Analytical vs numerical differentiation

[Walter]

Gordy,

It’s interestIng that there is a lot of information available regarding error calculation for statistics (sampling) and numerical analysis. Not so much for analytical calculus.

[Walter]

Gordy,

You’re getting way too hung up on the issue of errors. We are doing digital measurements of an analog process. Errors are inherent. That doesn’t make the data we gather useless or the resulting analysis wrong. It just means we need to be aware of it. I think we are all familiar with this dataset.

IMG_0750.jpeg (76.68 KiB) Viewed 117 times

I would say that, depending on what you are analyzing, this dataset may or may not have value.

What is more important imo is the sort of “what if” analysis we can do with an underlying function. I’ll give a very simple example. Suppose I have a known mass and measure its displacement over time. Given that data I can certainly calculate velocity and acceleration as well. But if I asked you to tell me the force that was applied and you haven’t measured the force you can estimate force using the simple equation F=ma. But what if we are in pre Newton days and you don’t know that formula? Take it one step farther and I’ll ask you what would happen if I doubled the force or the mass. Without the underlying formula you would have to repeat the experiment with a new set of parameters. If you have the underlying formula you can predict what will happen in our simple experiment without even having to do the experiment. You can apply that formula to different situations such as sliding down a hill or determine motion based on multiple external forces.

That’s not to say formulaic analysis is somehow better or more accurate. It’s a different tool. I provided an example of how such an analysis can be used in the evaluating the haul thread discussing rotation of the rod and passive hauling. But formulaic analysis has its own issues. I have no idea if the formula I used is accurate. I intend to add refinements as I work through them and each refinement should improve the accuracy of the result but it’s “garbage in garbage out”. If I start with the assumption that constant acceleration, for example, is what everybody uses and refuse to change that (assuming it turns out to be necessary) after finding out that is an incorrect assumption then my analysis will have no value.

Different tools, different jobs. If the only tool you have is a hammer then everything will look like a nail.

[Mangrove Cuckoo]

Walter,

As an aside... thank you so much for summarizing what I remember from my days in calculus class!

If only my professors offered some real world examples - like application of functions and derivatives to fly casting - I may have stayed with it. Who knows, if so I may have turned out to be an engineer geek instead of a chemistry geek!

Unfortunately, I find very few ways to use chemistry to improve my fly casting.

[Walter]

Gary,

Thanks!