Students are led step by step through the development of introductory ideas in mathematical modeling with differential equations. They will encounter the fundamental ideas underlying unlimited population growth (exponential models), limited population growth (logistic models), and a coupled system of differential equations modeling a predator-prey relationship. Furthermore, students will be reminded or introduced to key ideas such as particular and general solutions, interpretation of derivatives, qualitative analysis, graphical interpretation of P vs. t graphs as well as dP/dt vs. P graphs, phase lines, and phase planes. We use this activity on the first day of class, and its overall purpose is to:
- Get students introducing, talking, and working with one another.
- Set the tone for the level of qualitative analysis expected of students - no question is ``too basic" in their analysis of a situation and the mathematical translation thereof.
- Review ideas from Calculus (which they may not have seen for a year or more).
- Get students to shift their thinking to start with considering the rate of change instead of starting with a given function and differentiating it.
- Introduce students to the utility of differential equations.
- Introduce students to a few of the many tools that will be used throughout the course.
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