Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, refer to Table \(8 .\) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array} $$ Write the exponential function as an exponential equation with base \(e\).

Step-by-Step Solution

Answer

The exponential function is approximately \( f(x) = 555e^{-0.2x} \).

Step 1: Understand the problem

We need to express the given data table as an exponential function of the form \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants.

Step 2: Observing the trend

From the table, we observe that \( f(x) \) is decreasing as \( x \) increases. This suggests we are looking for a function where the exponential term involves a negative exponent.

Step 3: Calculate the ratio

Calculate the ratio \( \frac{f(x+1)}{f(x)} \) for different values of \( x \). For example, from 1 to 2, this is \( \frac{383}{555} \approx 0.69 \), and from 2 to 3, \( \frac{307}{383} \approx 0.80 \). Verify whether these ratios indicate an exponential trend.

Step 4: Estimating the base and the decay constant

Since the calculated ratios are not very consistent, take an average, which suggests how fast the function decays. Assume the exponential decay form \( e^{bx} \) where \( b \approx -1 \) based on the rapid decay between the terms.

Step 5: Estimating the initial value

The initial value \( a \) can be estimated using \( f(1) = 555 \). So, assuming the trend, \( f(1) = a e^b \) gives \( 555 = a e^b \). This can be used to refine \( a \) once \( b \) is somewhat accurately determined.

Step 6: Form the Exponential Equation

Using rough estimates, an exponential function that fits the data takes the form \( f(x) = 555 e^{-0.2x} \). Further refinement would require fitting using non-linear regression methods.

Key Concepts

Exponential Decay

Exponential decay describes how quantities decrease rapidly at a rate proportional to their current value. In our exercise, the function \( f(x) \) shows exponential decay because it decreases as \( x \) increases. This is typical in situations such as radioactive decay, cooling of a hot object, and interest payments on a declining balance.

In mathematical terms, a function experiencing exponential decay can be expressed as \( f(x) = a \cdot e^{bx} \), where \( b < 0 \) because the exponential term’s exponent is negative. This causes the exponential function's value to diminish as \( x \) increases.

Key features of exponential decay include:

• A constant percentage decrease over equal intervals
• The graph of the function is a downward-sloping curve
• As time progresses, the rate of decrease becomes less steep

In our problem, the decay pattern suggests that an equation like \( f(x) = 555 e^{-0.2x} \) is suitable, where \( -0.2 \) represents the decay rate, indicating a relatively brisk decline.

Non-linear Regression

Non-linear regression is a technique used to model data that do not fit a straight line. Instead, it provides a way to fit more complex curves to the data. In the context of our exercise, while we estimate the initial decay parameters, a precise fit would require non-linear regression.

Non-linear regression helps in:

• Finding a more accurate model by accounting for the curvature in the data
• Providing parameter estimates that best fit the given data points
• Evaluating how well the model predicts the data

For the given data table, the change in function output \( f(x) \) is not constant, indicating that a simple linear model is inadequate. Using non-linear regression would help refine an equation like \( f(x) = 555 e^{-0.2x} \), ensuring the selected parameters \( a \) and \( b \) accurately reflect the behavior observed in the data.

Base e

The number \( e \) is a fundamental mathematical constant, approximately equal to 2.71828. Known as Euler's number, it plays a critical role in various branches of mathematics, especially in continuous growth or decay processes.

Why is \( e \) important?

• It is the base of natural logarithms
• \( e \) arises naturally in situations involving continuous growth or decay
• Used in various real-world phenomena, including finance, biology, and physics

In exponential functions like \( f(x) = a e^{bx} \), utilizing the base \( e \) simplifies both the differentiation and integration processes, crucial for analyzing changes over time. Specifically, in our problem, employing \( e \) allows us to model the decaying process naturally and efficiently, reflecting the underlying mathematics of decay in real-world contexts.