Relates summations, integrals, graphs, functions via a short story about Christmas trees.

1. Oh, Tannenbaum, oh Tannenbaum

2. Oh, Tannenbaum We’re about to look at two stories involving Christmas trees (Tannenbaums). We will model the two stories by creating tables of data, graphing and writing a mathematical equation (function) for each story. Both of the stories involve a man named Hans Brinker who makes his living cultivating and selling Tannenbaums. Since he makes his money off the trees he sells, he likes to keep track of just how many trees he has available. The first story is at a time when Hans was just getting into the Baum business and he had little experience. He bought a farm that had 6000 trees and he got a contract to provide 600 trees per year to a vendor near Oberammergau (tickets are now on sell for the 2010 Passion Play at Oberammergau). Hans has asked me, the local tree counter, to determine how many trees he has as the years tick by.

3. Oh, Tannenbaum Figure 1: A picture of Hans as he appears today, still working his farm in Bavaria.

4. Oh, Tannenbaum Knowing that Hans is a visual learner, I decided to make a table of data and to graph it. I pointed out to Hans that the difference here is 600, the number of trees he sells each year. Figure 2: Table showing number of trees Hans has vs the number of years in business. Figure 3: Graph of number of trees Hans has vs the number of years in business.

5. Oh, Tannenbaum Somewhere in my education, I had to graph many types of functions and from that experience, I recognized that this situation can be modeled with a linear equation. It goes like this: I recognized immediately, because I used units, that this is a tree story and it should yield a tree equation (function).

6. Oh, Tannenbaum I then wrote this. Uh oh. The units don’t work. You can’t add terms that have different units.

7. Oh, Tannenbaum But I know this is a tree story so somehow I needed to get rid of those ‘year’ units in the denominator of the 2nd term. I then used the ol’ up-down, up-down rule. You can cancel down-units by multiplying by up-units. I wrote this. Life was sweet because those year units canceled, leaving me with units of trees all the way through the equation. Hans paid me hans-omely.

8. Oh, Tannenbaum Hans was in high cotton for a while. Many of Hans’ neighbors found this to be strange, however, since Hans was a tree farmer. Then, sometime around 2 AM in November of the 10th year, Hans woke with a start. “Whoa!”, he said. “I’m almost out of trees.”

9. Oh, Tannenbaum Hans needed a new business plan quick. But he had not really planned well and was not very liquid at the time. That was not surprising, because during winter in Bavaria few things are liquid. Hans recalls what a good job I did last time and rang me up. I had been waiting for this day and had been planning for it. I proposed to the now frantic Hans that he should start slow and plant 600 trees. And I had already talked to the vendors and they were willing to buy 10% of the trees that Hans had in the ground around November each year. I was about to give to Hans a very enduring income, and set out to show Hans what to expect.

10. Oh, Tannenbaum I started explaining to Hans like this: Hans starts with 600 trees that he plants in November. In Hans starts with 600 trees in the ground and let’s them grow for 1 year. Next November, Hans has the original 600 trees, he sells 10% and plants another 600. There’s a pattern here. We add 600 trees to 90% of the previous year’s total trees in the ground.

11. Oh, Tannenbaum Working with this pattern, I wr0te the following: My hard work was paying off. I then recognized another pattern and wrote:

12. Oh, Tannenbaum Well, that’s nice and compact. I tested it with Mathematica and compared it with what we were getting earlier. I entered this into Mathematica to get the total number of trees in the ground for some year like year 6 for instance. 600(1+NSum[.9^n, {n,1,6}] Figure 4: Comparison of data showing trees in ground calculated long hand and using my derived summation

13. Oh, Tannenbaum That’s nice! I have an easy way to calculate the number of trees Hans has in the ground at any time. So, I then used this tool to calculate it out for 40 yrs and then graphed it. Figure 4: Comparison of data showing trees in ground calculated long hand and using our derived summation

14. Oh, Tannenbaum I used this data to generate this graph. These data and this graph shows that at some time many years from now, the number of trees approaches a value of 6000. I would say that the number of trees will be limited to 6000 trees.

15. Oh, Tannenbaum Hans is baffled! He’s amazed!! Then he asked me a very simple question. He pointed at this equation.. and asked, “ The number of trees increases every year, yet the farm quits growing at about 6000 trees. I mean, there’s only plus signs in that equation. Why don’t I get billions and billions of trees?

16. Oh, Tannenbaum I reminded Hans that 10 years ago I wrote an equation for him that looked like this. And how when we graphed it, it gave a straight line graph the showed that after 10 years he would have zero trees. I explained that that equation gives him Trees he had in the ground after Y years. Then I showed him this expression that looks similar that relates to the current case.. Number of trees that Hans will sell every year when ‘T’ is the number of trees Hans has in the ground. Number of trees Hans plants every years.

17. Oh, Tannenbaum I explained that in this equation that the 6000 is a static number in the equation and that the only action happens with the 2nd term. And then I explained that in this expression, both terms are action terms. The 600 is trees he is adding and the .1T term is how fast he sells them. These two terms compete with each other. This equation has a totally different feel.

18. Oh, Tannenbaum The units on each term in this equation is Trees. This equation gives the number of trees in the ground for Han’s first go at farming. And, I pointed out that the ‘T’ in this equation, which is the number of trees in the ground, is itself dependent on how fast he plants and how fast he sells. This is much more complicated that it at first looks.

19. Oh, Tannenbaum Hans still didn’t quite understand why his farm would quit growing at 6000 trees. I then wrote this: Notice that we make sure the units here are the same.. …as the units here.

20. Oh, Tannenbaum And then explained that while the first equation written 10 yrs ago gave you the number of trees at any time, the 2nd equations tells the rate at which the farm is growing or shrinking. I told Hans that this is called a difference equation.

21. Oh, Tannenbaum Hans remarked that that 600 in this equation did that same sort of thing. The 600 was how fast he was removing trees from his farm. I congratulated Hans on being so observant and then told him there must be an equation that this 2nd equation is in some way part of and that in that equation this equation plays the same role as the 600 in the 10 yr old equation. Hans was so enthralled at this point that he wanted to leave the farm and become a mathematician. I gently reminded Hans that this is not math, its engineering.

22. Oh, Tannenbaum I then reasoned this with Hans: the first term increases the number of trees and the 2nd term decreases the number of trees. You may wonder, then, which term wins out, and if it does, does it always win out? For instance, if the first term (the one that makes the function larger) is always larger than the 2nd term (which makes the function decrease), then this function keeps increasing. Likewise, if the 2nd term is always larger than the 1st term, the function will always decrease.

23. Oh, Tannenbaum Hans pointed out that the first term will always be larger than the 2nd if he has fewer than 6000 trees and that the farm would grow. I then pointed out that it he has more than 6000 trees the 2nd term will be larger than the first term and the farm would shrink. Hans got me a beer, and remarked that he’s got it. Because if he has exactly 6000 trees, he will sell 600 trees and plant 600 trees and the farm neither grows nor shrinks. As long as he doesn’t change either the planting rate or the selling rate his farm will be stable at 6000 trees. He will have an income forever. Hans asks for a graph. I asked for a shot of Jaeger Meister and walked over to the computer.

24. Oh, Tannenbaum I pointed out that when Trees is less than 6000, change is in the positive region of the graph and when trees is greater than 6000 change is negative. I used Excel to generate this data and this graph.. Hans recognized that the line crosses at 6000.

25. Oh, Tannenbaum The slope is -.1 and has units of yr-1. Wonder what that means. This intercept is the number of trees Hans adds to his farm each year. When the line crosses this axis, the change in the number of trees goes to zero. The farm isn’t growing or shrinking.

26. Oh, Tannenbaum Even after I explained all of that to Hans, he still felt there was something magical that keeps… ..from increasing without bounds.

27. Oh, Tannenbaum Hans was impressed that I could use this equation.. ..to explain that doesn’t lead to him having all the trees in the universe. He was a bit disappointed, but still OK with it. Hans asked if I could determine how many trees he has at the end of any given year, starting with this same equation. The beer was good so I agreed to try. Making data sets seems to help, so I decided to make a chart using this equation.

28. Oh, Tannenbaum This term is the number of trees sold each year. That is column 3 in the table. This term tells you how many trees is planted each year. That is column 2 in the table. ‘T’ in this term is the number of trees in the ground. That is column 4 in the table.

29. Oh, Tannenbaum That action represents -.1T The numbers in this column, ‘T’, are the sum of all the trees planted minus all the trees sold by December 25th of each yr. For example, at the end of year 6, Hans would have planted 4200 trees and would have sold 1070. The difference is 3130.

30. Oh, Tannenbaum I used this data to generate this graph. These data and this graph shows that at some time many years from now, the number of trees approaches a value of 6000. We would say that the number of trees will be limited to 6000 trees.

31. Oh, Tannenbaum This data and this chart is what we got before with that fancy summation equation. Hans was gleeful. I pointed out that the slope of this curve is not constant. That slope represents the change in trees with respect to time (ΔT/Δt)=600-.1T

32. Oh, Tannenbaum I then used the previous table to generate this table. It gives ΔT/Δt as a function of yrs. I then used this data to generate this graph. These data and this graph show that the rate of growth of the farm slows with time. At some point the growth will stop. Growth of the farm approaches a limit of 0.

33. Oh, Tannenbaum I explained to Hans that these data and this graph are intimately related to this equation.

34. Oh, Tannenbaum These two graphs were graphed using the data we produced using the original equation… …and are, therefore, intimately related.

35. Oh, Tannenbaum I was about to show Hans something wondrous that ties all of this together.

36. Oh, Tannenbaum I told Hans that this graph is the derivative of the graph below, And this graph represents the anti-derivative of the graph above. Hans just shook his head, thinking ‘Was fur eindingeist %#$&*%@# derivative?”

37. Oh, Tannenbaum Hans wanted a little more for all the beer I was drinking. “Look”, he said, “this equation with the summation is kind of strange to me still. I mean it isn’t apparent on first glance that that really gives me the trees at any time. Can you work that a little so there’s a 6000 in it some place so it at least looks right?” It was late, but I was having fun, what with the beer and the polka music and all. Polka is known to help mathematical reasoning.

38. Oh, Tannenbaum Then I recalled being in another Polka stupor while I was deriving the derivative of ax. It was during that stupor that I derived something I called ‘e’. And in the middle of that derivation I had to be clever and say that any positive real number can be expressed as another positive real number raised to a power. I guess using a slide rule for so many years and being aware of logarithms that thought was just buried some place in my Polka mind.

39. Oh, Tannenbaum It was at that point that I thought to try expressing .9 as a power of e. What I found startled me: e-.1= .9!!! That just couldn’t be a coincidence, so I began anew with greater confidence. I rewrote the summation like this: Now, the way I derived that summation equation was by determining the number of trees I had in the ground for each of several years and following the pattern to get that equation. So that equation tells me the number of trees in the ground at the end of each year.

40. Oh, Tannenbaum It was time to plug in some numbers. So to make it easy, I just chose year 1. Here is what happened. Because I want to get an expression that has 6000 in it to satisfy Hans, I multiplied by 10) I stumbled around in the dark here for a while, but I knew I wanted to get to something on the left side of the = that would equal what I started with. Ultimately, I came up with this. That looked nice. The 6000 was in there AND so was the -.1

41. Oh, Tannenbaum I did this for years, 2 3 and 4, enough to convince myself I could write: It was time to get visual with graphs and charts.

42. Oh, Tannenbaum

43. Oh, Tannenbaum Too much beer. Too much Polka, Hans had fallen fast asleep. I woke him up and summarized what we had done. We had started with a difference equation: We derived a summation equation to give us Trees as a function of yrs. We rewrote .9 as e-.1 And finally we worked some algebra magic to get:

44. Oh, Tannenbaum I stumbled home aglow with mathematical wonderment and slept soundly for almost two days. By then Albert and Niels had reviewed what I had done and were interested in moving it further. Over the next couple of weeks, we determined that nature had many situations that could be modeled by equations similar to And that their solutions were all similar to

45. Oh, Tannenbaum I became so famous that I couldn’t go any place without people wanting to live that moment with me. So, I moved to the US. Changed my name from Ivan Zaborowski to John Saber, and settled near some Christmas trees in Minnesota. It was during my sojourn there that I developed something I called Integration. But, that’s another story for another day.

46. Oh, Tannenbaum Das Ende