Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question

A) Determine the derivative of I

B) The approximate change in infected people is dI ____ people

C) The actual change in infected individuals is Delta (I) _____ people

### Mathematical Modeling of Infection Spread #### Infection Growth Function The function \( I(t) \) representing the number of infected individuals over time \( t \) (in days) is given by: \[ I(t) = \frac{1000}{1 + 999e^{-6t}} \] Here: - \( t \) is time in days. - \( I \) represents the number of infected individuals. #### Description of the Graph - The graph plots \( I(t) \) on the y-axis and \( t \) on the x-axis. - The y-axis ranges from 0 to 1000, representing the number of infections. - The x-axis ranges from 0 to 25, representing time in days. - The curve starts at the origin (approximately zero infected individuals at \( t = 0 \)). - There is a steep increase in infections between \( t = 5 \) and \( t = 15 \). - The curve then gradually levels off approaching 1000 infected individuals as \( t \) increases beyond 20 days, indicating a saturation point. #### Purpose The objective is to estimate the change in infections from \( t = 10 \) to \( t = 11 \) using differentials. This model and graph can help visualize how infections grow over time and assist in understanding the critical periods of infection growth, which is crucial for planning healthcare responses and interventions.
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Transcribed Image Text:### Mathematical Modeling of Infection Spread #### Infection Growth Function The function \( I(t) \) representing the number of infected individuals over time \( t \) (in days) is given by: \[ I(t) = \frac{1000}{1 + 999e^{-6t}} \] Here: - \( t \) is time in days. - \( I \) represents the number of infected individuals. #### Description of the Graph - The graph plots \( I(t) \) on the y-axis and \( t \) on the x-axis. - The y-axis ranges from 0 to 1000, representing the number of infections. - The x-axis ranges from 0 to 25, representing time in days. - The curve starts at the origin (approximately zero infected individuals at \( t = 0 \)). - There is a steep increase in infections between \( t = 5 \) and \( t = 15 \). - The curve then gradually levels off approaching 1000 infected individuals as \( t \) increases beyond 20 days, indicating a saturation point. #### Purpose The objective is to estimate the change in infections from \( t = 10 \) to \( t = 11 \) using differentials. This model and graph can help visualize how infections grow over time and assist in understanding the critical periods of infection growth, which is crucial for planning healthcare responses and interventions.
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Step 2: Approximate change in 'I' at t=10 to t=11
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Step 3: The actual change in I(t) from t= 10 to t=11
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