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).

yes

Well, it looks like you've put in the effort and done the calculations correctly. However, I must say that estimating the value of C'(1990) as the average between the values of C'(1985) and C'(1995) is a bit like trying to find the average between a banana and a watermelon. It might not be the most accurate approach.

I mean, if you think about it, a lot can happen in 5 years, and the rate of change might not be linear. Plus, we don't have information about the values of C'(1991), C'(1992), C'(1993), and C'(1994), which could potentially affect the estimate.

But hey, math isn't always about being super precise. Sometimes a little estimation is good enough. So, I'll give you a round of applause for your effort and your sense of humor in attempting to interpret and estimate the value of C'(1990). Keep up the good work!

To check if you calculated the value of C'(1990) correctly, let's first calculate the differences between consecutive values of C(t) for the given years.

1985 - 1980: 187.3 - 129.9 = 57.4
1990 - 1985: 271.9 - 187.3 = 84.6
1995 - 1990: 409.3 - 271.9 = 137.4
2000 - 1995: 568.6 - 409.3 = 159.3

Now, let's calculate the differences between the years:

1985 - 1980 = 5
1990 - 1985 = 5
1995 - 1990 = 5
2000 - 1995 = 5

Now, let's calculate the rates of change using the difference quotient formula:

C'(1990) = (C(1990) - C(1985))/(1990 - 1985)
= (271.9 - 187.3)/(5)
= 84.6/5
= 16.92 billion

So, based on the calculations, the value of C'(1990) is 16.92 billion, which matches your answer. Therefore, you calculated it correctly.

To check if your solution is correct, we can indeed use the difference quotient formula to find an estimate for the value of C'(1990). The difference quotient is given by:

C'(1990) = lim (t -> 1990) [C(t) - C(1990)] / [t - 1990]

By plugging in the values from the table, we can calculate:

C'(1980) = [C(1980) - C(1990)] / [1980 - 1990] = (129.9 - 271.9) / (-10) = 14.2 billion
C'(1985) = [C(1985) - C(1990)] / [1985 - 1990] = (187.3 - 271.9) / (-5) = 16.92 billion
C'(1995) = [C(1995) - C(1990)] / [1995 - 1990] = (409.3 - 271.9) / 5 = 27.4 billion
C'(2000) = [C(2000) - C(1990)] / [2000 - 1990] = (568.6 - 271.9) / 10 = 29.67 billion

Since C'(1990) is the rate of change of C with respect to t when t = 1990, it represents the average rate of increase of the value of US currency in 1990.

To estimate C'(1990), we can take the average of the values calculated for C'(1985) and C'(1995), which is:

C'(1990) ≈ (C'(1985) + C'(1995)) / 2
= (16.92 + 27.4) / 2
= 22.16 billion

So, your estimate for C'(1990) is indeed 22.16 billion, which falls in between the values calculated for C'(1985) and C'(1995).