5. In some biological applications, population size P as a function of time t may be modeled as A 1+ Be-ct* where A, B, c> 0. Evaluate the following. P (t) (a) P (0) (b) lim P (t)
5. In some biological applications, population size P as a function of time t may be modeled as A 1+ Be-ct* where A, B, c> 0. Evaluate the following. P (t) (a) P (0) (b) lim P (t)
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 Population Size in Biological Applications
In certain biological applications, the population size \(P\) as a function of time \(t\) can be modeled using the following equation:
\[
P(t) = \frac{A}{1 + Be^{-ct}}
\]
where \(A\), \(B\), and \(c\) are positive constants (\(A, B, c > 0\)). The task is to evaluate the following:
1. \( P(0) \)
2. \( \lim_{{t \to \infty}} P(t) \)
### Key Points to Consider:
- \( P(t) \) represents the population size at time \( t \).
- \( A \), \( B \), and \( c \) are parameters that affect the shape and behavior of the population curve.
#### (a) Evaluate \( P(0) \)
To find the population size at time \( t = 0 \):
Substitute \( t = 0 \) into the equation:
\[
P(0) = \frac{A}{1 + Be^{-c \cdot 0}}
\]
Simplify the exponent:
\[
P(0) = \frac{A}{1 + B \cdot 1}
\]
Thus:
\[
P(0) = \frac{A}{1 + B}
\]
#### (b) Evaluate \( \lim_{{t \to \infty}} P(t) \)
To find the population size as time \( t \) approaches infinity:
Evaluate the limit as \( t \to \infty \):
\[
\lim_{{t \to \infty}} P(t) = \lim_{{t \to \infty}} \frac{A}{1 + Be^{-ct}}
\]
As \( t \to \infty \), \( e^{-ct} \to 0 \) (since \( c > 0 \)):
\[
\lim_{{t \to \infty}} P(t) = \frac{A}{1 + B \cdot 0}
\]
Thus:
\[
\lim_{{t \to \infty}} P(t) = \frac{A}{1} = A
\]
### Summary
- The population size at time \( t = 0 \) is \( P(0) = \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6484a2ba-7b6c-4144-bd9b-1f2d784a131b%2F40628f8a-1c51-4ec6-89e3-c403cdb79a38%2F7rvh8p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Modeling Population Size in Biological Applications
In certain biological applications, the population size \(P\) as a function of time \(t\) can be modeled using the following equation:
\[
P(t) = \frac{A}{1 + Be^{-ct}}
\]
where \(A\), \(B\), and \(c\) are positive constants (\(A, B, c > 0\)). The task is to evaluate the following:
1. \( P(0) \)
2. \( \lim_{{t \to \infty}} P(t) \)
### Key Points to Consider:
- \( P(t) \) represents the population size at time \( t \).
- \( A \), \( B \), and \( c \) are parameters that affect the shape and behavior of the population curve.
#### (a) Evaluate \( P(0) \)
To find the population size at time \( t = 0 \):
Substitute \( t = 0 \) into the equation:
\[
P(0) = \frac{A}{1 + Be^{-c \cdot 0}}
\]
Simplify the exponent:
\[
P(0) = \frac{A}{1 + B \cdot 1}
\]
Thus:
\[
P(0) = \frac{A}{1 + B}
\]
#### (b) Evaluate \( \lim_{{t \to \infty}} P(t) \)
To find the population size as time \( t \) approaches infinity:
Evaluate the limit as \( t \to \infty \):
\[
\lim_{{t \to \infty}} P(t) = \lim_{{t \to \infty}} \frac{A}{1 + Be^{-ct}}
\]
As \( t \to \infty \), \( e^{-ct} \to 0 \) (since \( c > 0 \)):
\[
\lim_{{t \to \infty}} P(t) = \frac{A}{1 + B \cdot 0}
\]
Thus:
\[
\lim_{{t \to \infty}} P(t) = \frac{A}{1} = A
\]
### Summary
- The population size at time \( t = 0 \) is \( P(0) = \frac{
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