Problem 76 - Brief Applied Calculus

Question

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter \(4 .\)

Step-by-Step Solution

Answer

The results approach a limiting value of approximately 2.71828, which is the constant \(e\).

Step 1: Enter the Function

Turn on your graphing calculator and access the function entry screen. Enter the function \( \left( 1 + \frac{1}{x} \right)^x \) for evaluation.

Step 2: Access Table Setup

Locate the 'TABLE' or 'TBLSET' option on the calculator menu. This feature allows you to set parameters for evaluating functions at specific values of \(x\).

Step 3: Set Table Parameters

In the table setup, set your starting value for \(x\) as 100. Decide an appropriate increment for future evaluations. You may choose increments like 10,000 or 100,000 to evaluate for higher values as indicated in the problem.

Step 4: View and Record Table Values

After setting up the table, press the Calculate (CALC) or Table button to generate the values. Record the evaluated function values for \(x = 100\), \(x = 10,000\), and \(x = 1,000,000\) from the table displayed.

Step 5: Analyze the Results

Observe the trend in the evaluated values. As \(x\) becomes larger, the value of \( \left( 1 + \frac{1}{x} \right)^x \) should approach a constant number. Compare these values to determine if they seem to approach a particular limiting value.

Step 6: Estimate the Limiting Value

Upon observation, the values should come closer to approximately 2.71828 for large values of \(x\). This number is a well-known mathematical constant, denoted as \(e\).

Key Concepts

Limiting Value

A limiting value is the value that a function approaches as the input approaches some point. For the function \( \left(1+\frac{1}{x}\right)^{x} \), the input \( x \) becomes very large, we anticipate that the outcomes are getting closer and closer to a particular number. The idea is simple:

• As \( x \) increases, the term \( \frac{1}{x} \) becomes tiny, approaching zero.
• The expression \( 1 + \frac{1}{x} \) nears exactly 1, but slightly greater.
• Exponentiating this small increment over large \( x \) results in a value that nears a constant.

An understanding of limiting values helps in identifying behaviors of functions over large domains and is crucial in calculus, particularly when dealing with infinite series or approaching extreme values.

Mathematical Constant e

The mathematical constant \( e \) is an irrational number approximately equal to 2.71828. It occurs frequently in mathematics, particularly in the context of exponential growth and complex analysis. Here’s why \( e \) is important:

• It is the base of natural logarithms, denoted as \( \ln(x) \).
• It's key in modeling exponential and logistic growth, such as population models or compound interest calculations.
• The number \( e \) emerges in calculus through the limiting expression \( \left(1+\frac{1}{x}\right)^{x} \).

Understanding the constant \( e \) begins with recognizing it as the "natural" growth factor for continuously compounding scenarios in both pure and applied mathematics.

Function Evaluation

Function evaluation is the process of calculating the value of a function for a specific input. A graphing calculator is a handy tool for this task, allowing you to input functions and specify ranges of values for which you want the output.How does function evaluation work?

• First, you input your desired function into the calculator. For instance, \( \left(1+\frac{1}{x}\right)^{x} \).
• You then specify the values of \( x \) to evaluate the function. This might be numbers like 100, 10,000, and 1,000,000.
• Using the calculator, observe how outputs change as these inputs grow. This helps identify pattern trends or limiting behaviors.

The practice of function evaluation is essential for visualizing the behavior of algebraic and transcendental functions, particularly when assessing limits and verifying continuity.