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

Derivatives and integrals are mathematically opposite, in calculus. To help you visualize, teachers may draw a parallel to some geometry concepts, but the geometry visualizations are not going to be conceptually opposite.

Unless you look at the geometry from a calculus point of view. Calculus studies “change” rather than fixed objects.

So if you imagine [a parabola](https://www.math.net/img/a/algebra/functions/quadratic-function/parabola.png), the derivative of it is not the slope of A tangent line, it’s what ALL the tangent lines do. It’s a function that describes the behavior of ALL tangent lines, which is that their slopes decrease to 0 and continue to decrease into the negative.

And the integral, “area under the curve”, again it’s not just the TOTAL area, it’s as you go along, from x=-2 to x=3 for example, it’s a description of what the function does, when you look at each point along the line (the area increases over time).

And that’s perhaps where your “opposite” hides: the slopes decrease and the area increases.

But in general, calculus is about systems that change, and trying to understand change with “pictures” (geometrical shapes are “fixed” in time) is detrimental, you can’t take the analogy very far, it loses too much in the translation.