This is exactly the kind of content I love to see -- interesting technical topic, completely outside of any area of expertise of mine, clearly presented, interesting, and pretty obviously useful. Please post more!
Social scientist here, and I have no clue of what you're talking about. Calculus destroyed my engineering aspirations.
Then come derivatives. So you have a function graph, and you're trying to determine the slope at a certain point. It's really simple for say y = x, where you just take any two points, but what if the graph is curvy? You'd take the point and then a point close to it, and calculate the slope for that. If the function is fairly smooth, that's fairly close to the actual slope. So you push the other point closer to the original point, and the slope you calculate gets closer to the real slope. You edge closer and closer, and suddenly it sounds a lot like a limit, where your initial point is x, your second point x + h, and h becomes smaller and smaller, so h -> 0.
Here's what's cool though. Since you didn't use 5, or 7.23, but x, you can put any point into this, so you've got a function that maps x onto the slope at x of the original function (at least if I remember correctly). You play around with a couple of different kinds of functions and arrive at various rules for differentiating things without having to do the whole limit thing, like x^2 -> 2x, x^3 -> 3x^2, e^x -> e^x, etc.
And derivatives are really handy. Say in kinematics, velocity is the derivative over time of position, acceleration is the derivative over time of velocity, and suddenly a whole bunch of things make a lot of sense.
Integrals are pretty neat too. So you want to estimate the area under a function from point a to point b. so you draw a box, and there's an area you're missing, or an area you're including but shouldn't, but it's close to the actual area underneath the function. So you think...hrm, I could use two boxes of half the width, that'd be closer to what I want to find. Then three, ten, a hundred, with the sum of the area of boxes getting closer to the actual area. Said more mathy, it'd be a to b, with a width of b-a, and each box gets a width b-a/h, and h gets infinitely large. Then we make that little limit jump again, and get a function. We do that for a couple of functions, deduce some rules, so we don't need to go through the entire process whilst integrating simple functions. a -> x + C, x -> 1/2x^2 + C for example.
With some understanding of the FTC (fundamental theorem of calculus), which roughly states that derivatives and integrals are inverse operations of each other (isn't that neat?), it again helps us make sense of things. For example, look at the equations in physics for constant accelerations: v = v_i + at, x = x_i + vt + 1/2at^2. See how v is the integral of a, and x is the integral of v? For the first one, "v_i" is the constant factor C, and "at" is what comes from the factor a. For the second one, "x_i" is the constant factor C, "vt" is "v_i" integrated, and "1/2at^2" is the integral of "at".
Calculus is super neat.
It generally worked well, with the exception of false negatives for cartoon nudity and false positives for pastrami.
https://recogmission.wordpress.com/2009/05/01/adult-image-de...