Problem 75 - 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}{cc} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline\(-1\) & \(0.398\) \\ \(-0.5\) & \(1.26\) \\ 0 & 4 \\ \(0.5\) & \(12.68\) \\ 1 & \(40.18\) \\ \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 relationship is exponential: \(y = 4 \times 16^x\).

Step 1: Understand the Problem

We are given a set of data points and need to determine if the relationship between \(x\) and \(y\) follows an exponential function (\(y = ab^x\)) or a power function (\(y = ax^b\)). To do this, we will apply logarithmic transformations that linearize these types of functions.

Step 2: Transform to Linear Form

For an exponential function \(y = ab^x\), taking the natural logarithm of both sides gives \(\ln(y) = \ln(a) + x\ln(b)\), suggesting a linear relationship between \(x\) and \(\ln(y)\). For a power function \(y = ax^b\), we take the logarithm to get \(\ln(y) = \ln(a) + b\ln(x)\), suggesting a linear relationship between \(\ln(x)\) and \(\ln(y)\).

Step 3: Apply Transformation to Data

Since we need to test both functions, first calculate \(\ln(y)\) for each \(y\): - For \(x = -1\), \(y = 0.398\), \(\ln(0.398) = -0.92\), - For \(x = -0.5\), \(y = 1.26\), \(\ln(1.26) = 0.231\), - For \(x = 0\), \(y = 4\), \(\ln(4) = 1.386\), - For \(x = 0.5\), \(y = 12.68\), \(\ln(12.68) = 2.54\), - For \(x = 1\), \(y = 40.18\), \(\ln(40.18) = 3.692\).

Step 4: Plot and Analyze the Graph

Plot the points \((x, \ln(y))\) to check for linearity. If these points form a straight line, the function is exponential. If not, plot \((\ln(x), \ln(y))\). In this scenario, for an exponential check, the points \((-1, -0.92), (-0.5, 0.231), (0, 1.386), (0.5, 2.54), (1, 3.692)\) should lie on a straight line.

Step 5: Determine the Functional Relationship

Upon plotting \((x, \ln(y))\), the points form a straight line, indicating an exponential relationship. The estimated slope and intercept can be used to find \(a\) and \(b\) in \(\ln(y) = \ln(a) + x\ln(b)\). Using linear regression on these points, find that \(\ln(b) \approx 2.772\) and \(\ln(a) \approx 1.386\), thus \(b \approx e^{2.772} \approx 16\) and \(a \approx e^{1.386} \approx 4\). The function is \(y = 4(16)^x\).

Key Concepts

Logarithmic Transformation

Logarithmic transformation is a mathematical technique used to linearize certain types of relationships in data, making them easier to analyze. In particular, this technique is useful when dealing with exponential and power functions.

• For an exponential function of the form \(y = ab^x\), the transformation involves taking the natural logarithm of both sides, leading to \(\ln(y) = \ln(a) + x\ln(b)\). This converts the equation into a linear form relating \(x\) and \(\ln(y)\).
• For a power function such as \(y = ax^b\), a similar approach applies: taking the natural logarithm gives \(\ln(y) = \ln(a) + b\ln(x)\). This produces a linear relationship between \(\ln(x)\) and \(\ln(y)\).

The key to using logarithmic transformation is recognizing when nonlinear relationships are present and how converting them can simplify the analysis, particularly when using statistical tools like linear regression.

Functional Relationship

Understanding the functional relationship between variables is fundamental to analyzing data. In many problems, as with the given exercise, we aim to identify whether the relationship is best described as exponential or a power law.

• Exponential Functions: These are characterized by multiplicative growth patterns, usually described by equations like \(y = ab^x\). As \(x\) changes, \(y\) multiplies by a constant factor \(b\).
• Power Functions: These represent relationships where one variable is a power of another, typically \(y = ax^b\). Here, changes in \(x\) lead to a polynomial scaling in \(y\).

Identifying the correct type of functional relationship helps in predicting future values, understanding growth rates, and making sense of scientific phenomena. In the given exercise, plotting transformed data helps reveal the linearity that suggests the correct model.

Linear Regression

Linear regression is a statistical method used to model and analyze the relationships between variables by fitting a linear equation to observed data. It is particularly powerful when working with data that has undergone a logarithmic transformation.After applying the transformations, we expect a linear relationship if the data follows an exponential or power law. Linear regression will help us to:

• Find the slope and intercept of the linear equation, which are directly related to the parameters of the original functions.
• Determine goodness of fit, which indicates how well the model represents the data. A high correlation coefficient suggests a strong linear relationship, confirming the model's validity.

In the exercise, by plotting the transformed data points \((x, \ln(y))\), we applied linear regression to estimate \(\ln(b)\) and \(\ln(a)\). This gave us the exponential function \(y = 4(16)^x\), accurately representing the data.