Problem 77 - Calculus for Biology and Medicine

Question

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 \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline \(0.1\) & \(0.045\) \\ \(0.5\) & \(1.33\) \\ 1 & \(5.7\) \\ \(1.5\) & \(13.36\) \\ 2 & \(24.44\) \\ \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 function is likely a power function: \( y = 4.5 \cdot x^{2.5} \).

Step 1: Understand Exponential and Power Functions

An exponential function has the form \( y = a \cdot b^x \) and a power function has the form \( y = a \cdot x^b \). We will use logarithmic transformations to identify which type this function most likely represents.

Step 2: Logarithmic Transformation for Exponential Function

For an exponential function \( y = a \cdot b^x \), we take the natural logarithm of both sides to get \( \ln(y) = \ln(a) + x \cdot \ln(b) \). Plot \( \ln(y) \) against \( x \) to see if it forms a straight line, which indicates an exponential relationship.

Step 3: Logarithmic Transformation for Power Function

For a power function \( y = a \cdot x^b \), taking the natural logarithm gives \( \ln(y) = \ln(a) + b \cdot \ln(x) \). Plot \( \ln(y) \) against \( \ln(x) \); if it forms a straight line, the relationship is a power function.

Step 4: Calculate Logarithms

Calculate \( \ln(y) \) for each \( y \) in the table and \( \ln(x) \) for each \( x \) in the table. Use these to plot both \( \ln(y) \) vs \( x \) for exponential and \( \ln(y) \) vs \( \ln(x) \) for power functions.

Step 5: Analyze the Graphs

Plot both sets of transformations. If the \( \ln(y) \) vs \( \ln(x) \) graph is linear, it indicates a power function. If the \( \ln(y) \) vs \( x \) graph is linear, it indicates an exponential function.

Step 6: Determine the Function Type

Based on the linearity of the transformed data, determine whether the function is exponential or power. In this case, the \( \ln(y) \) vs \( \ln(x) \) plot will provide the function type.

Step 7: Find the Parameters

Using the linear equation from the linear plot, determine \( a \) and \( b \) for the power function. The slope of the line, \( b \), and the intercept, \( \ln(a) \), can be used to find the parameters.

Key Concepts

Logarithmic Transformation

When faced with unfamiliar data that may arise from exponential or power relationships, logarithmic transformation is a powerful tool. It simplifies comparison by linearizing the data.

An exponential function, which can be expressed as \( y = a \cdot b^x \), becomes \( \ln(y) = \ln(a) + x \cdot \ln(b) \) after a logarithmic transformation. This transformed equation reveals a linear relationship between \( \ln(y) \) and \( x \), helping us identify exponential functions through straight line graphs.

Similarly, a power function represented by \( y = a \cdot x^b \) turns into \( \ln(y) = \ln(a) + b \cdot \ln(x) \) when transformed. The plot then becomes a straight line between \( \ln(y) \) and \( \ln(x) \), indicating power functions. By transforming both types of functions, we can use the linearity of their plots to differentiate between the two possibilities.

Graph Analysis

Graph analysis is the next crucial step after getting your logarithmic transformations ready. It provides a straightforward visual means of distinguishing between exponential and power functions.

Once you have calculated \( \ln(y) \) and \( \ln(x) \) for the data points, it's time to plot:

• Plot \( \ln(y) \) against \( x \) for exponential functions.
• Plot \( \ln(y) \) against \( \ln(x) \) for power functions.

By examining the two resulting graphs, we look for the one with a linear trend. A linear graph in the \( \ln(y) \) vs \( x \) plot indicates an exponential relationship. Conversely, a linear relationship in the \( \ln(y) \) vs \( \ln(x) \) plot confirms a power function. This graphical approach simplifies the decision-making process by clearly showing which type of function best models the data.

Functional Relationship Determination

After identifying the correct type of function through graph analysis, the next step is determining the precise functional relationship. This involves extracting the parameters \( a \) and \( b \).

For a graph found to represent a power function, the equation is \( y = a \cdot x^b \). Here:

• The slope of the linear plot corresponds to \( b \).
• The y-intercept, when back-transformed using exponentiation, gives \( a \).

Similarly, if the function is exponential, expressed as \( y = a \cdot b^x \), then:

• \( \ln(b) \) is the slope of the line.
• \( \ln(a) \) is the intercept, again needing exponentiation to find \( a \).

With both \( a \) and \( b \) determined, you can construct the exact functional relationship fitting your data. This thorough understanding ensures you not only grasp the concept but can apply it to other datasets confidently.