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

Slope indicates how rapidly the function changes its value as it goes on.

Derivative of the integral (i.e area) describes how rapidly the area grows, i.e. the initial function itself (since the larger the function value is, the more it adds to the area).

Integral of the derivative means adding all those little slopes together. At every point the slope points to where the function is going next, so integrating them will, again, trace the initial function.