Module 22  Differential Equations and Euler's Method  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
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 Y_{1} 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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