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

Opposite isn’t quite their right work. They are inverse functions. In an ideal work, if we have a function f(x), integral of the derivative of f(x) = derivative of the integral of f(x) = f(x). This is what people mean when they say that the derivative and integral are opposite. It’s the same idea that addition and subtraction, multiplication and division, or exponents and logarithms are opposites. Apply one to the other should get you the same result.

There is a problem though: the derivative is a “lossy” function. If we imagine a triangle and the same triangle on top of the box, the slope of the line at the top of the triangle is the same, so they would have the same derivative, but the derivative loses the y position of a curve, if we take the integral of that derivative, we would get the wrong result for one of the two curves.