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I solved this problem, can someone check it to make sure I did it right. There are two parts to this intrepation and estimate of the value.

Let C(t) be the total value of the US currency (coins and banknotes) in circulation at time, t. The table gives values of this function from 1980 to 2000, as of September 30, in billions of dollars. Intrepret and estimate the value of C'(1990).

Interpretation is that C'(1990) signigies the rate of change of C with respect to the twhen t=1990. Basically the rate of increase of value of US currency in 1990.

t 1980 1985 1990 1995 2000
C(t) 129.9 187.3 271.9 409.3 568.6

I solved this using the difference quotient
C'(1990) =
lim-->1990 C(t)-C(1990)/t-1990
I calculated these values by plugging into the above equation for each year in the table.

I got 14.2 billion for 1980, 16.92 for 1985, 27.4 for 1995 and 29.67 for 2000. C'(1990) would fall in between the 1985 and 1995 values so I averaged them to get the 1990 value at 22.16 billion for C'(1990).

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