if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?

Question

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

Answer 1

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.