Problem 38 - Precalculus Mathematics for Calculus

Question

The Definition of \(e\) Illustrate the definition of the number \(e\) by graphing the curve \(y=(1+1 / x)^{x}\) and the line \(y=e\) on the same screen, using the viewing rectangle \([0,40]\) by \([0,4].\)

Step-by-Step Solution

Answer

The graph shows how \( \left(1+\frac{1}{x}\right)^x \) approaches \( e \) as \( x \to \infty \).

Step 1: Understanding the Problem

We need to graph the function \( y = \left(1 + \frac{1}{x}\right)^x \) along with the horizontal line \( y = e \) using the provided viewing rectangle of \([0, 40]\) for \( x \) and \([0, 4]\) for \( y \). The task is to visually demonstrate how as \( x \) increases, \( y = \left(1 + \frac{1}{x}\right)^x \) approaches the value of \( e \), which is approximately 2.718.

Step 2: Setting Up the Graph

To graph the function, we first need an appropriate software tool or graphing calculator that allows us to input the function and set the viewing window. Ensure you're able to input exponential functions and expressions with fractions.

Step 3: Plotting the Function

Enter the function \( y = \left(1 + \frac{1}{x}\right)^x \) into the calculator or graphing software. Adjust the settings to display the graph between \( x = 0 \) and \( x = 40 \) and \( y = 0 \) to \( y = 4 \). This will show how the function behaves as \( x \) increases.

Step 4: Adding the Line for \(e\)

Input the line \( y = e \) (use the value 2.718 for graphical purposes) to the same graph. This horizontal line will represent the value of \( e \) across the same viewing window. It provides a reference to compare where the curve approaches \( e \).

Step 5: Analyzing the Graph

After plotting, observe that the curve \( y = \left(1 + \frac{1}{x}\right)^x \) rises from below and approaches the line \( y = e \) as \( x \) becomes larger, but never actually touches it for smaller \( x \). This visual representation illustrates the definition of the constant \( e \) as the limit of \( \left(1 + \frac{1}{x}\right)^x \) as \( x \) approaches infinity.

Key Concepts

Limit of a Function

Understanding the limit of a function is essential, as it describes the behavior of the function as the input approaches a certain value. In the context of the function \( y = \left(1 + \frac{1}{x}\right)^x \), the limit as \( x \) approaches infinity is of particular interest. This function illustrates the approximation of Euler's number \( e \) in an elegant way. When we say the limit of \( y = \left(1 + \frac{1}{x}\right)^x \) as \( x \rightarrow \infty \) is \( e \), it means that for very large values of \( x \), the value of the function gets closer and closer to \( e \), which is approximately 2.718. However, it never really becomes exactly \( e \); instead, it hovers increasingly near this value as \( x \) grows larger and larger. Thinking about limits helps us understand concepts of convergence, where a sequence or function approaches a specific value. This is crucial in calculus and mathematical analysis, allowing mathematicians to define continuity, derivatives, and integrals in a precise manner.

Constant e (Euler's Number)

The constant \( e \), also known as Euler's number, is a fundamental mathematical constant approximately equal to 2.71828. It is not just an arbitrary number but holds a significant place in mathematics, much like pi (\( \pi \)). Euler's number \( e \) is the base of the natural logarithm and is key in growth processes, from compound interest in finances to population growth in biology. This number emerges when calculating the limit mentioned before: \( y = \left(1 + \frac{1}{x}\right)^x \) as \( x \rightarrow \infty \). \( e \) is unique because of its properties in calculus. For example, the derivative of \( e^x \) with respect to \( x \) is \( e^x \), which is an exceptional feature making it invaluable for exponential models in science and engineering.

Graphing Functions

Graphing functions allows us to visualize how functions behave across different values. It's not just about plotting points, but about understanding the overall shape and behavior of a function. In this exercise, we graph \( y = \left(1 + \frac{1}{x}\right)^x \) and compare it to a horizontal line at \( y = e \). To graph these functions:

• Choose a reliable graphing tool or calculator that supports plotting functions and viewing windows.
• Input the function \( y = \left(1 + \frac{1}{x}\right)^x \) and adjust the viewing window to suitable ranges for \( x \) and \( y \) as given \([0, 40] \times [0, 4]\).
• Overlay the line \( y = e \), using its numerical approximation 2.718, as a reference line across the graph.

Graphing helps in understanding the dynamics of functions. By observing how the function approaches \( y = e \), students gain insights into limits and continuity. This visualization reinforces theoretical knowledge by providing practical, observable evidence of complex mathematical behavior.