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Re: [Phys-l] scope of the course

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.

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 ......