Problem 42 - Algebra 2

Question

Without graphing, determine whether each equation represents exponential growth or exponential decay. $$ s(t)=5 e^{t} $$

Step-by-Step Solution

Answer

The given exponential function \(s(t)=5 \cdot e^{t}\) represents exponential growth.

Step 1: Identify the Structure of the Function

The given function is \(s(t)=5 \cdot e^{t}\). In this function, the base of the exponent (e) is Euler's number, a special number that's approximately equal to 2.718.

Step 2: Analyze the Base of the Exponent

In terms of the standard exponential function \(f(x) = a \cdot b^{x}\), our base 'b' is 'e' which equals approximately 2.718. As this base 'b' is greater than 1, it indicates that the function represents exponential growth.

Step 3: Conclusion

By analyzing the base of the exponent, we conclude that as \(b > 1\), the function \(s(t)=5 \cdot e^{t}\) represents exponential growth.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions can describe a variety of real-world scenarios such as population growth, radioactive decay, and even interest calculations. The general form of an exponential function is given by \( f(x) = a \cdot b^x \), where:

• \( a \) is a constant that represents the initial value or y-intercept of the function.
• \( b \) is the base, a positive real number which dictates the growth or decay of the function.
• \( x \) is the exponent, which is typically a variable.

If the base \( b \) is greater than 1, the function models exponential growth. If the base is between 0 and 1, it models exponential decay. Recognizing these patterns is key to understanding the behavior of exponential functions.

Euler's Number

Euler's number, commonly denoted as \( e \), is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is found abundantly in natural phenomena.
One important aspect of \( e \) is that it serves as the base in continuous growth processes. For example:

• Compound interest calculations.
• Population dynamics in biology.
• Decay processes in physics.

The exponential function \( e^x \) provides a continuous growth model. Despite its complex appearance, the function is incredibly useful due to its properties of continuous growth rate and its presence in calculus as the solution to the differential equation \( \frac{d}{dx} e^x = e^x \).

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. This means that as time progresses, the rate of decay slows, but never entirely stops. The general form is similar to exponential growth but with a base \( b \) between 0 and 1, given by \( f(x) = a \cdot b^x \).
Common examples of exponential decay include:

• Radioactive decay of elements, where unstable nuclei lose energy over time.
• Depreciation of the value of assets or technologies over time.
• Cooling processes where hot objects lose heat to their surroundings.

Understanding exponential decay helps in predicting how quantities diminish over time and is essential for fields like finance, physics, and environmental science.