stability of equilibria.
• Find a linear approximation for simple algebraic functions at a point (e.g., for stability analysis) – essentially definition of derivative without limits
• Apply the limit definition of derivative and the rules of differentiation
• Interpret the derivative as a rate of change, and apply rules of derivatives outside of the context of formulas, including related rates.
• Interpret inequalities involving the derivative regarding monotonicity and concavity
• Develop and interpret differential equations as models
• Perform basic qualitative analysis of nonlinear differential equations, including identi- fication of equilibria and their stability.
• Use derivatives to find extreme points (optimization and points of inflection)
• Understand the definite integral as accumulated change given a rate of change, leading to the Fundamental Theorem of Calculus.
• Interpret the definite integral as area.

Examples of modeling situations might include:

• Proportional and power relations (density vs volume, area of similar shapes, allometric relations)
• Exponential relations (populations, money)
• Periodic relations (tides, monthly temperatures)
• Modeling populations with simple growth, migration, nonlinear growth (discrete and continuous)
• Drug dosage and metabolic clearance
• Rate of flow in/out of reservoir, evaporation

Textbook

Calculus for the Life Sciences by S. Schreiber, K. Smith and W. Getz (Wiley, 2014). A custom printing (excluding Chapter 7) will be used to reduce student textbook costs.