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Chapter 4: Problem 5

Investigate the effect of the parameter \(C\) on the logistic curve $$ P=\frac{10}{1+C e^{-t}} $$ Substitute several values for \(C\) and explain, with a graph and with words, the effect of \(C\) on the graph.

Short Answer

Expert verified
Higher values of C result in a steeper and faster initial rise in the logistic curve.

Step by step solution

01

Understanding the Logistic Curve

In the logistic curve equation \( P = \frac{10}{1 + C e^{-t}} \), \( P \) represents the population or quantity of interest over time \( t \), and \( C \) is a parameter that influences the shape of the curve. It affects how quickly \( P \) approaches its maximum value, which is 10 in this case. The function has a characteristic S-shaped curve.
02

Substituting Values for C

Let's substitute different values of \( C \) into the logistic curve equation and observe the changes. We'll consider \( C = 0.5, 1, 2, 5 \). This will help us understand how \( C \) affects the steepness and position of the curve.
03

Calculating the Curve for C = 0.5

For \( C = 0.5 \), the equation becomes \( P = \frac{10}{1 + 0.5 e^{-t}} \). This creates a curve that rises quickly and approaches its maximum at a relatively slow rate compared to higher values of \( C \).
04

Calculating the Curve for C = 1

With \( C = 1 \), the equation is \( P = \frac{10}{1 + e^{-t}} \). Here, the growth is moderate, and \( P \) approaches the maximum value of 10 at a typical rate without too much early steepness or slowness.
05

Calculating the Curve for C = 2

When \( C = 2 \), the formula becomes \( P = \frac{10}{1 + 2e^{-t}} \). The curve now rises more steeply early on and reaches closer to 10 more quickly than when \( C = 1 \). It indicates a faster initial growth rate.
06

Calculating the Curve for C = 5

For \( C = 5 \), the equation is \( P = \frac{10}{1 + 5e^{-t}} \). This results in a very steep rise after a slow start, quickly reaching close to the maximum value, which demonstrates the effect of a large \( C \) value on increasing early growth rates.
07

Analyzing Results

As \( C \) increases, the logistic curve's growth becomes faster initially, reaching close to the maximum value quicker. For smaller values of \( C \), the growth is slower, and the curve takes more time to approach 10. This shows \( C \) alters how sharply the curve turns (inflection point) and the initial growth rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logistic Curve
The logistic curve is a fundamental concept in mathematical modeling of population growth. It is described by the equation \( P = \frac{10}{1 + C e^{-t}} \). Here, \( P \) stands for the population size or any quantity of interest as a function of time \( t \). The curve is characterized by an S-shape, which mirrors how many populations grow.Initially, growth is slow. As time passes, it speeds up until it reaches a point of maximum growth rate called an inflection point. After that, growth slows and levels off as it gets closer to a maximum carrying capacity, which in this context is 10.The equation beautifully encapsulates the natural tendency of populations to experience rapid growth when resources are abundant, followed by a plateau as resources become limited.
Parameter Influence
The parameter \( C \) in the logistic curve equation has a profound impact on the curve's features. It dictates how quickly the curve approaches its maximum value.
  • When \( C \) is small, such as 0.5, the curve rises rapidly but slows as it approaches the maximum. The inflection point is lower and occurs later.
  • For moderate values like \( C = 1 \), the curve rises at a moderate pace throughout.
  • Higher values of \( C \) like 2 or 5 result in a steeper rise earlier in the timeline, followed by a quick leveling off.
Essentially, the larger the value of \( C \), the steeper the initial curve, indicating a faster initial growth rate and shifting the inflection point earlier in time.
Graph Interpretation
Understanding the graph of the logistic curve helps visualize how different values of \( C \) influence growth.Substituting values of \( C \) into the equation provides insight into these effects:
  • For \( C = 0.5 \), the curve gradually increases and takes longer to near the maximum value of 10.
  • With \( C = 1 \), the growth rate from start to finish is balanced, without any drastic increases or slowdowns.
  • When \( C = 2 \) or \( C = 5 \), the curves steeply incline early on, nearly touching the maximum much sooner.
In summary, altering \( C \) affects the steepness and timing with which the curve inches towards its plateau. The graphs clearly illustrate these variances, making the concept easier to comprehend.

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