I have a graph where I need to find a particular point based on sampling I currently have a graph which looks something like this: 



Now I have discrete data that represents the Number of Clusters(the x values), giving us the Average Cluster Distance (the y values). There is only a single elbow point, after which the values just descend to 0. Finally, I do know the maximum amount that K can take (the x domain). Based on this I have three questions: 


Is there a way I can find the value of the elbow point through differentials (first or second order)? 
In which case, what is the smallest sampling rate that will allow me to get a graph with the desired characteristics? I don't care too much about the accuracy as of right now (though I would like to know some kind of a tool that lets me measure the error). If I cannot use differentials, I am still interested in understanding how I could recreate the graph with the lowest sampling rate, and thus find the elbow point with the least amount of processing (For E.g. Nyquist Theorem, I would have 1/2 the amount of elements and still recreate the signal).
Finally, would incorporating random selection into the selection of the signal help me in any way?
Based on the graph you posted and the elbow defined, all you have to do is to take the first difference of the signal and compare it to a threshold value, assuming that the signal is monotonically decreasing. Such as this:

if( (y[k] - y[k-1]) &lt; Th )
          elbow = k-1


if the signal is not monotonically decreasing, you should look for other measures as well.

Sorry but I cannot tell the required sampling rate for this graph as you stated it comes from an already discrete data...?