[BlindMath] Average Rate of Change problem

[Jonathan Fine]

The pre-calculus problem is:

> Find the average rate of change of f(x) = x2 – x + 4 from x_1  = 2 to x_2=6 .

The answer is the fraction f(6) - f(2)  over 6 - 2 end fraction. And
then some algebra.

But to prepare properly for calculus, you need to know why is this the
average rate of change. So it's good that you ask this question. It
makes you a better student.

Rate of change is a bit like speed. If I leave at one, and arrive at
four, and cover 120 miles, then my average speed is 120 / 3 = 40 miles
per hour.

You can determine average speed using a tape measure and a clock (or
watch). This physics requires only pre-calculus math. My advice now is
to think of average rate of change as being like average speed.

Advanced math diversion

Calculus provides a way of defining and calculating the speed at a
particular instant of time.

You need a different tool to measure speed right now. For example,
traffic cops might use a radar speed gun. This uses what's called the
Doppler effect. The requires measuring a difference in frequency. And
frequency is a shorthand for "speed of repetition".

Once you have calculus, there's a different definition of average rate
of change. But by design and construction, it gives exactly the same
answer as the pre-calculus definition. And it relies on the
pre-calculus definition.
-- 
Jonathan