Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. \(y=3742(e)^{0.75 t}\)

Step-by-Step Solution

Answer

The equation represents continuous growth due to the positive exponent.

Step 1: Identify the Base of the Exponential Function

In the given equation, identify the base of the exponential function. The equation is written as \( y = 3742(e)^{0.75t} \). Here, \( e \) is the base and it represents the natural exponential base, approximately equal to 2.718.

Step 2: Determine the Exponent Sign

Look at the exponent of \( e \) in the exponential function, which is \( 0.75t \). The coefficient of \( t \) in the exponent is \( 0.75 \). This coefficient is positive, indicating the nature of the exponential expression.

Step 3: Conclude Based on Exponent

Since the coefficient of \( t \) in the exponent is positive (\( 0.75 \)), the function represents continuous growth. In a continuous growth scenario, exponents with positive coefficients result in increasing values as \( t \) increases.

Key Concepts

Exponential Function

When we talk about an exponential function, we're discussing a type of mathematical expression where a constant base is raised to a variable exponent. This function follows the general form of:

• \( y = a imes b^{x} \), where \( a \) represents a constant coefficient, \( b \) is the base of the exponential function, and \( x \) is the variable in the exponent.

In the provided exercise, the exponential function is expressed as \( y = 3742(e)^{0.75t} \). Here, \( 3742 \) acts as the initial value or starting amount, \( e \) is the base, and \( 0.75t \) is the exponent. Exponential functions are widely used in modeling growth and decay processes. They can exhibit features of continuous growth when the exponent's coefficient is positive, or continuous decay if it's negative. Thus, understanding how these elements interact is crucial for interpreting real-world behaviors.

Natural Exponential Base

One of the most fascinating choices for a base in exponential functions is the natural exponential base, denoted as \( e \).

• \( e \) is an irrational number approximately equal to 2.718 and it possesses unique mathematical properties.
• It frequently appears in natural processes like compound interest, population growth, and radioactive decay, making it a "natural" base for exponential functions.

In our equation, \( e \) is used as the base of the exponential expression. The special property of \( e \) is that when it is used as a base, exponential functions can model continuous growth or decay elegantly. Its ubiquitous nature in calculus stems from how it simplifies derivatives and integrals of exponential functions, being seamlessly connected to the concept of rate of change.

Exponent Coefficient

The exponent coefficient is a critical part of the exponential expression since it directly affects the behavior of the function.

• In the equation \( y = 3742(e)^{0.75t} \), the exponent \( 0.75t \) includes the coefficient \( 0.75 \).
• This coefficient determines whether the function experiences continuous growth or decay.

A positive exponent coefficient, such as \( 0.75 \) here, leads to an increase in \( y \) as \( t \) increases. This happens because the base \( e \), when raised to a power with a positive coefficient, results in increasingly larger values. On the other hand, if the exponent coefficient were negative, it would indicate continuous decay, as the function's value would decrease over time. Thus, examining this coefficient helps us predict the direction of change in the function values.

Continuous Decay

Continuous decay is an important concept when working with exponential functions. It refers to a scenario where the value of a function decreases gradually over time, based on an exponential function with a negative exponent coefficient.

• Characteristics: In mathematical expressions, continuous decay is represented by a negative coefficient in the exponent.
• For instance, if our equation were \( y = 3742(e)^{-0.75t} \), the outcome would be continuous decay.

In continuous decay, as the variable \( t \) increases, the overall value of \( y \) diminishes. This concept is often used in contexts such as depreciation, radioactive decay, and cooling processes. Understanding the negative coefficient in the exponent is key to identifying decay functions. However, in the provided exercise \( y = 3742(e)^{0.75t} \), the positive coefficient of 0.75 signifies continuous growth, not decay.