oh tannenbaum
TRANSCRIPT
Oh, Tannenbaum, oh Tannenbaum
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.
Oh, Tannenbaum
Figure 1: A picture of Hans as he appears today, still working his farm in Bavaria.
Oh, Tannenbaum
Years Gone By Number of Trees in the Ground
0 6000
1 5400
2 4800
3 4200
4 3600
5 3000
6 2400
7 1800
8 1200
9 600
10 0
Knowing that Hans is avisual learner, I decided tomake a table of data and tograph it.
Figure 2: Table showing number of trees Hans has vs the number of years in business.
0
1000
2000
3000
4000
5000
6000
7000
0 5 10 15
Tre
es
in t
he
Gro
un
d
Years
Trees in the Ground
Trees in the Ground
Figure 3: Graph of number of trees Hans has vs the number of years in business.
I pointed out to Hans that the difference here is 600, the number
of trees he sells each year.
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).
The story starts Something happens = Result
Hans Plants 6000 trees - Sells 600 trees/yr = Trees left in the ground
Oh, Tannenbaum
I then wrote this.
Uh oh. The units don’t work. You can’t add terms that have different units.
The story starts Something happens = Result
Hans Plants 6000 trees - Sells 600 trees yr = Trees left in the ground
T rees6000 T rees - 600 T rees in the ground
Y ear
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.
T rees6000 T rees - 600 T rees in the ground
Y ear
Trees ( )6000 Trees - 600 Trees in the ground
year
Y years
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.”
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.
Oh, Tannenbaum
I started explaining to Hans like this:
Hans starts with 600 trees that he plants in November.In
600
600 .1(600) 600 1140
1140 .1(1140) 600 1626
1626 .1(1626) 600 2063
.9 (2063) 600 2457
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.
(.9)600+600
[.
T rees at the end of each ye
9{ }+6
ar is given
for each year by the
00]
[.9
(.9)600+600
(.9)600+600{ }+60
expressions below :
yr1
yr2
yr3 [.9 +0]
[.9{ (.9)60[.9 0+6
60
00
0]
y }+6r4 [.9 00]+
2
3 2
4 3 2 1
600]
:
yr1 .9(6 00) + 600
2 .9 (600) .9(600) 600
3 .9 (600) .9 (600) .9(600) 60
+600
0
4 .9 .9 .9 .9 (600) (6
]
00)...
Sim plifying
yr
yr
yr
Working with this pattern, I wr0te the following:
My hard work was paying off. I then recognized another pattern and wrote:
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
Oh, Tannenbaum
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
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}]
Year Long Hand
Summation
0 600 600
1 1140 1140
2 1626 1626
3 2063 2063
4 2457 2457
5 2811
6 3130
Figure 4: Comparison of data showing trees in ground calculated long hand and using my derived summation
Oh, Tannenbaum
Year Long Hand
Summation
0 600 600
1 1140 1140
2 1626 1626
3 2063 2063
4 2457 2457
5 2811
6 3130
Figure 4: Comparison of data showing trees in ground calculated long hand and using our derived summation
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.
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
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.
Oh, Tannenbaum
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
Year Trees in ground
0 600
1 1140
2 1626
3 2063
4 2457
5 2811
6 3130
7 3417
8 3675
9 3908
10 4117
11 4305
12 4475
13 4627
14 4765
15 4888
16 4999
17 5099
18 5189
19 5271
20 5343
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?
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
Oh, Tannenbaum
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.
600 .1trees trees
Tyr yr
Then I showed him this expression that looks similar that relates to the current case..
6000 600Trees Y
Number of trees Hans plants every
years.
Number of trees that Hans will sell every year when ‘T’ is the
number of trees Hans has in the ground.
Oh, Tannenbaum
600 .1trees trees
Tyr yr
6000 600Trees Y
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.
Oh, Tannenbaum
600 .1trees trees
Tyr yr
6000 600Trees Y
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.
Oh, Tannenbaum
Hans still didn’t quite understand why his farm would quit growing at 6000 trees. I then wrote this:
600 .1T trees trees trees
Tt yr yr yr
Notice that we make sure the units here are
the same..
…as the units here.
6000 600Trees Y
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.
600 .1T trees trees trees
Tt yr yr yr
6000 600Trees Y
Oh, Tannenbaum
600 .1T trees trees trees
Tt yr yr yr
6000 600Trees Y
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.
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.
600 .1T trees trees trees
Tt yr yr yr
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.
600 .1T trees trees trees
Tt yr yr yr
Oh, Tannenbaum
600 .1T trees trees trees
Tt yr yr yr
I used Excel to generate this data and this graph..
trees in the ground change in # of trees
0 600
500 550
1000 500
1500 450
2000 400
2500 350
3000 300
3500 250
4000 200
4500 150
5000 100
5500 50
6000 0
6500 -50
7000 -100
7500 -150
8000 -200
8500 -250
9000 -300
9500 -350
10000 -400
-600
-400
-200
0
200
400
600
800
0 2000 4000 6000 8000 10000
Ch
an
ge
in
Nu
mb
er
of
tre
es(
tre
ss/y
r)
Trees in the Ground
Oh, Tannenbaum
Change in # of trees vs number of trees in the ground
Hans recognized that the line
crosses at 6000.
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.
Oh, Tannenbaum
600 .1T trees trees trees
Tt yr yr yr
-600
-400
-200
0
200
400
600
800
0 2000 4000 6000 8000 10000
Ch
an
ge
in
Nu
mb
er
of
tre
es(
tre
ss/y
r)
Trees in the Ground
Oh, Tannenbaum
Change in # of trees vs number of trees in the ground
The slope is -.1 and has units of
yr-1. Wonder what that means.
When the line crosses this axis, the change in the number of trees goes to zero. The farm isn’t growing or shrinking.
This intercept is the number of
trees Hans adds to his farm each
year.
Even after I explained all of that to Hans, he still felt there was something
magical that keeps…
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
..from increasing without bounds.
Oh, Tannenbaum
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.
600 .1T trees trees trees
Tt yr yr yr
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
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.
Year Trees planted Trees sold Trees in ground
0 600 0 600
1 600 60 1140
2 600 114 1626
3 600 163 2063
4 600 206 2457
5 600 246 2811
6 600 281 3130
7 600 313 3417
8 600 342 3675
9 600 368 3908
10 600 391 4117
11 600 412 4305
12 600 431 4475
13 600 447 4627
14 600 463 4765
15 600 476 4888
16 600 489 4999
17 600 500 5099
18 600 510 5189
19 600 519 5271
20 600 527 5343
600 .1T trees trees trees
Tt yr yr yr
This term tells you how many trees is planted each year.
That is column 2 in the table.
This term is the number of trees sold each year.
That is column 3 in the table.
‘T’ in this term is the number of
trees in the ground. That is column 4 in the
table.
Oh, Tannenbaum
Year Trees planted Trees sold Trees in ground
0 600 0 600
1 600 60 1140
2 600 114 1626
3 600 163 2063
4 600 206 2457
5 600 246 2811
6 600 281 3130
7 600 313 3417
8 600 342 3675
9 600 368 3908
10 600 391 4117
11 600 412 4305
12 600 431 4475
13 600 447 4627
14 600 463 4765
15 600 476 4888
16 600 489 4999
17 600 500 5099
18 600 510 5189
19 600 519 5271
20 600 527 5343
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.
Oh, Tannenbaum
For example, at the end of year 6, Hans would have planted 4200 trees and would have sold 1070.
The difference is 3130.
That action represents -.1T
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
Year Ttrees planted Trees sold Trees in ground
0 600 0 600
1 600 60 1140
2 600 114 1626
3 600 163 2063
4 600 206 2457
5 600 246 2811
6 600 281 3130
7 600 313 3417
8 600 342 3675
9 600 368 3908
10 600 391 4117
11 600 412 4305
12 600 431 4475
13 600 447 4627
14 600 463 4765
15 600 476 4888
16 600 489 4999
17 600 500 5099
18 600 510 5189
19 600 519 5271
20 600 527 5343
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.
Oh, Tannenbaum
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
Year Ttrees planted Trees sold Trees in ground
0 600 0 600
1 600 60 1140
2 600 114 1626
3 600 163 2063
4 600 206 2457
5 600 246 2811
6 600 281 3130
7 600 313 3417
8 600 342 3675
9 600 368 3908
10 600 391 4117
11 600 412 4305
12 600 431 4475
13 600 447 4627
14 600 463 4765
15 600 476 4888
16 600 489 4999
17 600 500 5099
18 600 510 5189
19 600 519 5271
20 600 527 5343
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
Oh, Tannenbaum
This data and this chart is what we got before with that fancy summation equation. Hans was gleeful.
I then used this data to generate this graph.
Year Trees in ground ΔT/Δt
0 600
1 1140 540
2 1626 486
3 2063 437
4 2457 394
5 2811 354
6 3130 319
7 3417 287
8 3675 258
9 3908 232
10 4117 209
11 4305 188
12 4475 169
13 4627 153
14 4765 137
15 4888 124
16 4999 111
17 5099 100
18 5189 90
19 5271 81
20 5343 73
I then used the previous table to generate this
table. It gives ΔT/Δt as a function of yrs.
0
100
200
300
400
500
600
700
0 5 10 15 20 25
Ch
an
ge
in
Nu
mb
er
of
Tre
es
Years
Change in Number of Trees vs Time (Yrs)
Change in Number of Trees vs Time (Yrs)
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.
Oh, Tannenbaum
Year Trees in ground ΔT/Δt
0 600
1 1140 540
2 1626 486
3 2063 437
4 2457 394
5 2811 354
6 3130 319
7 3417 287
8 3675 258
9 3908 232
10 4117 209
11 4305 188
12 4475 169
13 4627 153
14 4765 137
15 4888 124
16 4999 111
17 5099 100
18 5189 90
19 5271 81
20 5343 73
0
100
200
300
400
500
600
700
0 10 20 30
Ch
an
ge
in
Nu
mb
er
of
Tre
es
Years
Change in Number of Trees vs Time (Yrs)
Change in Number of Trees vs Time (Yrs)
Oh, Tannenbaum
600 .1T trees trees trees
Tt yr yr yr
I explained to Hans that these data and this graph are intimately related to this equation.
Oh, Tannenbaum
0
100
200
300
400
500
600
700
0 10 20 30
Ch
an
ge
in
Nu
mb
er
of
Tre
es
Years
Change in Number of Trees vs Time (Yrs)
Change in Number of Trees vs Time (Yrs)
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
These two graphs were graphed using the data we
produced using the original equation…
…and are, therefore, intimately
related.
Oh, Tannenbaum
0
100
200
300
400
500
600
700
0 10 20 30
Ch
an
ge
in
Nu
mb
er
of
Tre
es
pe
r y
ea
r
Years
Change in Number of Trees vs Time (Yrs)
Change in Number of Trees vs Time (Yrs)
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
600 .1T trees trees trees
Tt yr yr yr
I was about to show Hans something wondrous that ties all
of this together.
Oh, Tannenbaum
0
100
200
300
400
500
600
700
0 10 20 30
Ch
an
ge
in
Nu
mb
er
of
Tre
es
Years
Change in Number of Trees vs Time (Yrs)
Change in Number of Trees vs Time (Yrs)
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50
Nu
mb
er
of
Tre
es
on
Fa
rm
Time (yrs)
Number of Trees on Farm vs Years
Number of Trees on Farm vs Years
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 ein
dinge ist %#$&*%@# derivative?”
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.
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
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.
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
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:
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
.1
1
T = 600 600
forever
n
n
e
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.
Oh, Tannenbaum
It was time to plug in some numbers. So to make it easy, I just chose year 1.
Here is what happened.
.1
1
.1
.1
.1
.1
.1 .1
.1 .1
.1 .1
.1 .1
.1
T = 600 600
1140 600 600
11400 6000 6000
5400 6000
5400 6000
6000 6000 600
6000 6000 600
6000 5400 600 600
6000 5400 600 600
6000 5400
forever
n
n
e
e
e
e
e
e e
e e
e e
e e
T e
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
Oh, Tannenbaum
I did this for years, 2 3 and 4, enough to convince myself I could write:.1
1
.1
.1
.1
.1
.1 .1
.1 .1
.1 .1
.1 .1
.1
T = 600 600
1140 600 600
11400 6000 6000
5400 6000
5400 6000
6000 6000 600
6000 6000 600
6000 5400 600 600
6000 5400 600 600
6000 5400
forever
n
n
e
e
e
e
e
e e
e e
e e
e e
T e
.16000 5400
tT e
It was time to get visual with graphs and charts.
Oh, Tannenbaum
.16000 5400
tT e
Year Summation ‘e’
0 600 600
1 1140 1114
2 1626 1579
3 2063 2000
4 2457 2380
5 2811 2725
6 3130 3036
7 3417 3318
8 3675 3574
9 3908 3805
10 4117 4013
11 4305 4202
12 4475 4374
13 4627 4528
14 4765 4668
15 4888 4795
16 4999 4910
17 5099 5014
18 5189 5107
19 5271 5192
20 5343 5269
21 5409 5339
22 5468 5402
23 5521 5459
24 5569 5510
25 5612 5557
26 5651 5599
27 5686 5637
28 5717 5672
29 5746 5703
30 5771 5731
Oh, TannenbaumToo 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:
600 .1T trees trees trees
Tt yr yr yr
1
T rees in the ground after each year
= 600 1 .9
forever
n
n
.1
1
T = 600 600
forever
n
n
e
.16000 5400
tT e
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
600 .1T trees trees trees
Tt yr yr yr
.16000 5400
tT e
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.
Oh, Tannenbaum
Das Ende