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 not so much “opposite” as they are the reverse operation. If curve B is the area under curve A then curve A will equal the slope of curve B.

For example. Say curve A is a line leaving from the origin. At a given value of x (horizontal axis) the area under will be a triangle with base length x and height=slope*x. So area=0.5*slope*x^2. So curve B will be a parabola. A parabola has an ever increasing slope. Curve A represents that increasing slope as the line. Derivative is the process to go in one direction. Integral is the process to go back in the other direction.