# What exactly is calculus?

I’m an engineering student so yea I got calculus. My concern is, while I can solve calculus I don’t really understand what it is really all about. I want to understand it in like objective manner (idk if this make sense).

In: Mathematics

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On the largest scales, the universe acts like a thing that can be sliced and carved into the differentials you spend a year or two learning about in school. Right?

No. Every good law has its exceptions. There is only one constant in the universe, and that is the speed of light. From there, things get weird.

Nothing says math has to make sense.

Quite a few people fried their brains trying to prove otherwise. We can discover laws and watch reality adhere *beautifully* to those laws.

Good luck finding out the *why* behind those physical laws… and if you’re on some sort of search for ultimate meaning, don’t delve too deep. I’ve seen how that movie ends. You end up carving out your eyes and screaming at Laurence Fishburne.

*Liberate… me… ex inferis*

I’ll throw my hat in the ring. Calculus is about *limits*. Without limits, we can estimate a lot of things. We can estimate how fast a car is going by measuring where it is at a couple different times and seeing how far it has gone. We can estimate how much water is in a lake by measuring the distance across the lake at a bunch of places and adding up rectangles. We can estimate a lot.

Calculus allows us to take those estimates and see what happens to them if we allow the time step or the rectangle width to get really really small. As small as possible. It turns out that we can do math with those really small numbers, and that we can use that math to solve really important problems like how to send a spaceship to mars or how to make a better car engine.

Calculus allows us to take estimation, which we know how to do pretty well, and turn it into exact results, which is much harder.

It’s the branch of math that Isaac Newton invented. He wanted to find the slope of a single point on a parabola. It’s useful when you want to find the instantaneous speed of an accelerating object.

Calculus is about slicing things (like time, distance and area) up into smaller and smaller pieces, until those pieces are infinitesimally small, and being able to use those infinitesimally small pieces to make extremely precise calculations. For example, when we say a car travels at 100 km per hour, we can start by measuring the 100 km distance the car traveled in one hour, but that doesn’t tell us how fast the car was travelling at any particular point in its journey unless we assume a single uniform speed (and real systems simply don’t work that way). If we take two measurements – one at 30 min and another at 60 min – we can now understand how fast the car traveled on average during each half of it’s trip. Maybe it was 90 km per hour for the first 1/2 hour, and 110 km hour for the second half, so overall it would be 100 km per hour. Now we have more precise information. We can take measurements at 4 intervals, or 8 or 16 or 32, etc., with each measurement getting closer and closer to knowing the speed of the car at any particular instant. Taking this to its extreme (but logical) conclusion, we can determine the speed of the car at every instant through calculus, and it turns out this process gives us a lot of very useful information and insights about how things work in the real world (including how the speed is changing at any particular moment).

it’s basically like instant math. we want to know at any point on a function how fast that function is changing in real time.

Do you remember having to memorize formulas for area and volume of various 2d and 3d shapes? Calculus will let you figure out those formulas for yourself, when you run across a shape nobody gave you the formula for, or just can’t be bothered to look it up.

From an engineering perspective, you might need to figure out the volume of a tank or displacement of a ship from its shape, or you might have the measured power output of an engine and need to figure out how much energy it produced over the whole test run, so you can calculate efficiency.

calculus is the math behind change. Right now, you’re just learning the math, not how to use it. That’ll come later in your engineering classes (if you let me know what type i might be able to give you some examples, i’m working on my masters in mechanical) and differential equations.

It’s about understanding rates of change, and how those rates of change affect each other. Valuable when trying to understand complex systems of interacting forces.