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*From*: John Denker <jsd@av8n.com>*Date*: Sun, 05 Feb 2012 13:26:04 -0700

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 ......

**Follow-Ups**:**Re: [Phys-l] scope of the course***From:*Bernard Cleyet <bernardcleyet@redshift.com>

**References**:**[Phys-l] block sliding on a skateboard***From:*"LaMontagne, Bob" <RLAMONT@providence.edu>

**Re: [Phys-l] block sliding on a skateboard***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] block sliding on a skateboard***From:*Bernard Cleyet <bernardcleyet@redshift.com>

**[Phys-l] scope of the course***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] scope of the course***From:*Bernard Cleyet <bernardcleyet@redshift.com>

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