Problem 55 - Algebra and Trigonometry

Question

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

Graph \( y = \left(1 + \frac{1}{x}\right)^{x} \) and the horizontal line \( y = e \) over the specified range; observe the curve approaching the line.

Step 1: Introduction to the Problem

We are tasked with illustrating the definition of the number \( e \) by graphing two functions: \( y = \left(1 + \frac{1}{x}\right)^{x} \) and \( y = e \), within a given viewing rectangle of \( [0, 40] \) for the \( x \)-axis and \( [0, 4] \) for the \( y \)-axis.

Step 2: Graph the Function

To graph \( y = \left(1 + \frac{1}{x}\right)^{x} \), we need to plot this function while allowing \( x \) to take on values from 0 to 40. This will result in a curve that approaches the line \( y = e \) as \( x \) increases.

Step 3: Graph the Line

Next, graph the horizontal line \( y = e \). The number \( e \) is a mathematical constant approximately equal to 2.718. Therefore, this line will be horizontal across the plane at \( y = 2.718 \) within the viewing window of \( y \) from 0 to 4.

Step 4: Setting the Viewing Rectangle

Ensure the graphing tool is set to the viewing rectangle specified in the problem, with \( x \) ranging from 0 to 40 and \( y \) ranging from 0 to 4. This helps in capturing the key portion of the graph where \( y = \left(1 + \frac{1}{x}\right)^{x} \) starts approaching \( y = e \).

Step 5: Observing the Graphs

Once both graphs are plotted, observe how the curve \( y = \left(1 + \frac{1}{x}\right)^{x} \) starts from below the line \( y = e \) and approaches it as \( x \) increases, illustrating the limit definition of \( e \).

Step 6: Conclusion

The graph of \( y = \left(1 + \frac{1}{x}\right)^{x} \) approaches the horizontal line \( y = e \) as \( x \) increases, visually representing how the value of the expression approaches \( e \) as \( x \) tends to infinity.

Key Concepts

Graphing Functions

Graphing functions helps visualize mathematical relationships and understand complex ideas like trends and limits.
In this exercise, to illustrate the definition of the number \(e\), we graph two functions on the same coordinate plane:

• \(y = \left(1 + \frac{1}{x}\right)^{x}\)
• \(y = e\)

By plotting these functions, we observe how the curve of \(y = \left(1 + \frac{1}{x}\right)^{x}\) behaves as \(x\) changes.
It's best to think of this as taking a closer look at how values shift and move.
Utilizing the specific viewing rectangle

• \([0, 40]\) for the \(x\)-axis
• \([0, 4]\) for the \(y\)-axis

This range allows us to focus on how the function curves and meets the line \(y = e\). Over this range, the curve closely approaches the line—illustrating the process of approaching a limit.

Mathematical Constant

The number \(e\) is one of the most important mathematical constants and is approximately equal to 2.71828.
It is often referred to as Euler's number.
But what makes \(e\) so special?Much like \(\pi\) in geometry, which represents the ratio of the circumference of a circle to its diameter, \(e\) often appears in calculus and complex analysis, especially when dealing with rates of change and growth processes.Having a constant value means that \(e\) does not change; it's fixed and appears in various mathematical contexts, including exponents, logarithms, and certain probability theories.
This constant is central to the idea of continuous growth, making it essential in fields like finance, statistics, and natural sciences.

Limit Definition of e

The limit definition of \(e\) presents the number \(e\) as a value that the expression \(\left(1 + \frac{1}{x}\right)^{x}\) approaches as \(x\) approaches infinity.
Essentially, as \(x\) takes larger values, the expression gets closer and closer to \(e\).But what does this limit describe?Think of it as approaching an ultimate boundary—the value gets incredibly near to \(e\) without surpassing it within the function’s operation field.

• This property is critical in understanding exponential growth.
• Appears in calculations involving compound interest and population growth.

By illustrating this process through graphing, we can see how subtle changes in \(x\) influence the behavior of the overall expression, manifesting as a curve that steadily aligns with a horizontal line at \(y = e\). This visual representation helps consolidate how mathematical functions and constants interact through the power of limits.