How do I estimate the derivative of this function?

Question

1. The following table shows the fnc: G=f(t)
t 1987, 1990, 1993, 1996 ,1999
f(t) 53, 70, 73, 74, 81
(the numbers correspond to their respective years.)

Estimate f ' (1999).
Then,
Use the estimated value of f ' (1999) to estimate f (2005).


Please tell me how to set-up and solve this problem.
Thanks!

Answer 1

The value of the derivative of a function is equal to the instantaneous slope of the function's curve. In this case, we can estimate the value of that slope by using the last two values, (1996, 74) and (1999, 81). The slope between these two points is (change in f(t)) / (change in t) = (81 - 74) / (1999 - 1996) = 7 / 3, which is our estimate of f'(1999). We can estimate that f(2005) = f(1999) + (2005 - 1999)*f'(1999) = 81 + 6*(7/3) = 81 + 14 = 95.