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 functional relationship is exponential: \(y = 4 \times 10^x\).

Step 1: Identify Possible Models

We need to decide whether the data fits an exponential model, of the form \(y = ab^x\), or a power model, of the form \(y = ax^b\). Depending on the model, we will apply different logarithmic transformations.

Step 2: Logarithmic Transformations

For the exponential model \(y = ab^x\), taking the natural log gives \(\ln y = \ln a + x\ln b\). For the power model \(y = ax^b\), taking the log base 10 gives \(\log y = \log a + b \log x\). We apply the transformations suitable for each model to the data.

Step 3: Transform Data for Exponential Model

Compute \(\ln y\) for each \(y\) value in the table: \[\begin{align*}\ln(0.398) & \approx -0.921, \\ln(1.26) & \approx 0.232, \\ln(4) & \approx 1.386, \\ln(12.68) & \approx 2.541, \\ln(40.18) & \approx 3.692.\end{align*}\] Then plot \(x\) against \(\ln y\). If the points form a straight line, the relationship is exponential.

Step 4: Transform Data for Power Model

Compute \(\log y\) and \(\log x\) for each pair \((x, y)\): \[\begin{align*}\log(0.398) & \approx -0.400, & \log(-1) & \text{not defined}, \\log(1.26) & \approx 0.100, & \log(-0.5) & \text{not defined}, \\log(4) & \approx 0.602, & \log(0) & \text{not defined}, \\log(12.68) & \approx 1.103, & \log(0.5) & \approx -0.301, \\log(40.18) & \approx 1.604, & \log(1) & = 0.\end{align*}\]Recall: The logarithm of negative or zero values is undefined, which eliminates the power model since some \(x\) values are negative.

Step 5: Conclusion from Graph

Since the logarithms for negative and zero \(x\) values are undefined, making graphing for the power function impossible, we focus on the graph of \(x\) versus \(\ln y\). In this case, plotting these points should reveal a straight line, confirming an exponential relationship.

Step 6: Determine Exponential Relationship

Once a linear relationship in the graph confirms an exponential model, use two points to find the slope \(\ln b\) and \ln a using \(\ln y = \ln a + x \ln b\). For example, using points \((0, 1.386)\) and \((1, 3.692)\):\[\begin{align*}1.306 & = 0 + \ln b, \3.692 & = 1.386 + 1\ln b,\end{align*}\]Re-solve equations for \(a\) and \(b\):\(b \approx 10\) and \(a \approx 4\). The exponential relationship is \(y = 4 \times 10^x\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model situations where a quantity grows or decays at a constant percentage rate over time. The classic form of an exponential function is given by the equation \(y = ab^x\), where:

• \(a\) is the initial value of \(y\) when \(x=0\).
• \(b\) is the base, which determines the rate of growth or decay; if \(b > 1\) the function represents growth, if \(0 < b < 1\) it represents decay.
• \(x\) represents the independent variable or time.

Exponential functions are widespread in real life, including in cases such as compound interest, population growth, and radioactive decay. This pervasive nature makes understanding how they function essential, especially when deciding if data fits an exponential model over another type. In this exercise, understanding how to express \(y\) in terms of \(ab^x\) allows us to see if our original table data can be reformatted into a straight-line form using logarithmic transformations, hence confirming exponential behavior.

Power Functions

Power functions are another type of mathematical relationship, often written as \(y = ax^b\), where:

• \(a\) is a constant that scales the function.
• \(b\) is the exponent which determines the function's degree, affecting the steepness of the curve.
• \(x\) is the independent variable raised to the power \(b\).

Power functions can depict phenomena where the rate of change is not constant but rather depends on the value of \(x\) itself. For instance, area and volume calculations based on diameter or radius employ power functions. However, in our specific problem where negative \(x\) values are presented, power functions face a limitation due to the undefined nature of taking logarithms of negative numbers or zero. This limitation is critical since it restricts the ability to linearize the data for a power function, helping us decide against its applicability in favor of an exponential model.

Logarithmic Transformations

Logarithmic transformations are crucial techniques used to linearize relationships, especially when dealing with exponential and power functions. These transformations change a multiplicative relationship into an additive one, making patterns clearer. For exponential functions, the transformation is by taking the natural logarithm of \(y\), converting \(y = ab^x\) into \(\ln y = \ln a + x \ln b\). This formula’s linearity is beneficial for graphing and analysis.
Similarly, for power functions, applying the transformation \(\log y = \log a + b \log x\) simplifies \(y = ax^b\) so we can analyze the linearity between \(\log x\) and \(\log y\). However, in our exercise with some \(x\) values negative or zero, this approach isn't feasible due to logarithmic limitations preventing accurate transformation and graphing for power functions. Logarithmic transformations become an analytical tool that assists in determining whether a relationship follows an exponential pattern by changing the nature of the graph based on the type of function we are examining.

Graphing Techniques

Graphing techniques help visually determine the nature of a relationship between variables. A powerful graphing method involves transforming data to assess if a linear relationship is present. In this case, by plotting \(x\) against \(\ln y\), we establish whether the data adheres to an exponential model. A graph that yields a straight line after transformation strongly signals an exponential connection.

• If the plots of \(x\) against \(\ln y\) reveal a clear straight line, then we're observing evidence of \(y = ab^x\).
• If instead, the graph were plotted as \(\log y\) against \(\log x\) and showed linearity (in an appropriate scenario where \(\log x\) is defined), the data would suggest a power function.

However, given our problem’s context with undefined \(\log x\) values, this is not pursued. By graphically confirming the line through points, we provide visual clarity and strengthen the interpretation that our data set most likely follows an exponential pattern.