Let P(x) be the number of people of height ? x inches in the US. What
is the meaning of P primed(66)? What are its units? Estimate P prime
of (66) (using common sense). Is P prime(x) ever negative? [Hint: you
may want to approximate P prime (66) by a difference quotient, using h
= 1. Also, you may assume the US population is about 250 million, and
note that 66 = 5 feet 6 inches.

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Clarification of Question by wannabeleader-ga on 11 Jul 2005 15:27 PDT

I need this question writen up, explaining each step.

The units of P(x) are people.  P(x) is a function that tells you the
NUMBER OF PEOPLE whose height is under a certain number of inches, so
the units are simply "people."

P'(66) means the rate of change of the P(x) function at x = 66; that
is, your regular old derivative [d/dx]P(x) at the point x = 66. 
Translated in terms of the problem, this means that P'(x) is the rate
at which the number of people under a certain height is increasing or
decreasing with respect to inches.
That is: d(number of people under x inches)/d(x inches).

I'm not sure how to "estimate" P'(66) without seeing some general
statistics on height, which is beyond the time I have to devote to
this.  I don't know exactly what point this portion of the problem is
trying to make at first glance.

P'(x) will never be negative.  Think of it this way:  Suppose the
number of people under 48 inches is 100 (in other words, P(48) = 100),
then the number of people under 49 inches will be greater than or
equal to P(48), because if someone is shorter than 48 inches, they
will certainly be shorter than 49 inches as well, and now you're
adding in the group of folks between 48 and 49 inches, so P(49) >=
P(48).  This is true for all values of x, so P(x) is an increasing
function.  This means that the rate of change of P with respect to x
is always positive.  In other words, that P'(x) >=0 for all x.

If this were not the case, it would be possible to have more people
shorter than 4 feet than there are people shorter than 5 feet, which
you should intuitively recognize as an impossibility.

Hope this helps.

Sorry, realized we're looking for units of P'(x).  This would be
"people per inch."

So if you look at the average rate of change between P(55) and P(56)
for instance, then dP/dx = [P(56)-P(55)]/[56-55] = [P(56)-P(55)]
people/inch.  This tells the number of people that are between 55 and
56 inches tall.