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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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).
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.
Nice try. How many flugelbinders are in a doohickey
It used to drive me crazy in high school that the derivative of the volume of a sphere (four thirds pi r cubed) is the surface area of a sphere (four pi r squared). No teacher ever addressed this “coincidence.”
But after thinking about it for awhile I realized that if a sphere was growing, the tiniest little instantaneous additional volume being added would be the size of its surface.
That helped me a lot. Maybe it helps you too.
Someone needs to translate this question like I’m 5.
Homie I don’t think that question goes here. I don’t even understand what the question is
Opposite isn’t quite their right work. They are inverse functions. In an ideal work, if we have a function f(x), integral of the derivative of f(x) = derivative of the integral of f(x) = f(x). This is what people mean when they say that the derivative and integral are opposite. It’s the same idea that addition and subtraction, multiplication and division, or exponents and logarithms are opposites. Apply one to the other should get you the same result.
There is a problem though: the derivative is a “lossy” function. If we imagine a triangle and the same triangle on top of the box, the slope of the line at the top of the triangle is the same, so they would have the same derivative, but the derivative loses the y position of a curve, if we take the integral of that derivative, we would get the wrong result for one of the two curves.
I like the intuition that area under a curve is like summing the areas of a bunch of rectangle under the curve, ie. area = height * width. Then notice that slope is “rise over run” ie. slope = height / width. Then of course multiplying and dividing by width are “opposites” in the sense that they undo each other.
This is why the notation for integrals is “∫ f(x) dx)” where we multiply by “dx” (a small width of x) and the notation for derivatives is “df(x)/dx” where we divide by dx.
In this way we can see that area is the “opposite” of slope in the same way multiplication is the opposite of division.
If you take two values on the x-axis and vary the larger one, the rate of change of the area under the curve between them is the value of the function itself at that point.
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