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If you have ever taken a calculus class, you would know that it’s the most fascinating topic you could study. The fabulous discovery of calculus is the reason you are reading this post. Algebra and geometry can’t send rockets to space, or even make a Nintendo Switch without the help of calculus. This and the next couple posts will show you how calculus is the language of the universe.

The Dawn of Calculus

Back in the 17th century, Pierre de Fermat and Rene Descartes brought algebra and geometry together, into analytic geometry. This is where the coordinate plane, or the $xy$ plane came. They realized they could write equations on a grid. Let’s say Sam ate walnut bread for breakfast today. Each walnut bread has 300 calories. If we write an equation, we get $y = 300x$ where $y$ is the number of calories and $x$ is the number of walnut breads eaten. The slope of this equation is 300.

This linear relationship can be graphed; at $x=1$, $y=300$. At $x=2$, $y=600$, etc. They soon realized they can write an equation for any shape. For a circle, it’s $x^2 + y^2 = 4$. For an ellipse, it’s $x^2 + 2y^2 = 4$. For a hyperbola, it’s $xy$ = 1, and so on. Pierre de Fermat realized and found techniques to solve difficult problems with graphs, like finding the area under it or finding the part of the graph where the slope is 0. As you’ll soon find out, these techiques led to calculus.

The Fundamental Theorem

Calculus has two main parts: integral calculus and differential calculus. The fundamental theorem of calculus suggests these are inverse operations. Differentiation breaks apart a problem into infinite pieces. Integration adds them all together. This is also known as the infinity principle. Let’s say we have a curve, $x^2$. Straigh lines are easier to work with than curves. Thankfully, calculus solves that.

Differentiating the curve gives us infinitely many straight lines, each with an infinitesimally small length. We use that to find the instantaneous slope of the graph at any given point. Integrating the curve, however, adds up the areas of infinitely thin rectangles (the length times the width $dx$). That gives you the area under the curve from any given two points. These topics will be covered later in the next posts.

Conclusion

Calculus solves the mystery of curves, motion, and change - those are what brought us here to this day. Without calculus, we wouldn’t have phones, iPads, or any modern technology that we have today. We wouldn’t even have some of the medicine that saves our bodies from illnesses. None of that is possible without differential and integral calculus.

“Calculus is the language God talks”
- Richard P. Feynman

Nobel Prize Winner On Quantum Electrodynamics

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