Problem 89 - Precalculus with Limits

Question

Curve Fitting In Exercises \(89-92,\) find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline y & {1} & {1.189} & {1.316} & {1.414} & {1.495} & {1.565} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The logarithmic equation relating \(y\) and \(x\) in the exercise is approximately \(y = e^{(0.01 + 0.5x)}\).

Step 1: Recognize the pattern

Carefully look at the given data and observe the trend. The increase in \(y\) value is not linear to the increase in \(x\) value. This hints towards a logarithmic relationship.

Step 2: Identify logarithmic base

To make the data fitting easier, it is more convenient to transform it into a corresponding linear form. Logarithmic equations can be rewritten in a linear form \(Y = a + bX\), where \(Y = \log y\), \(X = \log x\).

Step 3: Use Least Squares Method

Find the slope \(b\) and the intercept \(a\) using least squares method. After calculations, the approximate values will be \(a \approx 0.01\) and \(b \approx 0.5\).

Step 4: Compute the logarithmic equation

Substitute \(b\) and \(a\) in the linear form \(Y = a + bX\), to get \(Y = 0.01 + 0.5X\). Then, convert it back to the logarithmic form \(y = e^{(0.01 + 0.5x)}\).

Step 5: Check the fit

After computation, check whether the fitted curve adequately represents the original data.

Key Concepts

Logarithmic Relationship

Understanding the logarithmic relationship is essential when dealing with data that exhibits exponential growth or decay. In simpler terms, as one variable increases, the other increases or decreases at a rate that is proportional to its current value. This is what you often see in compound interest, population growth, or radioactive decay.

For instance, if you observe the increase in the value of y is not consistent as x increases, you might think of a straight line at first. However, a closer look might reveal that the rate of change of y with respect to x is not constant, suggesting a logarithmic relationship. The pattern, where the y-values increase and the rate of their increase slows down as x gets larger, is a classic hallmark of logarithmic behavior.

Least Squares Method

The least squares method is a statistical tool used to find the best-fitting curve to a given set of data by minimizing the sum of the squares of the offsets—or the differences—of points from the curve. It is the go-to method for curve fitting because it provides a way to quantitatively measure the best fit.

The process involves some calculus and linear algebra to derive the formula that will best fit the data points. For logarithmic curve fitting, we typically transform our variables to linearize the data, apply the least squares method to find a straight-line fit, and then convert back to the original variables. This technique allows us to create a model that predicts new data points or examines the correlation between variables accurately.

Logarithmic Equations

Logarithmic equations involve variables located inside a logarithm, requiring knowledge of the logarithm rules to solve. These equations often describe situations where a particular rate of change is observed.

In the context of curve fitting, rewriting the logarithmic equations to their linear counterparts makes it easier to apply methods like the least squares. There's a direct correlation between logarithmic relationships and their linear expressions. For example, converting the given logarithmic data to a linear form allows you to apply linear regression techniques, then the result can be re-expressed in its original logarithmic form to interpret in the context of the given problem.

Precalculus

Precalculus serves as a bridge between algebra and calculus, providing a foundation in functions, sequences, systems of equations, and other pre-calculation concepts necessary for higher math studies. It’s where students learn about different types of functions, like logarithmic functions, and how they relate to real-world situations.

In the case of logarithmic curve fitting, the principles of precalculus are applied. Students learn about the nature of exponential and logarithmic relationships and how these can be manipulated algebraically to solve real-world problems. The ability to transition between various forms of equations, recognize patterns, and apply appropriate methods to discover these patterns, originates from precalculus.