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Chapter 4: Problem 12

Given a logistic growth function \(y=\frac{c}{1+a e^{-b t}},\) the limiting value of \(y\) is _____ .

Short Answer

Expert verified
The limiting value of \( y \) is \( c \).

Step by step solution

01

Identify the Logistic Growth Function

The logistic growth function is given as: \[ y = \frac{c}{1 + ae^{-bt}} \]
02

Calculate the Limiting Value as t Approaches Infinity

To find the limiting value, evaluate the function as \( t \to \infty \). Observe the behavior of the exponential term \( ae^{-bt} \).
03

Analyze the Exponential Term

As \( t \to \infty \), the term \( e^{-bt} \) converges to 0 because the exponent is negative. Therefore, \( ae^{-bt} \) also approaches 0.
04

Simplify the Expression

Substitute the approximations into the logistic function: \[ y = \frac{c}{1 + ae^{-bt}} \approx \frac{c}{1 + 0} = \frac{c}{1} = c \]

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Key Concepts

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

limiting value
The logistic growth function represents a population growth model constrained by a limiting factor. The function is defined as \(y = \frac{c}{1 + ae^{-bt}}\). As time \(t\) increases indefinitely, we want to determine the limiting value of \(y\), that is, where the population will stabilize. This involves evaluating \(y\) as \(t \to \infty\). In simplified terms, the limiting value is the value the function approaches when time keeps growing indefinitely.
As we progress through this, we'll see that the exponential term plays a crucial role in finding this value.
exponential term
Within the logistic growth function \(y = \frac{c}{1 + ae^{-bt}}\), the exponential term is \(ae^{-bt}\). This term helps us understand how quickly the population approaches its limiting value.
We know that as time \(t\) increases, the term \(e^{-bt}\) gets extremely small because of the negative exponent. Essentially, \(e^{-bt}\) will approach 0 as \(t\) approaches infinity. When substituting this behavior into the logistic function, the entire term \(ae^{-bt}\) approaches 0.
This simplification is crucial for analyzing the long-term behavior of the function.
infinite limit analysis
When performing infinite limit analysis, we look at what happens to the function as time \(t\) grows without bound. For the logistic growth function, this involves observing how each component behaves as \(t \to \infty\). As we determined earlier, the term \(ae^{-bt}\) approaches 0 in this limit. This tells us that the logistic function simplifies to \(y = \frac{c}{1 + 0} = \frac{c}{1} = c\).
Thus, through infinite limit analysis, we conclude that the limiting value of our logistic growth function \(y\) is \(c\). This represents the maximum population the environment can sustain.

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Most popular questions from this chapter

According to the CIA's World Fact Book, in \(2010,\) the population of the United States was approximately 310 million with a \(0.97 \%\) annual growth rate. (Source: www.cia.gov) At this rate, the population \(P(t)\) (in millions) can be approximated by \(P(t)=310(1.0097)^{t}\), where \(t\) is the time in years since 2010 . a. Is the graph of \(P\) an increasing or decreasing exponential function? b. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. c. Evaluate \(P(10)\) and interpret its meaning in the context of this problem. Round the population value to the nearest million. d. Evaluate \(P(20)\) and \(P(30)\). e. Evaluate \(P(200)\) and use this result to determine if it is reasonable to expect this model to continue indefinitely.

Use a calculator to approximate the given logarithms to 4 decimal places. a. The speed of light is \(2.9979 \times 10^{8} \mathrm{~m} / \mathrm{sec}\). Approximate \(\log \left(2.9979 \times 10^{8}\right)\) b. An elementary charge is \(1.602 \times 10^{-19} \mathrm{C}\). Approximate \(\log \left(1.602 \times 10^{-19}\right)\) c. Compare the value of the common logarithm to the power of 10 used in scientific notation.

Write the equation in logarithmic form. $$ 2^{5}=32 $$

Explain why the \(f(x)=x^{2}\) is not an exponential function.

Suppose that \(\$ 50,000\) from a retirement account is invested in a large cap stock fund. After 20 yr, the value is \(\$ 194,809.67\). a. Use the model \(A=P e^{r t}\) to determine the average rate of return under continuous compounding. b. How long will it take the investment to reach onequarter million dollars? Round to 1 decimal place.
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