Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The exponential model \(y=a e^{b x}\) represents a growth model when \(b>0.\)

Step-by-Step Solution

Answer

The statement is true. In the exponential model \(y=a e^{b x}\), a positive value of \(b\) indeed represents a growth model.

Step 1: Understanding the Growth model in mathematics

Exponential growth is a specific way that a quantity may increase over time. In an exponential growth model, this process occurs when the growth rate of the value in question becomes increasingly faster in proportion to its current size, often due to the process being a normal part of the system's nature. In other words, the model often describes natural phenomena that increase multiplicatively. The exponential function \(y=a e^{b x}\) where \(a > 0\), \(b > 0\), and \(x\) is a real number is a common example of exponential growth.

Step 2: Determine the role of \(b\) in the exponential model

In the exponential growth model, the parameter \(b\) represents the rate of growth. If \(b>0\), the quantity represented by \(y\) grows as \(x\) increases. If \(b<0\), the quantity represented by \(y\) would actually decrease as \(x\) increases, which means it would signify decay rather than growth.

Step 3: Conclude whether the statement is true or false

Since a positive value of \(b\) in the exponential function does represent a growth model, the statement 'The exponential model \(y=a e^{b x}\) represents a growth model when \(b>0\)' is indeed true.

Key Concepts

Exponential Model

An exponential model is a mathematical representation that describes how a quantity grows or shrinks over time.
The core of this model is the exponential function, written as \(y=a e^{b x}\).
Here, \(a\) is a positive constant, \(b\) represents the growth rate, and \(x\) is the variable, usually time.
In an exponential model, changes happen at a rate proportional to the current value of the quantity. This is fundamentally different from linear models, where changes occur at a constant rate.

This model is particularly helpful in describing real-world phenomena like population growth, radioactive decay, and compound interest.

• Population Growth: As more individuals are present, more births occur.
• Radioactive Decay: Each particle has an independent chance of decaying over time.
• Compound Interest: The more money you have, the more interest accumulates.

Growth Rate

Within the exponential model \(y = a e^{b x}\), the parameter \(b\) is crucial as it defines the growth rate.
When \(b > 0\), it indicates a growth scenario, suggesting that as \(x\) increases, \(y\) gets larger.
If \(b < 0\), the model reflects a situation of decay, where the quantity decreases over time.

To understand growth rate intuitively, consider how rapidly a situation escalates under this function.
For instance:

• If \(b\) is small but positive, growth is steady and gradual.
• With a larger \(b\), growth becomes rapid and more dramatic.

In practical terms, a business seeing an increase in its customer base may experience exponential growth if new customers attract even more customers.

Mathematical Justification

Mathematical justification involves validating why certain conditions lead to expected outcomes within an exponential model.
To justify that the statement about \(y=a e^{b x}\) represents growth when \(b>0\), consider:

• **Behavior of the Exponential Function:** In mathematics, \(e^{b x}\) increases as \(x\) increases, provided \(b\) is positive.
• **Real-World Observations:** Models fitting exponential growth constantly prove true in empirical data, like populations, finances, and decay processes.

Another layer of justification comes from calculus:

• **Derivative Analysis:** The derivative \(\frac{dy}{dx}=a b e^{b x}\) is positive when \(b>0\), confirming that \(y\) grows as \(x\) increases.

Thus, all observations, both theoretical and empirical, align with the statement when \(b > 0\), solidifying it as true.