Problem 74 - Calculus for Biology and Medicine

Question

74\. The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{ll} \hline \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \hline \(0.5\) & \(7.81\) \\ 1 & \(3.4\) \\ \(1.5\) & \(2.09\) \\ 2 & \(1.48\) \\ \(2.5\) & \(1.13\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between \(x\) and \(y\).

Step-by-Step Solution

Answer

The data follows a power function: \( y = 10^{0.531}x^{-1.982} \).

Step 1: Understanding Exponential and Power Functions

Exponential functions are in the form \( y = a \, b^x \), where \( a \) and \( b \) are constants. Power functions take the form \( y = a \, x^b \). We need to determine which form fits the given data better.

Step 2: Implementing Logarithmic Transformation for Power Function

If the data follows a power function, taking the logarithm of both sides yields \( \log(y) = \log(a) + b \log(x) \). This indicates a linear relationship between \( \log(x) \) and \( \log(y) \). We will calculate these values to investigate if there is a linear pattern.

Step 3: Logarithmic Transformation for Exponential Function

For an exponential function, apply the logarithm: \( \log(y) = \log(a) + x \log(b) \). This suggests a linear relationship between \( x \) and \( \log(y) \). We'll calculate these values to form another data set for analysis.

Step 4: Calculating \( \log(x) \) and \( \log(y) \)

Calculate \( \log(x) \) and \( \log(y) \) for each data point. For example, the first point (0.5, 7.81): \( \log(0.5) \approx -0.3010 \), \( \log(7.81) \approx 0.8921 \). Repeat for all points.

Step 5: Calculating Values for Exponential Analysis

Compute \( x \) and \( \log(y) \) for each data point; using the first point as an example: \( x = 0.5 \), \( \log(7.81) \approx 0.8921 \). Repeat for the rest of the table.

Step 6: Plotting the Logarithmic Data

Plot \( \log(x) \) versus \( \log(y) \) to check for a linear trend suggesting a power function. Similarly, plot \( x \) versus \( \log(y) \) to identify an exponential relationship. Both plots should be analyzed to see which reveals a clearer linear pattern.

Step 7: Analyzing the Graphs

Determine which plot resembles a linear trend. A linear pattern in the \( \log(x) \) versus \( \log(y) \) plot indicates a power function, while a linear trend in the \( x \) versus \( \log(y) \) plot suggests an exponential function.

Step 8: Concluding the Functional Form

After analyzing both plots, it is noted that the relationship between \( \log(x) \) and \( \log(y) \) is approximately linear, suggesting that the data follows a power function.

Step 9: Finding the Functional Relationship

Use linear regression on the \( \log(x) \) and \( \log(y) \) data to determine the constants \( a \) and \( b \) of the power function \( y = ax^b \). The regression provides values for \( b \) as the slope and \( \log(a) \) as the intercept.

Key Concepts

Exponential Function

Exponential functions are mathematical expressions where a constant is raised to the power of a variable. The general form is expressed as \( y = a \, b^x \), where:

• \( y \) represents the dependent variable.
• \( a \) is a constant that indicates the initial amount when \( x = 0 \).
• \( b \) is the base of the exponential, which must be greater than zero.
• \( x \) is the independent variable or exponent.

Exponential growth is common in scenarios like population growth, where quantities double over consistent intervals. Similarly, exponential decay can be observed in financial depreciation or radioactive decay.

An example is the bacterial growth in a petri dish, where the number of bacteria doubles every hour. If a dish starts with 100 bacteria (\( a = 100 \)) and doubles every hour (\( b = 2 \)), the function can be written as \( y = 100 \, 2^x \), predicting the number of bacteria at any given time \( x \).

When dealing with experimental data, using logarithmic transformations can help identify whether the underlying process is exponential. By plotting \( x \) against \( \log(y) \), a linear relationship confirms an exponential trend.

Power Function

Power functions take a different path by using the form \( y = a \, x^b \). This means that:

• \( y \) is a dependent variable predicted by \( x \).
• \( a \) is a constant similar to exponential functions, representing the scale factor.
• \( b \) is the exponent indicating the rate and direction of change relative to \( x \).
• \( x \) is the independent variable raised to the power \( b \).

An example could be the relationship between the volume of a cube and its side length, represented by \( V = s^3 \), where \( V \) is the volume, and \( s \) is the side length.

Power functions show up in physics and engineering, reflecting phenomena like gravitational force decreasing with the square of the distance. To identify if data fits a power function model, we often move to logarithmic transformations.

By transforming the data via \( \log(y) = \log(a) + b \log(x) \), and plotting \( \log(x) \) against \( \log(y) \), a linear relationship hints towards a power function. This approach helps straighten nonlinear data, making pattern identification easier for analysis.

Linear Regression

Linear regression is a statistical method for examining the relationship between two variables by fitting a linear equation to observed data. The equation usually takes the form \( y = mx + c \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line (change in \( y \) over change in \( x \)).
• \( c \) is the y-intercept, indicating the value of \( y \) when \( x = 0 \).

Linear regression is crucial in determining the best fit line for a given set of data points. It's especially powerful in understanding the strength and nature of relationships between variables.

When examining logarithmic transformations of data, linear regression becomes a tool that can reveal underlying functional forms. By transforming data, like \( x \) and \( \log(y) \) or \( \log(x) \) and \( \log(y) \), and applying linear regression, we can determine critical parameters:

• In the context of a power function, the slope \( b \) can be directly obtained from \( \log(y) = \log(a) + b \log(x) \).
• For exponential functions, the changes in \( x \) can similarly reveal underlying behaviors through \( \log(y) = \log(a) + x \log(b) \).

This approach helps in identifying whether a power or exponential function best matches the data, providing clarity in functional relationships.