Explain Like I'm 5, lifted with permission from Reddit username: u/practical_cartoonist

Practical_Cartoonist:
What is Calculus, Explain Like I'm Five:
Imagine you've got a big bowl full of water, and there's a little hole in the bottom of the bowl that's letting the water out. How long does it take for all the water to pour out? This is the sort of question you frequently deal with in a calculus class.
Of course you can't answer it because I haven't given you any numbers yet. But even if I had given you numbers, you would probably have trouble thinking how to approach it without the help of calculus.
The reason is that I can't give you a fixed rate for how fast the water is pouring out. If I said it was pouring out at 0.2L/second and there were 20L of water, then it would be a trivial problem to solve: 20/0.2 is 100, so it would take 100 seconds.
But that's not how it works in the real world. In the real world, the water would pour out faster at the start, and slower towards the end. At the start, when the bowl is full of water, there's a lot of heavy water putting pressure on the water at the bottom of the bowl, pushing it out faster. Near the end, when the bowl is nearly empty, the water might just be dribbling out.
It might be that the water's coming out at 0.4L/s at the start and 0.1L/s near the end. Hm. That makes this quite a bit trickier to solve. What can we do about this?
Welcome to calculus!
Calculus was invented to reason about rates of change. If you drop a ball from a shelf, how long does it take to hit the ground? Well it's a bit tricky, because the ball is moving very slowly at the start, and very quickly at the end. It speeds up as it falls, which means we can't just do a simple "distance divided by speed" calculation to figure out how long it takes to hit the ground. We need calculus.
There are two directions to calculus.
"If I know how long it took to hit the ground, can I calculate how quickly it was speeding up?". That's called "derivation".
"If I know how quickly it's speeding up, can I calculate how long it takes to hit the ground?". That's called "integration".
Calculus is going from "rate of change" to "the effect of that change", or vice versa.
What you will learn in calculus is how to do this (in both directions) when your mathematical formula looks in certain ways. Does it look like a polynomial? Follow this rule. Does it look like an exponential? Follow this other rule. Uh oh, it's a fraction. Follow that rule. Oh it's got "sin" and "cos" things in it. Guess what, even more rules.
What you will learn in "pre-calculus" is:
1.        ⁠Practising how to do algebraic manipulations. This is important because in calculus you need to learn a lot of rules, and those rules need you to be comfortable with algebra.
2.        ⁠Limits, which are the theoretical foundation for calculus, and allow you to understand why the rules are actually correct.


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