
The derivative
Hi all, this is the problem that I don't know how to solve it. I think I should use the linear approximation
f(x) ~ f(a) + f'(a)(xa)
But I'm not sure. Some hints will help me a lot. Thanks so much.
The table below shows the number of student, N, admitted by the School of BA in year t
T N(t)
2004 27
2005 102
2006 95
2007 116
2008 89
Estimate as accurately as possible the value of the derivative N’(t) in the year 2005, 2006, 2007

Re: The derivative
If f(x) ~ f(a) + f'(a)(x  a), then:
f(x)  f(a) ~ f'(a)(x  a), and assuming x  a is sufficiently small, and f is continuous:
f'(a) ~ [f(x)  f(a)]/(x  a).
Put another way, you cannot actually find N'(t) = dN/dt, but you CAN find ΔN/Δt, where Δt = 1 year.
The graph of N will be composed of line segments (well, dots, actually, but we tend to "connect the dots").
You will actually get two slopes, one for the segment on each side of each t. It might be a good idea to average the two numbers you get for 2005,2006 and 2007 (and perhaps you see why you are not asked about 2004 and 2008).