# How much information about a function do you need to determine the function?

I am messing around with a program that computes definite integrals using Riemann sums and had questions about constructing graphs and determining functions.

So let's say my program gives me any 20 points I want of my indefinite integral. So I know the value of the function I want from 0 to 19. I also know the derivative of the function I want, as that was my original function. Would that be enough to a) construct a graph of the function? and b) determine the equation of the function?

Intuitively, I feel like this should be enough information. There must be some way to determine which function gives you the 20 exact points you know, especially if you know the rate of change of that function (as you have its derivative).

However I also can see it not being possible, because, if I remember correctly, there are an infinite number of equations that pass through a finite amount of points.

Perhaps you could approximate the equation? Is that enough information to create the equation? Or how much more would you need?

• The safe answer to the title question is "all of it". If it's a polynomial, you can uniquely determine it by knowing that it is a polynomial, knowing its degree $n$, and knowing $n+1$ points of it. Other functions have their own requirements for uniquely determining them. – abiessu Mar 27 at 4:53