oh tannenbaum

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


= 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


= 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

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of

Tre

es

on

Fa

rm

Time (yrs)

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


= 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.


Tt yr yr yr

6000 600Trees Y

Oh, Tannenbaum


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.


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.


Tt yr yr yr

Oh, Tannenbaum


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


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


= 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.


Tt yr yr yr

1


= 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


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

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Time (yrs)



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

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Time (yrs)



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)


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



Oh, Tannenbaum


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

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es

Years



0

1000

2000

3000

4000

5000

6000

7000

0 10 20 30 40 50

Nu

mb

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of

Tre

es

on

Fa

rm

Time (yrs)



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

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es

pe

r y

ea

r

Years



0

1000

2000

3000

4000

5000

6000

7000

0 10 20 30 40 50

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of

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es

on

Fa

rm

Time (yrs)



1


= 600 1 .9

forever

n

n


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

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in

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es

Years



0

1000

2000

3000

4000

5000

6000

7000

0 10 20 30 40 50

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rm

Time (yrs)



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


= 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


= 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


= 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:


Tt yr yr yr

1


= 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


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

oh tannenbaum

Education

number of trees hans

units of trees

hans brinker

picture of hans

hans neighbors

frantic hans

bynumber of trees

year units