MHB What is the purpose of Approximation?

[shoshuban]

I am studying in IGCSE and I learned simple techniques to find, say, approximate change in Area of a circle for a small change in its radius, making use of :
δy/ δx ≈ dy/dx .

or δA/ δr ≈ dA/dr .
or δA ≈ dA/dr x δr

so what I basically have to do is find the derivative of A ( πr^​2 ) , which is 2πr.
then I multiply that with the small change in radius, δr to get the approximate change in area, δA.

But my question here is, why need all of this at all, when I can simply put the old and new values of r in the original equation of A ( πr² ) and find the exact difference by subtracting :
Change in A = π(r_initial)² - π(r_final)²

Why go for calculus to find an approximate value while we can easily get the accurate value without calculus?

 

[Ackbach]

You're quite right that in this case, you can get the exact difference by computing it. But what if you didn't actually have the function that computed the area of something? Often it's the case that we have a differential equation governing a function, but don't actually know what the function itself is. You can still compute an approximation using the differential formulas you mentioned, without having to evaluate the function itself.

But more broadly, why is it at all useful to have more than one way to do anything? Well, I would argue that we need ways to check our work. Checking our work does not consist of going through the same steps we went through before to make sure they were correct. We did those steps already! Of course they are correct! (Or are they...?) So, if we have a completely separate, parallel way to solve a problem, or approximate its solution, we can check our other solution. I am big on my students checking their work. When they have done something correctly, I want them to know that, without asking me.

Does that answer your question?

 

[shoshuban]

ohh thank you so much. :D

 

[Deveno]

Here is another idea:

You say you "know" the area of a circle is $\pi r^2$. Let's say $r = 1$, because we made our "measuring stick" from a circle radius, and decided it was "the sacred unit".

So our circle has area $\pi$. Well, suppose someone asked you-is $\pi > 3$? How would you convince them of this?

Or suppose they asked you a subtler question:

"Which is closer to $\pi,\ \frac{22}{7}$ or $\frac{355}{113}$?"

Assuming you "know" the answer, how would you convince someone else your answer was indeed the truth?

There's really nothing special about "$\pi$", in these questions, we might well ask the same sorts of things about $\sqrt{2}$, or $\log(4)$, or $\cos(5)$.

Comparing rational numbers (fractions) is easy: we can "cross-multiply" to obtain a comparison of *integers*. Presumably, you know how to tell if one integer is bigger than another. But some "amounts" that we might compare aren't easily expressed as "fractions"...whatever are we to do to tell if one such "irrational" amount is bigger than another?

This, in a nutshell, is where "approximation" comes into its own-we may not be able to give an *exact* name to a certain "amount", but if we can approximate it, we can at least tell what it is less than, and what it is more than.

It's not *perfect*, but it's a start...