Problem 58 - Precalculus: Functions and Graphs

Question

The data in the table are generated by the function \(f .\) Graphically approximate the unknown constants \(a\) and \(b\) to four decimal places. $$f(x)=a \ln b x$$ $$\begin{array}{|cc|} \hline x & f(x) \\ \hline 1 & -8.2080 \\ 2 & -11.7400 \\ \hline 3 & -13.8061 \\ \hline 4 & -15.2720 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\( a = -5.0921 \), \( b = 1.7323 \)

Step 1: Understand the Problem

We have a function of the form \( f(x) = a \ln(bx) \) and a set of data points from a table. Our goal is to determine the unknown constants \( a \) and \( b \) so that the function accurately fits the data.

Step 2: Set Up the Equation for Known Points

Use the data points given in the table to set up equations. Starting with \( x = 1 \), the equation becomes \( a \ln(b \times 1) = -8.2080 \). Therefore, \( a \ln b = -8.2080 \). Similarly, for \( x = 2 \), the equation is \( a \ln(2b) = -11.7400 \). Continue setting up equations for each data point.

Step 3: Simplify the Equations

For each equation, simplify by using properties of logarithms. For example, the second equation becomes \( a (\ln 2 + \ln b) = -11.7400 \). Therefore, we can express it as \( a \ln b + a \ln 2 = -11.7400 \). This gives us a system of equations in \( a \) and \( \ln b \).

Step 4: Solve Using Two Data Points

Choose any two equations to solve for \( a \) and \( b \). Use the equations from the points \( x = 1 \) and \( x = 2 \). These are \( a \ln b = -8.2080 \) and \( a \ln b + a \ln 2 = -11.7400 \). Subtract the first from the second to get \( a \ln 2 = -3.5320 \). Solve for \( a \): \( a = \frac{-3.5320}{\ln 2} \).

Step 5: Calculate \( b \)

With \( a \) known, substitute back into \( a \ln b = -8.2080 \) to solve for \( b \). Compute \( \ln b = \frac{-8.2080}{a} \), and then \( b = e^{\ln b} \).

Step 6: Verify the Solution

Check that the calculated values of \( a \) and \( b \) satisfy all given data points in the table. If not, reassess the calculations or try a different pair of data points. Once satisfaction is obtained for all points, finalize \( a \) and \( b \).

Step 7: Approximate to Four Decimal Places

Round \( a \) and \( b \) to four decimal places to match the requirement. Ensure all calculations maintain this level of precision.

Key Concepts

Approximation

In mathematics, approximation is a fundamental concept where we find values that are close to the exact solution. Here, we are approximating the constants \(a\) and \(b\) in the logarithmic function \(f(x) = a \ln(bx)\). The goal is to get these constants as accurate as possible, down to four decimal places.
To achieve this, we use data points provided in a table. By setting up equations for each data point and performing manipulations, we estimate the values needed. Since rounding is necessary for practical usage, ensuring the values are computed precisely before any approximations are taken is crucial.
Whenever approximation is involved, consider numerical methods like iterative refinements, which can enhance accuracy. In practice, these methods ensure our calculations converge to the desired precision level, minimizing errors.

Solving Equations

To solve the logarithmic equations derived from the data points, we follow a systematic approach. We have equations involving \(a\) and \(\ln b\). For instance, with \(x = 1\), the equation is \(a \ln b = -8.2080\).
The process involves transforming and simplifying such equations using logarithmic identities. For example, properties like \(\ln(ab) = \ln a + \ln b\) help break down the problem into smaller parts.
The goal is to create a system of equations that isolates variables. By subtracting similar equations, you can solve for one of the constants, say \(a\). Then substitute back to find \(b\). Remember, the most reliable way to solve these types of problems involves accessing algebraic manipulation, ensuring each step logically follows from the previous.

Data Analysis

Data analysis in this context involves interpreting and using the table's information to derive mathematical formulas. The table lists different \(x\) values and their corresponding \(f(x)\) results, which the function \(f(x) = a \ln(bx)\) must fit.
By analyzing the change in \(f(x)\) as \(x\) varies, we can spot if the data follows a clear pattern indicative of such a function. This analysis helps in hypothesizing the correct transformations and manipulations necessary for solving for \(a\) and \(b\).
Additionally, effective data analysis ensures the end result consistently represents the data within the desirable accuracy. Always verify and, if necessary, iterate the process to refine our approximation.

Graphical Method

The graphical method is a visual approach to solving the problems with unknowns. Here, plotting \(f(x)\) against \(x\) for different values helps visualize if your calculated \(a\) and \(b\) give a reasonable approximation.
This is particularly useful when initial algebraic solutions might be inaccurate. By drawing a graph, you can compare the calculated function with actual data, adjusting the constants as needed.
Graphing tools or software can automate this comparison, providing a quick assessment of how well your model fits the data points. Adjustments can be iteratively made until the graph of your function aligns closely with the graph of actual data, confirming the reliability and accuracy of your approximations.