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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.

Before-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, such as 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 is the mysteries of curves. It consists of two main pillars: integral calculus and differential calculus. These are inverses operations. Differentiation is used when finding the slope of a curve, or rate of change, such as the speed of Usain Bolt in his 2009 100-meter race in Berlin. The rate of change of a straight line is the change in $y$ over the change in $x$. However, differentiation finds the instantaneous slope of a curve.

Integration finds the area under a curve. For example, if a train is moving at a non-linear speed, integration can find the distance the train traveled. Differentiation breaks a curve into infinitely many pieces, which are each infinitesimally small to find the slope at any given point. Integration adds up the areas of infinitely thin rectangles (the length $f(x)$ times the width $dx$), which gives you the area under the curve from any given point.

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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