Math 112 module 8 discussion

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1 MATH 112 Module 6 discussion 𝑥 The function 𝑦 = 𝑥 2+1 + 1 can be used to model the spread of a contagious disease over time. The independent variable x represents the time in days since the onset of the epidemic and the dependent variable y represents the proportion of infected individuals in a population. The practical interpretation of the function is in the range of x that represents the progression of time during the outbreak of the epidemic and its effect on the proportion of infected individuals in the population. In the real world, this would represent when the epidemic is actively spreading. −𝑥 2 +1 The first derivation of the function is 𝑦 ′ = (𝑥 2+1)2 and its graph is as below: The first derivative of the function represents the rate of change of the proportion of infected individuals with respect to time. The graph of the first derivative shows that the rate of change of the proportion of infected individuals is highest when x = 0 which indicates that the spread of the disease is at its peak during this time. The rate of change decreases as x increases which suggests the spread of the disease slows over time. In the context of the model, the contagious disease is spreading rapidly during the early stages with the rate of new infections slowing down with time. This aligns with real-world epidemic situations where the intensity of initial outbreaks is high before slowing down over time. 2 To find the values for which the first derivative of the function is 0, we replace y’ with 0; −𝑥 2 + 1 =0 (𝑥 2 + 1)2 −𝑥 2 = −1 𝑥2 = 1 Therefore, 𝑥 = √1 = 1, −1 The values of x for which the first derivative is 0 are 1 and -1. These values represent critical points in the spread of the disease. The rate of change of the proportion of infected individuals is zero. This indicates the turning points in the progression of the epidemic.

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