Problem 78 - 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 \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \hline \(0.1\) & \(1.72\) \\ \(0.5\) & \(1.41\) \\ 1 & \(1.11\) \\ \(1.5\) & \(0.872\) \\ 2 & \(0.685\) \\ \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 fits a power function: \( y = a \cdot x^k \).

Step 1: Understanding Exponential vs. Power Function

An exponential function can be described as \( y = a \cdot b^x \), while a power function is expressed as \( y = a \cdot x^k \). The best way to distinguish them is to apply logarithmic transformations.

Step 2: Logarithmic Transformation for Exponential

For an exponential function, taking a logarithm of both sides of \( y = a \cdot b^x \) gives \( \log(y) = \log(a) + x\log(b) \), suggesting a linear relationship between \( x \) and \( \log(y) \).

Step 3: Logarithmic Transformation for Power Function

For a power function, taking a logarithm of both sides of \( y = a \cdot x^k \) results in \( \log(y) = \log(a) + k\log(x) \), which implies a linear relationship between \( \log(x) \) and \( \log(y) \).

Step 4: Logarithmic Conversion of Data

Convert the given data to logarithms: Calculate \( \log(x) \) and \( \log(y) \) for each data point.

Step 5: Create Linearity Graphs

Plot the points \((x, \log(y))\) for exponential function testing and \((\log(x), \log(y))\) for power function testing.

Step 6: Analyze Graphs

Analyze which plot displays a linear relationship. If \((x, \log(y))\) aligns linearly, it's an exponential function. If \((\log(x), \log(y))\) aligns linearly, it's a power function.

Step 7: Determine Equation Parameters

Using the identified relationship, apply methods (like linear regression) to determine `a` and `b` for exponential or `a` and `k` for power function from the linear format results.

Key Concepts

Exponential Function

An exponential function is a mathematical expression of the form \( y = a \cdot b^x \). Here, \(a\) is the initial amount, \(b\) is the base or growth factor, and \(x\) represents the exponent or power that the base is raised to. This type of function is characterized by rapid growth or decay depending on the value of \(b\). If \(b > 1\), the function describes exponential growth, while \(0 < b < 1\) indicates exponential decay.
One way to identify an exponential function is through logarithmic transformation. By taking the logarithm of both sides of the equation, the exponential format can be converted into a form that's easier to analyze:

• Apply log to the entire equation: \( \log(y) = \log(a \cdot b^x) \)
• Simplify to get a linear-looking form: \( \log(y) = \log(a) + x \cdot \log(b) \)

This transformation helps to check if there exists a linear relationship between \(x\) and \(\log(y)\).
By plotting \((x, \log(y))\), if the points form a straight line, it suggests the data follows an exponential function pattern.

Power Function

A power function is structured as \( y = a \cdot x^k \), where \(a\) is the coefficient and \(k\) is the exponent. This type of function describes relationships where one variable is a constant multiple of another variable raised to a power. Power functions are ubiquitous, appearing in physics, engineering, and economics to show various proportional relationships.
To distinguish it from others, logarithmic transformation can be applied:

• Transform the power equation by applying logarithms: \( \log(y) = \log(a \cdot x^k) \)
• Simplify to a linear form: \( \log(y) = \log(a) + k \cdot \log(x) \)

This shows that when plotted, \(\log(x)\) and \(\log(y)\) should create a linear pattern if the data fits a power function.
To confirm, creating a graph with \((\log(x), \log(y))\) and observing a linear trend assures that the power function fits the data.

Linear Relationship

A linear relationship is one in which two variables are related in such a way that a change in one variable is directly proportional to a change in the other. In the context of transformations, a linear relationship is uncovered when plotting transformed data, such as \((x, \log(y))\) or \((\log(x), \log(y))\).
When you apply a transformation to data and then plot it:

• If the plot of \((x, \log(y))\) results in a straight line, it indicates a linear relationship derived from an exponential function.
• Conversely, if \((\log(x), \log(y))\) forms a straight line, it suggests a linear relationship from a power function.

The slope of these straight-line graphs can give you the exponent value or rate of change, depending on whether the function is exponential or a power function.
Therefore, recognizing a linear relationship between transformed data points is vital in identifying the underlying functional relationship between variables.