Problem 17 - Contemporary Precalculus

Question

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline y & 2 & 25 & 81 & 175 & 310 & 497 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Dataset: \((x, y): (1, 2), (3, 25), (5, 81), (7, 175), (9, 310), (11, 497)\) Answer: A power model is the most appropriate for this data.

Step 1: Set 1: \((x, \ln y)\)

Calculate the natural logarithm of \(y\) values and pair them with \(x\) values: $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\\ \hline \ln y & \ln 2 & \ln 25 & \ln 81 & \ln 175 & \ln 310 & \ln 497 \\\ \hline \end{array}$$

Step 2: Set 2: \((\ln x, \ln y)\)

Calculate the natural logarithm of both \(x\) and \(y\) values and pair them together: $$\begin{array}{|l|c|c|c|c|c|c|} \hline \ln x & \ln 1 & \ln 3 & \ln 5 & \ln 7 & \ln 9 & \ln 11 \\\ \hline \ln y & \ln 2 & \ln 25 & \ln 81 & \ln 175 & \ln 310 & \ln 497 \\\ \hline \end{array}$$

Step 3: Set 3: \((\ln x, y)\)

Calculate the natural logarithm of \(x\) values and pair them with \(y\) values: $$\begin{array}{|l|c|c|c|c|c|c|} \hline \ln x & \ln 1 & \ln 3 & \ln 5 & \ln 7 & \ln 9 & \ln 11 \\\ \hline y & 2 & 25 & 81 & 175 & 310 & 497 \\\ \hline \end{array}$$ Now that we have the three sets, we can create graphs for each of these sets and visually determine which one is closest to being linear.

Step 4: Create and analyze graphs

Observe the trend of each set of data points in a graphical representation. Check which set of data points is more linear than the others. Upon creating and analyzing the graphs, it's noticeable that Set 2: \((\ln x, \ln y)\) is closest to being linear. Therefore, a power model would be appropriate for this data since we have a linear relation between \(\ln x\) and \(\ln y\).

Key Concepts

Exponential Model

An exponential model is a mathematical representation where one of the variables grows exponentially with respect to the other. This type of model is expressed in the form \(y = ab^x\), where \(a\) and \(b\) are constants, and \(b\) is the base of the exponential function. In simple words, this model is suitable for datasets that show rapid increases or decreases and can be modeled as a constant raised to the power of the independent variable.

To determine if your data fits an exponential model, you would check the set \(\{(x, \ln y)\}\). This implies that if taking the natural log of \(y\) and plotting it against \(x\) results in a linear graph, then the relationship between \(x\) and \(y\) can be described by an exponential model.

Key characteristics of exponential models include:

• Exponential growth or decay.
• The rate of change increases or decreases over time.
• Common in processes such as population growth, radioactive decay, and interest calculations.

Observing data in the form of \(x, \ln y\)\ can quickly reveal if this is the type of model applicable. However, in our original exercise, this form did not yield the most linear set of data points.

Power Model

A power model is often used to describe relationships where one quantity varies as a power of another. The general equation for a power model is \(y = ax^b\), where \(a\) and \(b\) are constants and the exponent \(b\) can be any real number.

To test for a power model, you should analyze the \(\ln x, \ln y\) data set. If this set presents a linear relationship when plotted, then the data is likely best described by a power model. In simple terms, if both the independent and dependent variables need to be logarithmically transformed to uncover linearity, a power model fits the data appropriately.

Characteristics of power models include:

• Non-linear relationships that are straightened by logarithmic transformations.
• Useful for modeling phenomena where the effect does not increase linearly—such as intensity of sound or area versus size.
• Commonly used in multiplicative processes.

In the given original exercise, the \(\ln x, \ln y\) set was found to be the most linear, indicating that a power model is suitable for the data. Indeed, power models are versatile and handy in various scientific fields.

Logarithmic Model

Logarithmic models are used to describe relationships that initially grow quickly but level off over time. Mathematically, this model is represented as \(y = a + b \ln x\), where \(a\) and \(b\) are constants. This form suggests a steady progression, quickly decelerating as the independent variable increases.

To assess the suitability of a logarithmic model, you would inspect the set \(\ln x, y\). Here, the natural logarithm of \(x\) is paired with the \(y\) values. If this set exhibits linearity upon plotting, then a logarithmic model is likely appropriate.

Features of logarithmic models:

• Good for scenarios with a rapid initial increase that stabilizes over time, such as learning curves or diminishing returns.
• The rate of increase decreases as the variable \(x\) increases.
• Common in ecological or biological data where growth initially spikes and then plateaus.

In the context of the discussed dataset, the analysis did not favor the use of a logarithmic model. The other data transformations were more effective in revealing linearity for fitting an appropriate model.