Two sides of the same coin

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Problems like this come up frequently. Imagine that each household in a certain small town sends a total of one ton of garbage to the dump every year. Over time, the garbage accumulates in the dump, taking up more and more space.

Figure 1.3 Carl Friedrich Gauss (1777-1855)

a long time after graduating from elementary school

Let's label the years as n=1,2,3,..., and let the function 1 x(n) represent the amount of garbage that has accumulated by the end of year . If the population is constant, say 13 households, then garbage accumulates at a constant rate, and we have x(n)=13n .

But maybe the town's population is growing. If the population starts out as 1 household in year 1, and then grows to 2 in year 2, and soon, then we have the same kind of problem that the young Gauss solved. After 100 years, the accumulated amount of garbage will be 5,050 tons. The pile of refuse grows more quickly every year; the rate of change of is not constant. Tabulating the examples we've done so far, we have this:

rate of change	Accumulated result
13

The rate of change of the function can be notated as $\dot{x}$ . Given the function $\dot{x}$ , we can always determine the function for any value of by doing a running sum.

Likewise, if we know , we can determine $\dot{x}$ by subtraction. In the example where x=13n , we can find $\dot{x}=x(n)-x(n-1)=13n-13(n-1)=13$ . Or if we knew that the accumulated amount of garbage was given by (n^2+n)/2 , we could calculate the town's population like this:

$\frac{n^2+n}{2}-\frac{(n-1)^2+(n-1)}{2} \\ \begin{align*} &=\frac{n^2+n-(n^2-2n+1+n-1)}{2} \\ &= n \end{align*}$

Figure 1.4

$\dot{x}$ is the slope of x.

The graphical interpretation of this is shown in Figure 1.4: on a graph of x=(n^2+n)/2 , the slope of the line connecting two successive points is the value of the function $\dot{x}$ .

In other words, the functions and $\dot{x}$ are like different sides of the same coin. If you know one, you can find the other | with two caveats.

First, we've been assuming implicitly that the function starts out at x(0)=0 . That might not be true in general. For instance, if we're adding water to a reservoir over a certain period of time, the reservoir probably didn't start out completely empty. Thus, if we know $\dot{x}$ , we can't find out everything about without some further information: the starting value of . If someone tells you $\dot{x}=13$ , you can't conclude x=13n , but only x=13n+c , where is some constant. There's no such ambiguity if you're going the opposite way, from to $\dot{x}=13$ . Even if $x(0)\neq 0$ , we still have $\dot{x}=13n+c-[13(n-1)+c]=13$ .

Second, it may be difficult, or even impossible, to find a formula for the answer when we want to determine the running sum given a formula for the rate of change $\dot{x}$ . Gauss had a flash of insight that led him to the result (n^2+n)/2 , but in general we might only be able to use a computer spreadsheet to calculate a number for the running sum, rather than an equation that would be valid for all values of .

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Two sides of the same coin