Calculus II: Modeling With Differential Equations
An infectious disease spreads through a community: What is the most effective action to stop an epidemic? Populations of fish swell and decline periodically: Should we change the level of fishing allowed this year to have a better fish population next year? Foxes snack on rabbits: In the long term, will we end up with too many foxes or too many rabbits? Calculus can help us answer these questions. We can make a mathematical model of each situation, composed of equations involving derivatives (called differential equations). These models can tell us what happens to a system over time which, in turn, gives us predictive power. Additionally, we can alter models to reflect different scenarios (e.g., instituting a quarantine, changing hunting quotas) and then see how these scenarios play out. The topics of study in Calculus II include power series, integration, and numerical approximation, all of which can be applied to solve differential equations. Our work will be done both by hand and by computer. Conveniently, learning the basics of constructing and solving differential equations (our first topic of the semester) includes a review of Calculus I concepts. Conference work will explore additional mathematical topics. This seminar is intended for students planning further study in mathematics or science, medicine, engineering, economics, or any technical field, as well as students who seek to enhance their logical thinking and problem-solving skills. Prerequisite: Calculus I (differential calculus in either a high-school or college setting).