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.