A closer look at the area function

The area function

We spent meaningful time becoming acquainted with the Continuous Factorial Function (or, if you prefer, the gamma function), which required being exposed to the area function. Remember the area function was more of a background operation that performed its work of calculating an area to be input into the Continuous Factorial Function. In this section, we will focus completely on this area function and, hopefully, uncover some interesting nuggets that may be new to you. Our approach, as usual, will be rather playful, so bring your imagination with you! Remember, the area function has this form:

$y=\dfrac{x^c}{e^x},$

where $c$ is some constant we input.

In the previous section, we viewed this area graph for several constants. Three of the constants we analyzed were 4, 4.5, and 5. As a review, Figure 1 is the same graph for the area function using these 3 constants.

Figure 1. Review of our area function with constant 4, 4.5, 5

Let’s take a closer peek at these area graphs in hopes of uncovering features that make them special.

A different view of the area function

Notice the peak of the curve is a little less than 10 for the constant 4.5, while the peak for the constant 4 is a little less than 5 and the peak for the constant 5 is greater than 20. That means the curve for the constant 4.5 is closer to the curve for the constant 4 than to the curve for the constant 5. In Figure 2, let’s retrieve our microscope (so to speak) and view the area graphs for constants between 4 and 5 in increments of 0.1 $(4,4.1,4.2,4.3,…,5)$ .

Figure 2: A nice crescendo

Isn’t that pretty? It looks like a nice crescendo, like an orchestra playing a symphony as it moves to the climax. This is a much more interesting curve than simply drawing a square of length $x$ , which was our area graph for the function $y=x^2$ . The Continuous Factorial Function certainly wins the prize for the better-looking area graph.

Since we’ve taken the time to draw these beautiful curves, let’s understand how they get to be so interesting. Notice all the curves have similar shapes. They start and end small but have a huge peak in the middle. How do we get these crescendo-type peaks? Here is the formula for the largest curve, when the constant is 5: $y=\dfrac{x^5}{e^x}$ . Remember we identified the variable $x$ resides in both the numerator and denominator. Since a fraction is also division, we can think of this fraction as the numerator divided by the denominator. Let’s review this function by comparing the numerator and denominator as the input $x$ changes.

Battle of numerator vs. denominator

Observe that both the numerator and denominator involve a form of multiplication and both increase as $x$ increases. But which one increases faster? The beauty of a fraction is we can instantly identify whether the numerator or denominator is larger at any moment based on whether the fraction is greater than 1 or less than 1. If the fraction is greater than 1, then the numerator is larger, and if the fraction is less than 1, the denominator is “winning.”

Both the numerator and denominator have input $x$ , but both bring a “partner” with them. The “partner” $x$ works with in the numerator is 5, while the “partner” $x$ works with in the denominator is $e$ , which is a smaller partner at about 2.718. I playfully view this as a heavyweight fight of multiplication where the denominator is the underdog by using the smaller number 2.718. Continuing this theme of a battle between the numerator and denominator for dominance, let’s review the strategy for each position.

Notice the numerator and denominator both fight for dominance by using multiplication, but they use different strategies. The numerator’s pattern for multiplication is to take the input and multiply 5 copies of itself. However, the number of times the denominator multiplies depends on the input $x$ . That means as the input $x$ increases, the number of multiplied copies of $e$ increases. In short, as $x$ increases, the numerator's strategy is to increase the number that gets multiplied while the denominator's strategy is to increase the number of times $e$ is multiplied by itself.

Of course, this perspective only works for integer values of $x$ since we can’t think of repeated multiplication for $e$ , for example, 4.3 times. Let’s start this heavyweight fight assuming the constant is 5. Who wins this match? If we view the result after “round 1” when $x = 1$ , the numerator is $1×1×1×1×1$ , so it is just stuck while the denominator is $e^1$ , or 2.718.

Since the denominator's value of 2.718 exceeds the numerator's value of 1, round 1 goes to the denominator. Round 2, when $x=2$ , clearly goes to the numerator as it leverages its hefty partner 5 to produce $2×2×2×2×2=32$ while the denominator only produces $2.718×2.718=7.389$ . Rounds 3, 4, and 5 clearly go to the numerator as the numerator is at its best after round 5, producing an impressive $5×5×5×5×5=3,125$ while the denominator is $2.718×2.718×2.718×2.718×2.718=148.413$ . Notice the ratio at this peak point is $3,125/148.413=21.06$ , which is the peak to this area function.

Lose a round but win the fight

At the end of round 5, both numerator and denominator multiply a number 5 times but because the numerator is favored and has a bigger partner, it clearly wins. However, the denominator is not trying to win a round but win the fight, so it’s planning for future rounds. Notice, after round 5, momentum shifts to the denominator; even though the numerator clearly wins, the denominator is gaining momentum.

Finally, in round 13 (when $x=13$ ), the denominator takes over because it is multiplying $e$ by itself 13 times, while the numerator is growing but not as fast as it multiplies 13 five times. Then, every round after round 13, the underdog denominator wins. If we try to explain the reason the denominator wins, we can state it is using the magic of exponential growth.

If we view the graphs for the other constants between 4 and 5, notice the peak always occurs for the value when $x$ equals the constant. Thus, when the constant is 4, the peak of the graph occurs at $x=4$ . After $x=4$ , the denominator’s strategy of exponential growth gains momentum and increases faster than the numerator.

Multiplication vs. exponentiation

In summary, the numerator and denominator appear similar at first glance. Both have a base number and both have an exponent. The difference is the “variable” $x$ is the base in the numerator but it is the exponent in the denominator. This seemingly small difference is quite important. We know multiplication is repeated addition and exponentiation is repeated multiplication.

In the numerator, as $x$ increases, we are increasing the number that is being multiplied, but the number of times we multiply remains constant (5 if $c=5$ ). In the denominator, as $x$ increases, we are increasing the number of times we perform multiplication. Eventually, the number of times we perform multiplication will produce a larger result than just increasing the number that we are multiplying by a fixed number of times.

This exponential growth is often how viruses spread (think COVID-19), which is why it is difficult to contain the virus from spreading. Even though the number may start small, over time, the number infected has the potential to grow very quickly. This explains why it is difficult to win the fight against exponential growth.

We have completed our Expanse 1 journey through the jungle of the infinite. Of course, we have only scratched the surface of what is available in math to see what happens when we extend concepts without bound. So our next journey in Lazarus Math is to travel to another level, Expanse 2, where we unleash the infinite.

💡 The way we learn math so often focuses on the destination but can sometimes ignore the “journey.” In Expanse 1, we have been introduced to the infinite and seen it in action. Our goal has been to see the infinite move using the most basic “ingredients” of input and operation. In the first section, we explored the harmonic series, which is an infinite sum that used the positive integers as input. In the second section, we expanded from the discrete to the continuous version with the expression

\frac{1}{x}

. Since we can’t add a continuous expression like we can a discrete expression, we calculated the area under the graph of

y=\frac{1}{x}

. So instead of adding numbers, we are adding area. The area under this graph defined the natural log function. Then we took the difference between the continuous and discrete in Section 3 and found the Euler-Mascheroni constant. In Section 4, we moved from addition to multiplication with the factorial function. Again, we wanted to expand from the discrete to the continuous, so we connected the dots with the Continuous Factorial Function, which is just a small input shift from the gamma function. We generated the Continuous Factorial Function by using another area function, which was our focus in section 5. We created this area function by graphing the ratio of two expressions that apply different growth processes. The numerator used a multiplicative process and the denominator used an exponential process. The area under this graph over all positive real numbers produced the precise result we needed for the Continuous Factorial Function. Yes, through our journey in Expanse 1, we did arrive at two important “destinations” in math: the natural log function and the gamma function. Both are extremely important functions in math, so math people should know how to use these functions. But the process of creating these functions was our focus and, in my opinion, was filled with intrigue and beauty.