18. Public awareness of a new drug is modeled by 5.2t P(t) = +0.18 0.015t2 +0.342 where t is the number of months after FDA approval and P(t) is the fraction of people who are aware of the drug and its possible uses. a. Find the critical points for P(t). b. Sketch the graph of P(t). c. At what time, t, during the time interval 0
18. Public awareness of a new drug is modeled by 5.2t P(t) = +0.18 0.015t2 +0.342 where t is the number of months after FDA approval and P(t) is the fraction of people who are aware of the drug and its possible uses. a. Find the critical points for P(t). b. Sketch the graph of P(t). c. At what time, t, during the time interval 0
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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Question 18
![### Text Transcription for an Educational Website
#### Population Dynamics and Mathematical Models
**Graph Explanation:**
The graph shown is from a study by T.S. Bellows, titled "Descriptive Properties of Some Models for Density Dependence," published in the *Journal of Animal Ecology* in 1981. The graph depicts the dynamics of a population based on a mathematical model, with a curve illustrating changes over a particular range.
**12. Population Model Dynamics:**
- **Equation:** \( x_{n+1} = r f(x_n) \)
If we assume that each adult produces \( r \) eggs, then the population dynamics are represented by this equation.
- **Tasks:**
- a. Determine the equilibria and their stability for values \( r = 2, 4, 6 \).
- b. Simulate the model with \( x_0 = 0.1 \) for \( r = 2, 4, 6 \).
**15. Model for Eastern Pacific Yellowfin Tuna:**
- **Growth Model:**
\[
G(N) = 2.61N \left(1 - \left(\frac{N}{148}\right)^\theta\right)
\]
\( N \) = population size in thousands of tons, \( t \) = years, and \( \theta \) > 0 determines density dependence strength.
- **Tasks:**
- Determine the population size at maximum sustainable harvest rate.
- Discuss the effect of \( \theta \) on population size.
**17. Biochemical Switch Model:**
- **Equation:**
\[
x_{n+1} = \frac{3x_n^2}{1 + x_n^2}
\]
\( x_n \) = concentration of a biochemical at time \( n \).
- **Tasks:**
- a. Find equilibria.
- b. Verify \( f(x) = \frac{3x^2}{1 + x^2} \) is increasing for \( x > 0 \).
- c. Discuss solutions to the equation.
**18. Public Awareness of a New Drug:**
- **Model:**
\[
P(t) = \frac{5.2t}{0.015t^2 + 0.342} + 0.18
\]
\( t \) = months after](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4db1e84-75df-498d-9686-5cabfd0e0c2f%2F3f79470f-c2cd-4a9d-b845-86c074be4d11%2Fnz59zbf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Text Transcription for an Educational Website
#### Population Dynamics and Mathematical Models
**Graph Explanation:**
The graph shown is from a study by T.S. Bellows, titled "Descriptive Properties of Some Models for Density Dependence," published in the *Journal of Animal Ecology* in 1981. The graph depicts the dynamics of a population based on a mathematical model, with a curve illustrating changes over a particular range.
**12. Population Model Dynamics:**
- **Equation:** \( x_{n+1} = r f(x_n) \)
If we assume that each adult produces \( r \) eggs, then the population dynamics are represented by this equation.
- **Tasks:**
- a. Determine the equilibria and their stability for values \( r = 2, 4, 6 \).
- b. Simulate the model with \( x_0 = 0.1 \) for \( r = 2, 4, 6 \).
**15. Model for Eastern Pacific Yellowfin Tuna:**
- **Growth Model:**
\[
G(N) = 2.61N \left(1 - \left(\frac{N}{148}\right)^\theta\right)
\]
\( N \) = population size in thousands of tons, \( t \) = years, and \( \theta \) > 0 determines density dependence strength.
- **Tasks:**
- Determine the population size at maximum sustainable harvest rate.
- Discuss the effect of \( \theta \) on population size.
**17. Biochemical Switch Model:**
- **Equation:**
\[
x_{n+1} = \frac{3x_n^2}{1 + x_n^2}
\]
\( x_n \) = concentration of a biochemical at time \( n \).
- **Tasks:**
- a. Find equilibria.
- b. Verify \( f(x) = \frac{3x^2}{1 + x^2} \) is increasing for \( x > 0 \).
- c. Discuss solutions to the equation.
**18. Public Awareness of a New Drug:**
- **Model:**
\[
P(t) = \frac{5.2t}{0.015t^2 + 0.342} + 0.18
\]
\( t \) = months after
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