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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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.
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