Wednesday, February 24, 2016

What Is Calculus and Why do we Study it?


From an MIT site. Its all important but some is bold for emphasis.

1.2 What Is Calculus and Why do we Study it? 
Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. 
I have been around for a while, and know how things change, more or less. What can calculus add to that? 
I am sure you know lots about how things change. And you have a qualitative notion of calculus. For example the concept of speed of motion is a notion straight from calculus, though it surely existed long before calculus did and you know lots about it. 
So what does calculus add for me? 
It provides a way for us to construct relatively simple quantitative models of change, and to deduce their consequences. 
To what end? 
With this you get the ability to find the effects of changing conditions on the system being investigated. By studying these, you can learn how to control the system to do make it do what you want it to do. Calculus, by giving engineers and you the ability to model and control systems gives them (and potentially you) extraordinary power over the material world. The development of calculus and its applications to physics and engineering is probably the most significant factor in the development of modern science beyond where it was in the days of Archimedes. And this was responsible for the industrial revolution and everything that has followed from it including almost all the major advances of the last few centuries. 
Are you trying to claim that I will know enough about calculus to model systems and deduce enough to control them? 
If you had asked me this question ten years ago I would have said no. Now it is within the realm of possibility, for some non-trivial systems, with your use of your laptop or desk computer. 
OK, but how does calculus models change? What is calculus like? 
The fundamental idea of calculus is to study change by studying "instantaneous" change, by which we mean changes over tiny intervals of time.

It appears none of these principles are followed by the academe of economics.

For instance, so how is the idea of creating a variable T, which is in itself an ex post accounting result as it is "taxes net of xfers" in consonance with the fundamental idea of calculus which is to study INSTANTANEOUS change?


11 comments:

Dan Lynch said...

Right, modeling dynamic systems.

And also calculus teaches you to think, to concentrate.

I loved calculus.

Matt Franko said...

Dan its like this is what they ALL are doing:

Economist A: What do we do?

Economist B: I dont know.... I got it, lets see what happens and when that is over we can say that's what we did!

Bob said...

Curve fitting.

Ignacio said...

I like somewhere I read that physics was the study of movement, which is what essentially is.

P.S: A lot of calculus in physics...

Greg said...

"The fundamental idea of calculus is to study change by studying "instantaneous" change, by which we mean changes over tiny intervals of time. "

What this made me think of is how often we hear economists, monetarists especially, talk about the "long term" , when in fact the long term is just many sequences of short terms.

Matt Franko said...

Well good point Greg that is like doing an integral over deltaT....

I just dont know how these people's brains work....

They look at the G-T subtraction where T itself is an ex post subtractive result, 'taxes NET of xfers', but you dont know what that result is until AFTER it already has happened... yet they will speak of the variable T in present tense...

Might be like in physics, you would have an aircraft take off and get to 35,000 ft. then come back down to 10,000 and they would think it only went up to 10,000 ft as thats where it ended up... or fly around the world and land back at the same airport and they would think it didnt go anywhere...

No attention paid to time domain in economics...

Matt Franko said...

btw Mike is tracking some sort of discrepancy with this T variable within the economic data that is running at $500B ytd...

nivekvb said...

I wasn't good at maths at school and left with moderate grades, but good enough to get an apprenticeship. I did well at college studying craft subjected that my company met me do am ONC in electronics.

Well, I was freaking it because there was so much maths in the course, but I worked hard at it.

We did exams throughout the year and the first test came only after a couple of months, and it was a maths exam. Well, just before the exam I went cold with fear and I could hardly breath and I thought I'm never doing another course again, but when I got my results back a week later I had got 100%.

After that I started to fall in love with maths, and I was finding it much easier than I thought it would be.

I now realise that the block cut me about maths at school was really about 'learned helplessness'. I gave up too quickly and thought that I would never be able to mean that.

Calculus always looked so mysterious and I couldn't wait to learn it. We started it right at the end of the first year. An Introduction to Calculus, said my text book.

The teacher showed us a few simple questions on the board and then we had a go. Wow, it was nowhere near as hard as I thought.

I fell in love with calculus and became in awe of it. I thought it was incredible how you could when out the area under an irregular shape like that, and also apply to things like acceleration. I tried to figure out the proofs but that was more tricky. Afterwards I went on and did the Btec HNC.

Matt Franko said...

Kevin imo if you subject yourself to all of that for such a long period it ends up having permanent cognitive effects... which then you will apply generally in all of your cognitive activities...

MIA in the academe of economics for sure... they are screwing up wrt inclusion of the time domain in their "models" so-called BIG LEAGUE...

intajake said...

Does anyone know any free high level math/calculus video tutorials/courses.
Maybe at some point I'll work on it

intajake said...

Does anyone know any free high level math/calculus video tutorials/courses.
Maybe at some point I'll work on it