Introduction to Continuous Mathematical Modeling

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Mastery Concept Check-list

This page will be the synopsis of skills tested in quizzes required for mastery in the course.

1. Each student will understand the relation between a growth rate function and the corresponding slope field.
2. Each student will understand the relation between a time-homogeneous growth rate function and the phase-line, including the stability of equilibria.
3. Each student will know how to determine the dimension of parameters to make a differential equation dimensionally homogeneous.
4. Each student will know how to nondimensionalize a single differential equation with no free parameters.
5. Each student will know how to nondimensionalize a single differential equation with a free parameter.
6. Each student will show the ability to determine values where the concavity of a solution to an autonomous differential equation changes.
7. Each student will know how to determine the stability of an equilibrium using the derivative of the growth rate function.
8. Each student will know how to sketch a simple bifurcation diagram involving one free parameter.
9. Each student will know how to determine the stability of branches on a bifurcation diagram, recognizing transcritical, saddle-node, and pitchfork bifurcations.
10. Each student will know how to compute and sketch nullclines for a system of two first-order differential equations, and identify equilibria and the the direction of motion in the phase plane.
11. Each student will know how to determine the stability of a linear system of two differential equations.
12. Each student will know how to compute and evaluate the Jacobian of a system of two nonlinear differential equations.
13. Each student will be able to sketch a phase-plane for a conservative one-dimensional system given a potential energy function.