Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad942d81-d900-47bd-b390-ed511be993df%2F3232a0d2-98f8-49b4-99e7-72fdbe40e107%2Flvsd5u4_processed.jpeg&w=3840&q=75)
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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