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

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)