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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Nice try. How many flugelbinders are in a doohickey
Think of it like we’re drawing a mountain. You place your pencil on the paper, and you move it left to right.
The higher up you go, the more “massive” the mountain is that you’re drawing. Moving the straight across increases the mass of the mountain. However, if instead you move the pencil upwards and you move to the right, the mountain is getting even bigger, right?
And if you move the pencil diagonally down, that would make the mountain less massive than if you had moved straight across.
This is the intuition that shows the relationship between the slope of a curve, and the area underneath it. the “mass” of the “mountain” is the area.
Responding to your edit. Indefinite integrals are basically defined as anti derivatives, rather than area under a curve. They are defined basically as the function when you take its derivative you end up back where you started. But this isn’t helpful conceptually as it’s just by definition.
But you can also express indefinite integrals as an area under the curve, but you set one of the bounds of the integral to be a variable x’ instead of a constant (the other bound you can just choose an arbitrary constant). Now the slope of this integral is really asking, how fast does the integrated function change with x, or in other words, how fast does the area under the original curve change with x? The additional area you add under the original curve f(x) is just going to be f(x) dx which brings you back to the original function.
Instead of starting at a function f(x) and asking, how is df/dx the opposite of integral f(x) dx, rather think about taking the slope of [ integral f(x) dx] and you see you end up where you started.
To get a derivative you need to subtract and divide (find difference and rate). To get an integral you need to multiply and add (find areas and add them together).
Imagine a graph of a cars speed over time. The speed at any point in the graph is the cars speed at that moment. Now imagine a graph of that same car’s distance travelled. The sum of all the speed values ends up being the distance. (If you’re going 100 kilometers an hour, you’ll have travelled 100 kilometers after an hour.) The current distance is always the sum of all the speed so far.
Now think about it the other way. The change in distance is the speed. In other words, the slope of the distance line. The slope is a measure of how much things are changing, which is the derivative. If the slope of the distance is horizontal (zero) there is no speed. The steeper the slope, the more distance you’re adding at each step, which is to say speed.
It goes the same way for acceleration. Going back to the original graph of speed, if it’s completely horizontal there is no acceleration. But if the speed is increasing, the change (slope) is the acceleration. So if you graph acceleration it will be the derivative of speed. And speed is the integral of acceleration.
Both are characteristics of the same function. They are related, but they are not the same thing.
For example, if you are on a trip, it is possible to plot the velocity versus the time (how fast you are going at each moment).
The area under the curve is how far you traveled. The slope of the curve is how fast you are accelerating at that moment.
Different information/characteristics of the same thing.
The slope tells you whether the line is going to cover more or less area per x moving forward. If the rate of area coverage increases moving forward, then the line’s slope must be positive.
It’s probably easiest to break it down in terms of simple geometry.
Let’s draw two points. Now, write the equation for a slope between those two points:
(Horizontal Distance Between Points) / (Vertical Distance Between Points)
Now, write the equation for a rectangle that has those two points at opposite corners:
(Horizontal Distance Between Points) * (Vertical Distance Between Points)
In other words, the first (the derivative) is X / Y while the second (the integral) is X * Y. Does this make it easier to understand why they’re inverse operations?
They are not at all mathematically opposite in any way. Well… let me back up.
The derivative is the instantaneous rate of change of a function.
The definite integral is the bounded area under a curve of a function.
The indefinite integral is the family of antiderivatives of a function.
It’s a shame we called both an “integral” because there is nothing fundamentally connecting those two concepts. One is an area, the other is a family of functions. It just so happens that someone discovered that we can calculate the definite integral using the indefinite integral.
If you want to dive really deep though, look into Stoke’s theorem, the multi-dimensional version of the fundamental theorem of calculus. It basically says that the cumulative area of a function in a region can be determined by the value of the antiderivative of the function on the boundary of the region. Crazy stuff.
They are closer to _inverses_ than _opposites._ They somewhat undo each other.
Consider an example like this:
You drive 50 km/hr (the rate of change / tangent part) for 2 hrs (the bounds of the integral, 0 to 2). You traveled 100km (the area under the curve).
“Fifty kilometers per hour” and “One hundred kilometers” are not opposites.
(disclaimer: _inverse functions_ are unrelated to this)
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