Module 22 - Differential Equations and Euler's Method | ||||||||||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test | ||||||||||||||||||
Lesson 22.1: Logistic Growth Revisited | ||||||||||||||||||
Module 5 contained an experiment that simulated the spread of a rumor through a small population. In this lesson you will revisit that data and solve the related differential equation. Modeling the Data The following data from a simulation gives the cumulative number of people who know a rumor as it spreads through a population of 25 people.
![]() [0.3, 8.7, 1] x [-3.08, 29.08, 10] The shape of this scatter plot is called a logistic curve. The differential equation that describes the rate at which the rumor spreads is where y is the number of people who know the rumor at time t. t is time measured in days, M is the maximum possible number of people who can know the rumor, and k is a constant. For this simulation, M is 25. You will use the "Guess and Check" method to find an approximation for k that fits the data by graphing solutions to the differential equation for different values of k. Use an initial guess of 0.05 for k.
![]() ![]() The solution to the differential equation with k = 0.05 has the correct general shape but does not fit the data very well. 22.1.1 Modify the value of k in Y1 and run program EULERG again to get a better fit to the scatter plot. Continue to modify k and run EULERG until the graph of the solution to the differential equation approximates the scatter plot well. What value of k seems to work best? Click here for the answer. |
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