Problem 67 Describe the differences in the ... [FREE SOLUTION]

Chapter 3: Problem 67

Describe the differences in the graphs of an exponential function and a logistic function.

Short Answer

Expert verified

Exponential function shows continuous rapid growth or decay, while logistic function shows slowed growth approaching a maximum limit, forming an S-shaped curve.

Step by step solution

- Define Exponential Function

An exponential function can be represented as: \( f(x) = a \times b^x \), where \( a \) is a constant, \( b \) is the base of the exponential, and \( x \) is the exponent. It shows rapid growth or decay, depending on whether \( b \) is greater than or less than 1.

- Define Logistic Function

A logistic function can be represented as: \( g(x) = \frac{L}{1 + e^{-k(x-x_0)}} \), where \( L \) is the carrying capacity, \( k \) is the growth rate, and \( x_0 \) is the x-value of the function鈥檚 midpoint. This graph shows initial exponential growth that slows down and approaches a maximum limit (carrying capacity).

- Graph Shape Comparison

The graph of an exponential function is a continuously increasing (if \( b > 1 \)) or decreasing (if \( 0 < b < 1 \)) curve that never touches the x-axis. In contrast, the logistic function graph starts with exponential growth but slows down as it approaches the carrying capacity, forming an S-shaped curve also known as a sigmoid curve.

- Behavior at Extremes

For an exponential function, as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \) or \( f(x) \rightarrow 0 \) depending on whether \( b > 1 \) or \( 0 < b < 1 \). For a logistic function, as \( x \rightarrow \infty \), \( g(x) \rightarrow L \), and as \( x \rightarrow -\infty \), \( g(x) \rightarrow 0 \).

- Asymptotes

An exponential function has a horizontal asymptote at \( y = 0 \). A logistic function has two horizontal asymptotes: one at \( y = 0 \) and another at \( y = L \), where \( L \) is the carrying capacity.

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Key Concepts

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

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to its current value. This means the quantity increases exponentially over time. The standard form of an exponential growth function is represented as: \( f(x) = a \times b^x \) where \( a \) is a starting value, and \( b \) is a growth factor greater than 1.

An example of exponential growth is population growth without constraints.
The graph of an exponential growth function is a J-shaped curve that gets steeper and steeper as \( x \) increases.
Such functions continuously increase and do not level off.

Exponential Decay

Exponential decay, on the other hand, occurs when the growth rate is negative, leading to a decrease in the quantity over time. The functional form remains \( f(x) = a \times b^x \). However, in decay, the base \( b \) is between 0 and 1.

Radioactive decay and cooling of hot objects are typical examples of exponential decay.
The decay function decreases gradually and approaches zero as \( x \) increases, but never actually reaches it.
The graph is a downward-sloping curve, also known as a decaying exponential curve.

Sigmoid Curve

A sigmoid curve, often seen in logistic functions, is S-shaped and describes processes where growth starts exponentially but slows down as it approaches a maximum limit鈥攖he carrying capacity. The standard form of a logistic function is: \( g(x) = \frac{L}{1 + e^{-k(x-x_0)}} \), where:

\( L \) is the carrying capacity
\( k \) is the growth rate
\( x_0 \) is the x-value of the function's midpoint

The sigmoid curve starts with slow growth, then accelerates exponentially, and finally slows down as it nears \( L \).
This pattern is often seen in biological systems, such as population growth in an environment with limited resources.

Carrying Capacity

Carrying capacity, represented by \( L \) in a logistic function, is the maximum value that the function can reach. It signifies the upper limit of growth that an environment can sustainably support.

For example, the carrying capacity in a population growth model could be the maximum population size an environment can sustain.
Once the carrying capacity is reached, the growth rate slows down, and the population stabilizes around this value.
In the logistic function, \( g(x) \) asymptotically approaches \( L \) as \( x \) increases.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as \( x \) tends to either \( \infty \) or \( -\infty \). For exponential functions:

An increasing exponential function (\

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Describe the differences in the graphs of an exponential function and a logistic function.

Short Answer

Step by step solution

- Define Exponential Function

- Define Logistic Function

- Graph Shape Comparison

- Behavior at Extremes

- Asymptotes

Key Concepts

Exponential Growth

Exponential Decay

Sigmoid Curve

Carrying Capacity

Horizontal Asymptote

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