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Calculus - Exponential Distribution

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The following table gives the percent of the US Population living in urban Areas as a function of year2. ...
5. (a) Estimate f'(2) using the values of f in the table. ...
5. The thickness, P, in mm, of pelican eggshell depends ...
11. The quantity, Q mg, of nicotine in the body t minutes after a cigarette is smoked ...

[Please see the attached question file].

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Some questions:

(a) Average rate of change of the percent population = (69.9 - 20) * 100/20 = 249.5%
(b) P'(1960) = {P(1970) - P(1960)}/10 = (73.5 - 69.9)/10 = 0.36% per year
(c) P'(1830) = {P(1860) - P(1830)}/30 = (20 - 9)/30 = 0.367% per year
The rate at which the percent population was increasing in 1830 is slightly higher (almost the same) as that in 1960. We may assume therefore that the rate of percent influx has remained constant at these two time periods.
(d) The function is increasing (but at a declining rate).

Fitting a regression curve to the data in the table, we get the best estimate of the regression as
f(x) = 0.0313x^3 - 0.833x^2 + 5.95x + 9.9
(a) f'(x) = 0.0939x^2 - 1.666x + 5.95 and f'(2) = 0.0939(2)^2 - 1.666(2) + 5.95 = 3.
OR
[Direct method: f'(2) = {f(4) - f(2)}/(4 - 2) = {24 - 18}/2 = 3.
(b) f'(x) > 0 ==> 0.0939x^2 - 1.666x + 5.95 > 0 ==> 0.0939(x - 4.9556)(x - 12.7867) > 0
==> x < 4.95 or x > 12.79 (Approx)
==> f'(x) is positive for x < 4 and negative when x > 4.
...

Solution Summary

Neat, step-by-step solutions regarding exponential distribution in the problems are provided.

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