Unit 7 Summary

Differential Equations

I can …

  • Recognize a differential equation.
  • Test whether a given function is a solution to a given differential equation.
  • Solve a simple differential equation of the form \(\frac{dy}{dx} = f(x)\) using antiderivatives.
  • Explain why a differential equation has an infinite family of solutions.
  • Use an initial condition to turn a general solution into the unique solution to the differential equation.
  • Create a differential equation to model a written description of the rate of change.

Population Models

I can …

  • Solve the differential equation \(\frac{dP}{dt} = rP\) using \(P(t)=Ce^{rt}\).
  • Solve the differential equation \(\frac{dP}{dt} = r(P-A)\) using \(P(t)=A+Ce^{rt}\).
  • Solve the differential equation \(\frac{dP}{dt} = rP\left( 1- \frac{P}{K}\right)\) using the logistic function \(P(t)=\frac{K}{1+Ce^{-rt}}\).
  • Find the constant \(C\) in each of the above functions using an initial condition.
  • Solve a problem using Newton’s Law of Heating and Cooling.

Euler’s Method

I can …

  • Explain what Euler’s Method is, and why we must use numerical approximation to solve many differential equations.
  • Manually perform a few steps of Euler’s Method.
  • Use RStudio to find an Euler’s Method solution to a differential equation with initial conditions.

Slope Fields

  • Explain what a slope field is.
  • Draw trajectories on a slope field and describe the dynamics of those solution curves.
  • Use a slope field to determine the long-term behavior of a solution curve.
  • Match a slope field to its differential equation.

SIR Model

  • Describe the Susceptible-Infected-Removed model for the spread of disease.
  • Explain the various terms in the differential equations of the SIR model.
  • Define the infection rate \(a\) and the removal rate \(b\).
  • Create an SIR slope field using RStudio, and interpret what I see.
  • Create an SIR trajectory plot using RStudio, and interpret what I see.
  • Explain what the epidemic phase of a disease outbreak is
  • Identify the threshold population by looking at an SIR slope field, looking at an SIR trajectory, or calculating \(b/a\) in the SIR differential equations
  • Describe the effect of various actions to mitigate the spread of disease, and how that changes the SIR Model.