Re: [Phys-L] calc probs for physics --> Taylor approximations

[John Denker <jsd@av8n.com>]

The following two items are related:

In another thread, on 12/19/2013 09:07 AM, Bill Nettles wrote:

> I think it's important to teach the art of estimation.  [1]

Yes.

On 12/21/2013 01:31 PM, Paul Lulai wrote:
 
> > Does anyone have a favorite resource from which they select
> > calc-based problems for physics students?             [2]

Item [2] is related to item [1] in various ways, most conspicuously
by Taylor series.

Sometimes a Taylor series is the only possible solution, because
an exact solution would be intractable.  However, even when it
is not strictly necessary, a Taylor series is a valuable option,
alongside (not merely in place of) an exact solution.  The lowest-
order expansion can provide insight about trends and relationships.

This provides a whole zoo of answers to question [2].  Almost
any nontrivial formula has some sort of interesting lowest-order
expansion.  It is standard good practice in real-world physics,
whenever you derive an equation, to expand it to lowest order to
see what is going on.  This is sort of like checking the dimensions,
but more powerful and more interesting.

  My favorite example of this involves boosting the [energy, momentum]
  4-vector and then expanding to lowest order in velocity ... at 
  which point the classical kinetic energy pops out, as a lowest-order 
  approximation to the real energy (i.e. the fully relativistic energy).
  This demonstrates that physics is both simpler and more powerful than
  most people ever imagined.

  At the high-school level, students can memorize the lowest-order 
  expansion for cosh(x) and/or sqrt(1+x), and convince themselves that 
  it works by doing a few numerical examples and plotting the results.

  In the calculus-based course, there is no need to memorize the 
  details of the Taylor expansion, because you can rederive it in 
  less time than it takes to tell about it.

There are gazillions of other examples.  More important than any
particular example is forming the /habit/ of doing this.