A country's population, according to a news article, is currently 300 million, and it is gaining approximately 2 people every 5 seconds. If f(t) is the population (in millions) of this country, and t is the time in years since the article was published, find f (0) and f '(0).

Question

anonymous · Answer

I found f(0) which is 300 million. but I do not understand how to approach f ' (0)

dumbcow · Answer

use limit definition of derivative which essentially is the slope formula
$$f'(x_0) = \frac{f(x_1) - f(x_0)}{x_1 - x_0}$$
as x1 approaches x0
since we are talking about a difference of 5 seconds, this is a very good approximation for slope at t=0
$$f'(0) = \frac{3,000,002 - 3,000,000}{5^*-0} = \frac{2}{5^*}$$
however t is in years, so we must convert 5 sec into years
there are 3600*24*365 seconds in 1 year ....
$$f'(0) = \frac{2}{\frac{5}{3600*24*365}} = \frac{2}{\frac{1}{3600*24*73}} = 2*3600*24*73$$