# The Marvelous Real-World Applications of Calculus

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## Chapter 1: The Foundations of Calculus

Calculus has been described in various enlightening ways throughout history. Archimedes once referred to it as a revolutionary way of thinking, while Richard Feynman famously claimed it to be the language of the universe. Both perspectives hold truth, as calculus serves not only as a thought process but also as a tool for elucidating complex phenomena. Essentially, language is a verbal expression of our thoughts, and calculus translates abstract concepts into comprehensible forms.

Since Leibniz introduced calculus, mathematicians and physicists have harnessed its principles to explore laws of physics that were yet to be verified. Those with a solid grasp of mathematical reasoning have utilized calculus to tackle intricate problems in their surroundings. Thus, it's fair to assert that calculus is intertwined with nearly everything in our universe.

Calculus focuses on the dynamics of change. It provides us with the analytical tools to understand continuous transformations and aids in discovering optimal solutions to problems arising from these changes. The calculus we study today consists of two main branches: differential calculus and integral calculus, which can be viewed as complementary aspects of a single concept. Taking the derivative of a function and subsequently integrating it returns us to the original function.

Differential calculus is concerned with understanding rates of change—providing insight into instantaneous variations over specific intervals. This branch of calculus is essential for modeling various changing phenomena, such as blood pressure, heart rates, stock market fluctuations, and more. When graphed, these scenarios often yield curves or parabolas. By zooming in on a point on such a graph, we can identify the slope or average rate of change at that instant. For instance, when observing Usain Bolt in action, we can use differential calculus to determine his speed at any given moment.

In contrast, integral calculus involves calculating the area under curves between defined points. While calculating the area of a square is straightforward, determining the area of a curve requires partitioning it into numerous rectangles—potentially an infinite number—and summing their areas for an accurate estimate.

In the following sections, I will delve deeper into the practical applications of calculus, a topic I have passionately researched. I owe much of my understanding to inspiring lectures by distinguished mathematicians like Ali Nesin and Steven Strogatz. Through this article, I aim to convey their teachings in my own words.

### Section 1.1: The Educational Impact of Mathematics

During my university studies, I had the privilege of learning from the brilliant Ali Nesin. A true enthusiast for mathematics, he consistently encouraged us to adopt a mathematical mindset. He asserted that everyone, from the local baker to the nation's politicians, could benefit from this way of thinking. While this notion may seem idealistic, there are indeed politicians who embody this approach.

A notable example of applying mathematical thinking in real life is Abraham Lincoln's fascination with Euclid's *Elements*. This ancient text, often mistaken for merely a geometry book, elucidates fundamental truths through geometric principles. Lincoln's engagement with Euclid transformed his thought processes and communication skills, ultimately influencing his leadership of a nation.

This narrative, taken from Carl Sandburg's *Abraham Lincoln, the Prairie Years and the War Years*, alongside William Dunham's *Journey Through Genius*, exemplifies how mathematics can bridge theoretical knowledge and practical application. In a well-functioning education system, students should be able to relate their learning to real-world contexts, enhancing its significance in their lives.

As a mathematics educator, I found this perspective captivating. Throughout history, mathematicians and physicists have sought to use mathematics to explain our universe and the divine. The ancient Greeks' interest in geometry stemmed from their quest to comprehend the cosmos. This illustrates that the essence of a sound educational framework has been understood for millennia, and contemporary mathematics educators are fortunate to inherit this legacy.

Unfortunately, statistics reveal a disheartening reality: mathematics remains a source of anxiety for many students. According to the Institute of Education Sciences, approximately 93 percent of Americans experience some degree of math-related stress. Yet, mathematics serves as a crucial link to our tangible world.

The idea that students cannot apply what they learn in math class to real life is unfounded. So why do we struggle to achieve this in education? Several factors contribute to this issue, including teacher performance relative to salaries, excessive administrative work, and ineffective teaching methods. However, I believe the root cause lies within the curriculum itself. Teachers often present immediate results without exploring the underlying reasons behind mathematical concepts. When students ask, "Why are we learning this?" they frequently receive generic answers that fail to captivate their curiosity.

One of my greatest frustrations is that students are introduced to the concept of proof far too late in their education. The notion of proof should be introduced as soon as students begin to develop abstract thinking skills. For instance, when students learn to calculate the area of a circle using the formula πr², they often memorize it without understanding its derivation. This, in my view, constitutes a significant flaw in mathematics education. It's imperative that we teach students the foundational principles of mathematical thinking, or in essence, calculus.

### Section 1.2: The Essence of Calculus

Teaching students how to prove concepts is, in my opinion, a fundamental right, akin to access to essential resources. Moreover, imparting this method of thinking is not overly complex. For example, Steven Strogatz eloquently explains the origins of calculus and its applications in his book *Infinite Powers: How Calculus Reveals the Secrets of the Universe*.

Historically, calculating the area of regular shapes was straightforward. However, determining the area of curved shapes like circles posed a significant challenge that early mathematicians grappled with. Archimedes, recognized for his discovery of buoyancy principles, made perhaps his most remarkable contribution in developing mathematical reasoning to find the area of a circle.

Faced with a complex challenge, Archimedes approached the problem by simplifying it into manageable parts. He initially divided the circle into four segments, rearranging them into a new shape with an area equivalent to the circle. Subsequently, he increased the number of segments to eight, then 16, 32, and so forth. As he continued this process, the resulting shape increasingly resembled a parallelogram, allowing him to utilize the area formula for rectangular shapes. Ultimately, he concluded that by performing this operation infinitely, he could derive the area of the circle and develop the formula πr².

This example encapsulates the essence of calculus: breaking down complex problems into simpler components and addressing them individually before recombining the results. While calculus may seem daunting, its core principle revolves around this straightforward methodology. The unique aspect of calculus lies in the infinite division of problems, whether taking derivatives or integrals.

Despite its significance, this fundamental concept is often overlooked in education. It perplexes me that an educational system that champions real-world applications fails to convey the beauty and simplicity of calculus to its students. This mathematical framework serves as a language through which we can articulate the complexities of our surroundings.

As mentioned earlier, Richard Feynman encapsulated this sentiment by stating that "calculus is the language in which God speaks." While some may debate Feynman's beliefs, Steven Strogatz provides insightful explanations in his lecture titled *The Beauty of Calculus* at Yale University.

The second video title is *What is Calculus used for? | How to use calculus in real life - YouTube*. This video delves into the various practical uses of calculus, illustrating its importance in understanding and solving real-world problems.

In a conversation with Herman Wouk, a celebrated author, Feynman emphasized the necessity of learning calculus, asserting, "You had better learn it; it's the language God talks." Wouk, recognizing the significance of this advice, sought to familiarize himself with calculus, eventually authoring *The Language God Talks*, which explores the intersection of science and faith.

Calculus has been a cornerstone for scientists seeking to comprehend the universe. Notably, Isaac Newton used calculus to articulate the forces governing celestial bodies, while others have employed it to unlock the mysteries of electromagnetism, as exemplified by James Clerk Maxwell's groundbreaking equations.

In the realms of technology and medicine, calculus continues to play an integral role. For example, advanced graphics in computer animations rely on the principles of calculus for creating lifelike images. Similarly, medical training simulations utilize calculus-based models to ensure precision in surgical practices, as seen in the New York Face Transplant Program, where detailed anatomical replicas are crafted with mathematical accuracy.

The applications of calculus extend to analyzing instantaneous rates of change, such as speed. Police radar guns, for instance, employ calculus to measure vehicle speeds accurately, demonstrating how this mathematical discipline permeates everyday life.

Calculus also plays a pivotal role in sports analytics. When examining Usain Bolt's record-breaking sprint, we can use calculus to determine his maximum speed and analyze performance nuances during the race.

### Section 1.3: The Future of Calculus

As humanity aspires to explore outer space, understanding the dynamics of rocket propulsion becomes essential. Calculus enables scientists to model the changing mass and momentum of rockets, allowing for accurate predictions of their trajectories.

In meteorology, differential equations facilitate weather forecasting by accounting for the continuous changes in atmospheric conditions. The myriad examples of calculus in our daily lives underscore its vital role in advancing our understanding of the world.

In conclusion, I express my gratitude to pioneers like Archimedes, Leibniz, and Newton, whose contributions have shaped our understanding of mathematics and its applications.

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