Why Applications of Calculus in Real Life is Essential to Make Kids Love Math?

Differential calculus deals with the concept of rate of change. Meaning it aids us in finding an instantaneous change in a given timeframe. Examples of this include continually changing things that we need to make graphs of. Blood pressure, heart rate, stock markets, the weight of rockets, the speed of a runner, air pressure and temperature, the population of bacteria are some of the many vital examples. We use differential calculus when analyzing the curved graphs, or parabolas, that map these events to find instantaneous rate. That means that if we zoom in far enough into a parabola and see a linear line, we can find the average or the slope. An example of this is if you look at a photograph of Usain Bolt running, you don’t see him running. However, it is clear that he is running. To find the speed at which he is running at that exact instance (instantaneous rate), we use differential calculus.
In integral calculus, however, we find the curved area between two determined points. Any elementary student can find the area of a square. Yet, to find the area of a curved section, we have to split the graph into as many, even infinitely many, rectangles. After that, we find and add the areas of said rectangles together to find a very accurate estimate for the section’s area.
Left: mathematician Ali Nesin has built an oasis for math education in rural Turkey. For his efforts, he wins the 2018 Leelavati Prize. Source: Simon Foundation | Right: Steven Strogatz is a mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. Source: stevenstrogatz.com
As a young lawyer living in a dormitory, every night, when everyone retreated to their beds to sleep, Abraham Lincoln would read a book that possibly no other law student had read before. The book was written 2300 years ago by a Greek mathematician named Euclid, Euclid’s Elements. Contrary to popular belief, Element’s of Euclid wasn’t merely a book of geometry. Using triangles, lines, and circles, Euclid was explaining to humanity why certain things were absolute. After reading that book, Lincoln entirely changed the way he thought and even spoke. Lincoln associated the reason he read that book with wanting to learn how to respond to situations that he came across rationally. Soon after, he would come to lead a large nation, and the decisions he would make would affect millions’ lives. As a lawyer, he had to master rational thinking to impress the judges that he would go in front of.
Left: Abraham Lincoln was an American statesman and lawyer who served as the 16th president of the United States from 1861 until his assassination in 1865. — Wikipedia | Right: Lincoln’s favorite book, Elements of Euclid by TASCHEN

The odds of a student not being able to apply the things they learn in a math classroom outside of it are next to zero.

While it may seem absurd to you, I believe that a child learning how to prove something is one of their fundamental rights next to water, electricity, and access to the internet.

There was a complex problem in front of Archimedes, and he did not know how to overcome it. After a while, Archimedes figured out that he had to approach the problem from a different standpoint. First, he split the problem into smaller pieces. After all, managing small problems is a lot easier. He split the circle into four smaller pieces first and then arranged them as shown below. The area of the new shape had to be the same as that of the circle anyways.
Adapted from Steven Strogatz’s article, “Take It to the Limit” | Source
Archimedes had solved the problem. He quickly went one step further and split the circle into eight smaller pieces and arranged them as shown below. Archimedes was delighted. The resulting shape was pretty much a parallelogram. Meaning it was a shape he could use the base*height formula to find the area of.
Adapted from Steven Strogatz’s article, “Take It to the Limit” | Source
Archimedes then split the circle into 16, 32, and 64 pieces, which you may have expected if you approached the incident instinctively. The more pieces the circle was divided into, the more the resulting shape resembled a parallelogram. Archimedes concluded that if we were to infinitely do this operation, we would have the area of the circle and developed the πr² formula.
Adapted from Steven Strogatz’s article, “Take It to the Limit” | Source
I picked up and skimmed freshman texts in college bookstores, hoping to come across one that might help a mathematical ignoramus like me, who had spent his college year in the humanities — Herman Wouk, The Language God Talks, p.6
For example, they were curious about the people that lived on the other side of the world and invented wireless communication to reach them near instantly. They became curious about the Moon, which they saw every night, and sent a man up there. They wanted to free fall from the edge of space and were able to calculate the exact spot the free faller would land at. Invisible to the naked eye, they thought that the atom must have an incredible amount of energy, so they split the atom and discovered nuclear energy. They were tired of waiting nine months to see the gender of a baby, so they invented a way to know the baby’s state of health. They made maps detailed enough to see singular streets from the satellite images they took and effectively erased the struggle of finding your way home.
First, Maxwell took all of Faraday’s physics notes and mathematically formulated them with calculus’s help. Then, without changing them, he rewrote those formulas differently. Using one formula, he discovered others. He was trying to comprehend it using a different perspective. Finally, one of the equations he found was the answer to the question he was looking for. He turned all of Faraday's knowledge into a twenty differential equations with 20 equations. Those equations would explain how the magnetic field that Faraday described created electromagnetic waves. His work was later published as “On Physical Lines of Force” in March 1861.
Maxwell equations: A complete description of light, electricity and magnetism.
You can watch Tony DeRose’s “Pixar: The Math Behind the Movies.”
This technique came to be when Archimedes found the area of a curved shape by repeatedly dividing it into smaller sections. Meaning to find the area of a parabola, Archimedes drew a triangle as big as he could inside of it. Then, he drew more triangles, as big as possible, in the parabola’s empty spaces. Since he could find the area of those triangles, the combination of them must have been close to the area of the parabola. The more triangles he could fit into the parabola, the closer he would get to finding the area of it.
Polygons of increasing sides compared to a circle | Source Jason Ross
Many of you know that average speed is measured by dividing the distance traveled by the time it takes to travel said distance. Meaning if you travel 60 miles in one hour, your average speed would be 60 m/h. If you go 20 meters in 1 second, your speed would be 20 m/s. If you would like, given the required technology, you could find the average speed at one-thousandth of a second. Therefore, police radar machines measure the time it takes to travel between two points. They split that timeframe nearly infinitely many times to get your instantaneous speed. This operation, taking a derivative, is another branch of calculus.

The Mathematics Behind Usain Bolt’s Olympic Record

How can we find his maximum speed then? The time it took him to run the 100 meters gives us the average speed. Let’s shorten the distance and do the operation again then. For example, let’s make a measurement every 10 meters and take his average speed every 10 meters. Doing that shows us that at some parts, he was faster, while in others, he was slower. Will these measurements give us his maximum speed? If we look at it with the same logic, no. That is because every 10 meters is a long distance for finding the maximum speed. Therefore, if we take measurements every meter, we will have 100 measurements and have a more solid result. When you analyze the graph, you will notice that in some parts, he is slower. That is because of the friction that occurs whenever he puts his foot down. If we approach it instinctively, we know that we can find his average speed at every second or an even smaller amount of time. Of course, to do this, we would need very powerful machines.
Adapted from “Usain Bolt’s Split Times and the Power of Calculus” | Source

Calculus for Space

Meaning we have to monitor the rocket’s speed continually. However, countless factors constantly affect the speed of a rocket. As you may have thought, the rocket carries fuel and uses it to propel itself forward. The fuel’s mass is constantly changing, changing the mass of the rocket, affecting its speed. On the other hand, the rocket’s thrust is propelling it in the opposite direction with incredible force. That is because, according to Newton’s third law, for every action, there is an equal and opposite reaction. On another note, the rocket has to shed parts of itself at given moments. When we take all of these parameters into view, we need to find the rate of change of momentum for the rocket. We can only do that with derivatives.With the same logic, there is constantly changing temperature and pressure in the atmosphere. Today, using differential equations, meteorologists can estimate the weather.

The number of examples I have mentioned above can be multiplied unimaginably many times. In summary, we use calculus in our daily lives to mathematically model and analyze anything we want. When we make robots, video games, wind roses, talk about blood flow, analyze any piece of data, work on viruses, bacteria, and other organisms that spread rapidly, when we go to space, we always use calculus.

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better. alikayaspor@gmail.com