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
They are not at all mathematically opposite in any way. Well… let me back up.
The derivative is the instantaneous rate of change of a function.
The definite integral is the bounded area under a curve of a function.
The indefinite integral is the family of antiderivatives of a function.
It’s a shame we called both an “integral” because there is nothing fundamentally connecting those two concepts. One is an area, the other is a family of functions. It just so happens that someone discovered that we can calculate the definite integral using the indefinite integral.
If you want to dive really deep though, look into Stoke’s theorem, the multi-dimensional version of the fundamental theorem of calculus. It basically says that the cumulative area of a function in a region can be determined by the value of the antiderivative of the function on the boundary of the region. Crazy stuff.
Latest Answers