(6) Show all your work. (a) In logistic growth, the rate of growth of a population, R depends on the population size N as follows: N 1-K). R = r N where r and K are positive constants. Find the rate of change of the growth rate with respect to the population size.
(6) Show all your work. (a) In logistic growth, the rate of growth of a population, R depends on the population size N as follows: N 1-K). R = r N where r and K are positive constants. Find the rate of change of the growth rate with respect to the population size.
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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![### Problem 6
**Show all your work.**
**(a)** In logistic growth, the **rate of growth** of a population, \( R \) depends on the population size \( N \) as follows:
\[ R = rN \left(1 - \frac{N}{K}\right), \]
where \( r \) and \( K \) are positive constants. Find the **rate of change** of the growth rate with respect to the population size.
---
**(b)** Let \( f(x) = \begin{cases}
\frac{3}{x} e^{x+1}, & x < 1 \\
2x + 3b, & x \ge 1
\end{cases} \).
Find the constant \( b \) such that the function \( f \) is
\[ x = 1 \]
\[ 1 = \int_0^1 R \cdot L = f(x) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cc04b1e-803c-4d9d-b550-d2ae04bcc2b3%2F53ac1ad6-5fa6-4e76-a026-bc6087a88a8b%2Fyrppb47_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 6
**Show all your work.**
**(a)** In logistic growth, the **rate of growth** of a population, \( R \) depends on the population size \( N \) as follows:
\[ R = rN \left(1 - \frac{N}{K}\right), \]
where \( r \) and \( K \) are positive constants. Find the **rate of change** of the growth rate with respect to the population size.
---
**(b)** Let \( f(x) = \begin{cases}
\frac{3}{x} e^{x+1}, & x < 1 \\
2x + 3b, & x \ge 1
\end{cases} \).
Find the constant \( b \) such that the function \( f \) is
\[ x = 1 \]
\[ 1 = \int_0^1 R \cdot L = f(x) \]
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