Problem 18 - Thomas Calculus

Question

Make a table of values for the function \(F(x)=(x+2) /(x-2)\) at the points \(x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000\) \(x=10001 / 10000,\) and \(x=1 .\) a. Find the average rate of change of \(F(x)\) over the intervals \([1, x]\) for each \(x \neq 1\) in your table. b. Extending the table if necessary, try to determine the rate of change of \(F(x)\) at \(x=1.\)

Step-by-Step Solution

Answer

Create a table by evaluating \(F(x)\) at given points, then find average rates for intervals \([1,x]\), observing convergence towards an expected value as \(x\) nears 1.

Step 1: Calculate Function Values

Calculate the value of the function \(F(x) = \frac{x+2}{x-2}\) for each given \(x\): \(x = 1.2, \frac{11}{10}, \frac{101}{100}, \frac{1001}{1000}, \frac{10001}{10000}\). Use the function formula to substitute each \(x\) and simplify to get the values.

Step 2: Set Up the Table

Create a table with two columns. List each \(x\) value from Step 1 in the first column, and its corresponding \(F(x)\) value in the second column. This provides a clear overview of the calculated function values.

Step 3: Calculate Average Rate of Change

For each interval \([1, x]\), calculate the average rate of change using the formula \(\frac{F(x) - F(1)}{x - 1}\). Find \(F(1)\) by substituting \(x = 1\) into the function, and then use this value in the calculations for each \(x eq 1\).

Step 4: Extend the Table if Necessary

If needed, include more points closer to \(x = 1\) and repeat Steps 1 and 3 to get more accurate results and observe any trends. This step is crucial for understanding the rate of change at \(x = 1\).

Step 5: Analyze the Trends

Look at the average rates of change from Steps 3 and 4. Identify any patterns or convergence as \(x\) approaches 1. This will help infer the instantaneous rate of change at \(x = 1\).

Key Concepts

Function Values

Function values are the outputs you get when you plug in specific inputs into a function. In this case, you're looking at the function \( F(x) = \frac{x+2}{x-2} \). Each \( x \) value you choose will give you a different result when substituted into the function expression. To find these, substitute each of the given \( x \) values, like \( x = 1.2 \) or \( x = \frac{11}{10} \), into \( F(x) \). For example, when \( x = 1.2 \), substitute it into the equation to get \( F(1.2) = \frac{1.2 + 2}{1.2 - 2} \). Simplify this expression to find the specific function value at that point. Doing this process accurately is crucial because these function values form the basis for further calculations and help you understand how the function behaves at different points.

Table of Values

A table of values is an organized way to display the different inputs and their corresponding outputs for a function. It's helpful because it provides a quick reference to see how the function values change as \( x \) changes. For the function \( F(x) = \frac{x+2}{x-2} \), after calculating the function values for each given \( x \), you place those results into a table.

• Column 1 is for the \( x \) values, which might include numbers like \( 1.2, \frac{11}{10},\) etc.
• Column 2 is for the corresponding \( F(x) \) values you calculated.

This table not only makes it easier to see the outcomes but also helps when you move on to calculate the average rate of change. Tables help identify trends or patterns that might not be obvious when looking at standalone numbers.

Rate of Change

The rate of change gives you an idea of how much a function's output changes in relation to changes in its input. For functions, this is often referred to as the "average rate of change." To find the average rate of change of \( F(x) = \frac{x+2}{x-2} \) over the interval \([1, x]\), you use the formula \( \frac{F(x) - F(1)}{x - 1} \). This formula calculates how much \( F(x) \) increases or decreases per unit increase in \( x \) between the specific interval.Here's how you would apply it using the table of values:1. Find \( F(1) \) by substituting \( x = 1 \) into the function, which is a key point for these calculations.2. For each \( x \), calculate \( F(x) \) using the steps from the table.3. Substitute \( F(x) \) and \( F(1) \) into the rate of change formula to find the average rate for each interval.Knowing the rate of change is fundamental in understanding how dynamic the function is and how sensitive it is to changes in \( x \).

Instantaneous Rate of Change

The instantaneous rate of change at a specific point is like zooming in on the function to find the exact rate at which it changes at that point. This is different from the average rate, which is over an interval. Calculating the instantaneous rate of change can be tricky, especially when dealing with points where direct calculation may not be straightforward, like \( x = 1 \) in our function.In this scenario, you begin by taking a series of points that get closer and closer to \( x = 1 \). By calculating the average rate of change for these increasingly smaller intervals, you can observe the trend and infer the instantaneous rate.Seeing these rates converge towards a specific value indicates the instantaneous rate at that precise point. If you determine that the values are approaching a specific number as you extend and refine your table with values ever closer to \( x = 1 \), you can conclude this number is the rate of change at \( x = 1 \). This method connects to the concept of a limit in calculus, which formalizes the idea of an instantaneous rate of change.