if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?
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ELi5: College Calculus
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They are not so much “opposite” as they are the reverse operation. If curve B is the area under curve A then curve A will equal the slope of curve B.
For example. Say curve A is a line leaving from the origin. At a given value of x (horizontal axis) the area under will be a triangle with base length x and height=slope*x. So area=0.5*slope*x^2. So curve B will be a parabola. A parabola has an ever increasing slope. Curve A represents that increasing slope as the line. Derivative is the process to go in one direction. Integral is the process to go back in the other direction.
You’re jumping a step here. If you take the derivative of a function, you would have to integrate twice to get to the area. Integrating once only gets you back to the original function.
Ain’t no way a 5 year old can get any of that
My physics teacher in high school drilled the “d-a-v-t-u” functions into us – distance, acceleration, velocity, time, potential enegy. In particular, distance, velocity, and acceleration are related to each other through time and *calculus* relationships.
Velocity is the derivative of the position function Z(t), where Z are three Cartesian coordinates, i.e. Z'(t). Acceleration is the derivative of the velocity function, so it is Z”(t).
Conversely, velocity is the integral of the acceleration function, and position is the integral of the velocity function.
Think of what that means if you are running. The more ground you cover, the greater Z is getting. If you’re running through a forest, you might have to go slower on uphills, and faster on downhills. The slower you run, the less ground you cover each second, right? If you look at a chart of position v time, your line will start drooping to the right. As the slope of that line gets flatter, your speed is also slowing down. But remember, the slope of the line is also the derivative of the line; that is, the slope of your position line is your speed. So as you slow down, the *slope* of the line drops, even though you continue to get farther away from the origin.
The integral of the line – the total distance travelled – will continue to grow, but not as quickly once you’ve slowed down. And if we want to know how quickly you slowed down – your acceleration – we simply have to take the derivative of the velocity line.
Now, these are inverse relationships, so they can go the other way. If I told you that you start at 0,0, accelerate instantaneously to 1/ms, then after 1 second, accelerate again to 2 m/s, and after 4 seconds, decelerate to 1 m/s, and after 5 seconds, decelerate to zero m/s. How far did you travel?
Easy right? 1 metre the first second, 6 m in the next 3, 1 m in 5th, and then you’re stopped. 8 m in 5 seconds total. And in fact, that’s the integral underneath this chart, where each X represents one second horizontally and one metre vertically.
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So, once you have one of the three specified as some kind of function, you can figure out the other two. (OK, with integrals you have to account for initial conditions, e.g. what if you started at 2,4 instead of 0,0?)
It’s more that it’s the opposite treatment. You start with a line…removing a dimension (derivation) transforms that line into a single point. Adding a dimension (integration) transforms the line into a rectangle.
You can find equations that make it not quite a rectangle, but the idea is the same.
The function is you standing at the beach. Depending on where you look, you can see sand, water, sticks… All kinds of stuff.
It’s integral is a satellite image of the area. You can see entire continents and oceans, but good luck knowing where sticks are.
The derivative is a macro lens or microscope. Looking through it, you can identify individual grains of sand or creatures you couldn’t see before, but it’s difficult for you to understand where you are, whether the ground is level, how far the water might be.
If you liked through a satellite with a microscope lens attached to it, you might see something that looks similar to when you were standing on the beach. You can see sticks again, but not grains of sand. They cancelled out. Same with looking at a crazy high-resolution satellite image with a microscope.
These operations represent focusing on “smaller” or “bigger” pictures. You’re looking at the same stuff. Just using different tools/changing your perspective.
Integrals are about adding up values within a range. They look at the big picture/zoom out. Larger scope stuff.
Derivatives are about zooming so far in that things just look like a line. It’s a teeny view of the whole function.
A derivative hints at what the very next value could be. An integral “remembers” many values at once. Both take advantage of the fact that a function is made of infinitely many points that can be zoomed in on or counted.
I hope that helps a little.
Think of the three dimensional analog. It’s the same way that the volume under a surface is different from the tangential plane at coordinates on that same surface.
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