Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Modeling Research Article Production Over Time** The number of research articles in a prominent journal authored by researchers in Europe can be modeled by the function: \[ P(t) = \frac{7.0}{1 + 5.4(1.2)^{-t/2}} \] where \( t \) is time in years. Presented here are the graphs for \( P \), \( P' \), and \( P'' \). **Graph Descriptions:** 1. **Graph of \( P \):** - The graph shows a sigmoid curve, increasing over time. - Initially, growth starts slow, then accelerates, and levels off again. 2. **Graph of \( P' \):** - The derivative of \( P \), representing the rate of change, forms a bell-shaped curve. - It peaks at around \( t = 15 \) and decreases thereafter. 3. **Graph of \( P'' \):** - The second derivative graph depicts an oscillating curve crossing the axis. - Indicates a point of inflection where the graph shifts from concave up to concave down. **Inflection and Concavity Analysis:** - **Concave Up:** \( 0 < t < 18 \) - **Concave Down:** \( 18 < t < 40 \) - **Point of Inflection:** \( t = 18 \) **Interpretation:** The point of inflection at \( t = 18 \) signifies a pivotal moment in article growth dynamics. According to the graph of \( P' \), the derivative peaks, suggesting maximum growth rate for article production at \( t = 15 \). This means the rate of increase in articles authored by European researchers was greatest around this time.
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Transcribed Image Text:**Modeling Research Article Production Over Time** The number of research articles in a prominent journal authored by researchers in Europe can be modeled by the function: \[ P(t) = \frac{7.0}{1 + 5.4(1.2)^{-t/2}} \] where \( t \) is time in years. Presented here are the graphs for \( P \), \( P' \), and \( P'' \). **Graph Descriptions:** 1. **Graph of \( P \):** - The graph shows a sigmoid curve, increasing over time. - Initially, growth starts slow, then accelerates, and levels off again. 2. **Graph of \( P' \):** - The derivative of \( P \), representing the rate of change, forms a bell-shaped curve. - It peaks at around \( t = 15 \) and decreases thereafter. 3. **Graph of \( P'' \):** - The second derivative graph depicts an oscillating curve crossing the axis. - Indicates a point of inflection where the graph shifts from concave up to concave down. **Inflection and Concavity Analysis:** - **Concave Up:** \( 0 < t < 18 \) - **Concave Down:** \( 18 < t < 40 \) - **Point of Inflection:** \( t = 18 \) **Interpretation:** The point of inflection at \( t = 18 \) signifies a pivotal moment in article growth dynamics. According to the graph of \( P' \), the derivative peaks, suggesting maximum growth rate for article production at \( t = 15 \). This means the rate of increase in articles authored by European researchers was greatest around this time.
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