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Calculus 4 - Euler's Number
This maximum limit, when the interest is calculated an infinite number of times, is Euler's number, e. If the bank could calculate interest every tiny fraction of a second, the $1 you put in would grow to about $2.71828 at the end of the year. This number, e, is special because it's the limit of how much something can grow in a year with continuous compounding at 100% interest.
In other words... it was calculated at the end of the year, it would be 2 dollars. But if you keep calculating in every moment, it would be $2.71828
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Calculus 2 - Integral vs differential
"Calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus. This strategy can be used on anything that we can imagine slicing endlessly. Such infinitely divisible things are called continua and are said to be continuous"
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A metaphor
Imagine calculus as a journey through a vast landscape. Differential calculus is like examining the intricate details of the terrain as you take small steps, focusing on the changes in elevation and slope at each point. It's akin to closely inspecting the bumps and curves of the landscape.
On the other hand, integral calculus is like zooming out and calculating the total area under the entire landscape. Instead of focusing on individual details, integral calculus helps you understand the cumulative effect, grasping the overall extent and shape of the terrain. It's as if you're appreciating the entirety of the landscape from a broader perspective.
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Calculus Part 1 - The area of a circle
It all stared with the mystery of curves and the proof for the area of a circle (which is a big curve). This was first achieved by Archimedes. He broke the circle into slices and reassembles them as a rectangle. He pretends that there is no curves - just very small straights. This slight of hand gave birth to integral calculus (and is the earliest use of the infinity principles). The formula for area is A = rC/2.
It soon became clear that calculus was the key to understanding nature and all the aspects which acted like curves - changing motion, object in light following archs, growth generally.
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Figma - Creating multiple components and combining as variants
Then
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