if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?
In: 332
ELI5: this question
Think of a function as just the change in y. If it goes from 2 to 5, y is 3. Now, the slope is y amount (2 to 5) divided by the x amount it took to change from 2 to 5. If x changed from 0 to 6, that’s 6 for x, so the division (the derivative) is 0.5. For each x, y increases in 0.5.
Now let’s do the opposite. Instead of divide y by x, multiply it. What is the y amount multiplied by the x amount in our example? 18. Ok, good. Now what is x (width) multiplied by y (height). It’s the area!
Not ELI5 enough, but I feel like it needs to be said: The relationship between “Speed”, “Velocity” and “Acceleration” differs by integration in one direction and derivatives in the other direction.
ELI5: What the fuck did this guy just ask?
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.
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.
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.
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
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.
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