Solving B(t) & B'(t): Graphs & Table

[Meredith.Lord]

this question confuses me and my teacher hasnt been in school for the last 3 days and I am positive we have not learned anything like this. please help

let B(t) be the population of Belgium at time t. the table below gives the midyear values at B(t), in thousands, from 1980 to 2000. Fill in the rest of the table with the values for the derivatives of this function. DIscuss how you arrived at those values. Plot both B(t) and B'(t) on graph paper and compare the graphs.

T 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000
B(t) 9847 9856 9855 9862 9884 9962 10036 10109 10152 10175 10186
B’(t)

thats the information from the graph, but i don't know how its been gotten and then i have to fill in B'(t)

please help!

 

[mathman]

For data at a discrete set of points, you would approximate derivatives by difference quotients. There are a couple of ways to do this.

For the end points take the diference between the value at the end point and its nearest neighbor and divide by 2 (the difference in years). In both cases take the later year B minus the earlier year B.

For intermediate points carry out the analogous calculation on both sides and average.

A more sophisticated apporach would be fitting the data with splines and taking derivatives, but I suspect that would be above the level you're at.

 

[HallsofIvy]

The derivative is defined as
$$\lim_{h\rightarrow 0}\frac{f(a+ h)- f(a)}{h}$$
Since you are only given populations at integer years, you can't let h "go to 0" but you can take "h" as small as possible: 1 year. That is, approximate the derivative, as best you can, by B(a+1)- B(a). That is the "discrete quotient" mathman is talking about. It is, of course, just the change in population during the a^th year.