Re: [Phys-l] scope of the course

[John Denker <jsd@av8n.com>]

On 02/05/2012 12:25 PM, Bernard Cleyet wrote:
> I'm too lazy to see how, w/ algebra, one may derive the exponential
> relationship, please "show me".

1) Bacteria in warm milk.  Twenty minutes later, there are twice as
many.  Twenty minutes later, the number doubles again.
 -- Rate of increase in X is proportional to X.
 -- X is an exponential function of time.

No calculus required.  Draw the graph.  This is within the scope of 
the course ... in 5th grade.
  http://www.google.com/search?q=%225th+grade%22+exponential+growth

2) Ditto for radioactive decay.  Draw the graph.

3) More generally:  For any X that is an exponential function of t,
 calculate ΔX/Δt for any fixed finite Δt ... not necessarily equal to 
 the half-life.  No need to take the limit dX/dt; finite differences 
 work just fine.  Observe that the rate of change in X is proportional
 to X.

 This requires little more than the elementary algebraic properties of
 exponents, plus the distributive law of arithmetic.

===============

*) Even more generally:
  -- Force proportional to X^0 (i.e. independent of X) gives X as a 
     second-order polynomial.
  -- Force proportional to  X gives X as an exponential.
  -- Force proportional to -X gives X as sine and cosine.

... all of which is well within the scope of the algebra-based physics
course.  No calculus required.  Draw the graph already.

Obviously this assumes the students in the algebra-based physics course
actually know some algebra, which is not always a safe assumption ......