If I take an anti derivative of a function that gives quantity over time...what is the interpretation of the new function? Because you are treating the quantity function as a rate and putting it into a new function, what does the new one mean?

I just learned about anti-derivatives and I'm quite fascinated :)

Question

If I take an anti derivative of a function that gives quantity over time...what is the interpretation of the new function? Because you are treating the quantity function as a rate and putting it into a new function, what does the new one mean? 

I just learned about anti-derivatives and I'm quite fascinated :)

anonymous · Answer

Well it depends on both the "input" and "output" of the original function. Usually we take the antiderivative $with~respect~to$ the input (a very important phrase), but some functions have multiple inputs, so we often are required to specify which one.

So with your example, you have the output as a rate of something over time. Actually, this rate is probably a derivative, or an instantaneous change in something per an infinitesimal change in time. Then it's a function of time, that is, it gives the rate of the process at a certain point in time. Taking the antiderivative (with respect to time) gives a function which tells us the total progress of this process since some point in time. For example, taking the antiderivative of speed gives us distance. (That is how we find the arclength of vector valued functions, which you will learn in calculus 3).