Calculus in the

TMP Summer Institute, 2008

Rethinking

Time of Computers

Nature and nature's laws lay hid by night.

God said, "Let Newton be!" - and all was light.

- - Alexander Pope

How dare anyone suggest a "rethinking" of calculus!

Why such exaltation? De Revolutionibus Orbium Celestium

launches the Copernican Revolution.1543

1609 Kepler's Astronomia Nova postulates an elliptic orbit for Mars (and other planets).1633 Galileo is convicted of heresy.

1687 Newton shows that an inverse square law accounts for the Copernican model.

1800+ Calculus helped fuel an industrial revolution that re-shaped our very lives.

Why a "re-thinking"?• The standard calculus curriculum is neither necessary nor sufficient for an understanding of the role that mathematics has played in shaping our world.

• The world has undergone another revolution, one based on computer technology.

• Computer technology has profound consequences for the ways in which mathematics is done and applied in the modern world.

• Computer technology has profound consequences for the ways in which calculus can be conceived and taught.

1. Mary receives a piggy bank containing 10 pennies. She decides to save partof her weekly allowance in the bank and records her weekly savings in the tablebelow. She then challenges her brother Johnny to calculate the amountaccumulated at the end of each week. Can you solve Johnny's problem?

Week Pennies savedthat week

Pennies accumulatedat end of week

101 82 23 54 4

1820

2925

If a(n) denotes the amount saved in the n-th week

A(n) = A(0) + a(1) + a(2) + … + a(n-1) + a(n)

A(n-1)

[

and A(n) the amount accumulated at the end of the n-th week,

a(n) A(n)n

then A(0) = 10 and

A(n) = + a(n)

]

0

€

then a(n) = A(n) - A(n -1)

2. Johnny also receives a piggy bank containing 10 pennies and decides to savepart of his weekly allowance in the bank. However, he records the amount hehas accumulated at the end of each week. He then challenges his sister Mary tocalculate the amount he saved each week. Can you solve Mary's problem?

Week Pennies savedthat week

Pennies accumulatedat end of week

101 142 193 224 29

Having invoked the associative law of addition to conclude that A(n) = A(n-1) + a(n), Mary transposes the term A(n-1) to obtain

a(n) = A(n) - A(n-1)

4 5 3 7

Synthesis:

∆ A(n)

€

If A(n) = A(0) + a(k)k=1

k=n

∑

€

If F(x) = F(0) + f(t)dt0

x

∫

€

then f(x) = ddx

F(x)

n a(n) A(n)

"… the sums of the difference between successive terms, no matter how great their number, will be equal to the difference between the terms at the beginning and the end of the series."

Gottfried Leibniz in Historia et Qrigo

A(k) - A(k-1) = A(n) - A(0)

€

[A(k)k=1

k=n

∑ − A(k -1)]

a(k)

"Why an eight year old child could understand this. Someone go out and fetch me an eight year old child!"

-- Groucho Marx

"The idea out of which Leibniz's calculus grew was the inverse relationship of sums and differences in … sequences of numbers."

Victor Katz in A History of Mathematics

What is Calculus?A collection of powerful mathematical techniques that were developed and organized in the 17th century.

Leibniz - Acta Eruditorum Newton - Principia Mathematica

Newton's explanations relied on terms such as "fluxions" and "nascent and evanescent quantities". This led to questions of logic and rigor,

"… he who can digest a second or third fluxion … need not, methinks, be squeamish about any point in divinity."

Historia et origo calculi differentialis

most notably from Bishop Berkeley who lambasted "infidel mathematicians" and pointed out flaws in Newton's formulation:

-- George Berkeley

Such criticism did not prevent the calculus of Newton and Leibniz from being applied with great success by Euler, the Bernoullis, Simpson, d'Alembert, Euler, and others. It was not until 1821 that something resembling modern standards of rigor emerged in Cauchy's Cours d'Analyse,

• There is a distinction between proving and understanding. What should be their interplay?

• To what extent should instruction reflect the historical development of a subject

Issues:

• How does the emergence of computer technology figure into our answers?

standards that Cauchy did not impose in his teaching.

vs. its structure?

Where can such a re-thinking lead us?1. Anticipating the traditional calculus sequence, albeit in discrete form.

If the a(n) are given by formula, is there a corresponding formula for the A(n)? If so, what is it?

If the A(n) are given by formula, is there a corresponding formula for the a(n)? If so, what is it?

2. "Rules for Change" and Discrete Dynamical Systems.

Introductory calculus leads to the study of rules for change called "differential equations".

3. Applications and Modeling.

4. All of the above.

These can be studied in discrete form. Such study leads to remarkable new concepts!

Constant change leads to linear accumulation"Amos save 5 dollars a week. How many dollars will he accumulate in N weeks?""Alicia rides her bicycle at 8 miles an hour. How far will she ride in T hours?"

What does linear change lead to?"Juan saves 5 dollars the first week, 10 dollars the second, …, and 5n dollars in the n-th week. How many dollars will he accumulate in N weeks?"

1. Anticipating the traditional calculus curriculum.

What does quadratic change lead to?

"Keisha saves 1 dollar the first week, 4 dollars the second, … and n2 dollars in the n-th week. How many dollars will she accumulate in N weeks?"

An ad hoc approach to Juan's problem can be based on an oft repeated anecdote about little Karl Gauss:

1 + 2 + 3 + … + 100S =

2S

2S = N(N+1)

Returning to Juan's problem:

5 + 10 + 15 + … + 5N = 5(1 + 2 + 3 + … + N)

€

=5N(N +1)2

N-1 + N

= N+1+ N+1 + N+1 + N+1…

€

1 + 2 + ... + N =N(N +1)

2

S = N + N-1 + … + 2 + 1

5050

We begin by asking

"What leads to quadratic accumulation?"

Corresponding to this approach we consider the following:

"Discrete calculus" provides a more powerful formalism for solving such problems!

This is the approach we use in calculus!

€

It is the fact that ddx

x2 = 2x that leads

us to conclude that xdx = x2

2∫

"Cecilia wants her bank account to grow quadratically - i.e., to have accumulated n2 dollars at the end of the n-th week. How much does she have to save each week?"

+ c

???

???

1

4

9

n2

n2 -2n + 1

1

week a(n) A(n)

1

2

3

… … …

n-1

n

3

5

n2 - (n2 -2n +1)

If a(n) = 2n -1, then A(n) = n2

If a(n) = 2n, then A(n) = n2 + n

If a(n) = n,

€

then A(n) =n2 + n

2

2n -1

€

n(n +1)2

Wow!

We have arrived at 1 + 2 + 3 + … + N = N(N+1)/2

A more challenging problem is to find a formula for the sum of the first N perfect squares:

What is 1 + 4 + 9 + … + N2 ?

"Discrete calculus" gives us an algorithmic approach for solving such problems.

1. For A(n) = n3, find a(n) = A(n) - A(n-1).

2. Having found a quadratic expression for a(n) that leads to A(n) = n3, modify this expression to obtain a(n) = n2 and determine the effect of the modifications on A(n) .

1

8

27

n3

n3 -3n2 + 3n - 1

1

week a(n) A(n)

1

2

3

… … …

n-1

n

7

19

???

If a(n) = 3n2 - 3n + 1 then A(n) = n3

If a(n) = 3n2 - 3n

then A(n) =

If a(n) = n2 - n

If a(n) = n2

€

then A(n) = 2n3 + 3n2 + n

6

3n2 - 3n + 1

then A(n) =

This is not to suggest that "real calculus" is to be replaced by its discrete counterpart! However, in the time of computers it seems important for students (and their teachers) to realize that the concepts of calculus can be applied to sequences of numbers as well as to functions in abstract form. It is also important to acknowledge that there are more concrete and intuitive ways of approaching topics central to calculus than those used in standard texts.

"A month's intelligent instruction in the theory of numbers ought to be twice as instructive, twice as useful, and at least ten times as entertaining as the same amount of "calculus for engineers."

-- G. H. Hardy

2. Discrete Dynamical SystemsIn prescribing the values of a(n), we are in fact formulating a "rule for change." That is, if the a(n) are given by a functional rule f(n),

We have seen that f(n) = 5 leads to A(n) = 5n.

Similarly, f(n) = n leads to A(n) = A(0) + n(n+1)/2.

But there also exist rules for change of the form

A(n) - A(n-1) = f(n, A(n-1))

We encounter such "difference equations" in the study of exponential growth.

then

A(n) - A(n-1) = f(n)

A(0) + 5n

f(A(n-1))

Exponential Growth

Exponential growth can also be studied in a financial context, namely that of compound interest.

A(n) - A(n-1)

Let A(n) denote your bank balance at the end of the n-th year.If the bank compounds interest annually and the (decimal) interest rate is R, then the interest earned in the n-th year is given by

This difference equation can be written in terms of an "iterative rule"

A(n) = A(n-1) + RA(n-1)

= RA(n-1)

A(n) = (1+R)A(n-1)or

Exponential Growth (cont.)

The iterative rule A(n) = (1+R)A(n-1) leads to a closed form solution for A(n).

A(1) = (1+R)A(0)

If A(0) denotes the initial balance, we have:

and more generally,

A(n) = (1+R)nA(0)

Since n (time) appears as an exponent we refer to this as exponential growth.

A(2) = (1+R)A(1) = (1+R)2A(0)

A(3) = (1+R)A(2) = (1+R)3A(0)

Every difference equation of the form

A(n) - A(n-1) = f(A(n-1))

corresponds to an iterative scheme

A(n) = F(A(n-1))

and vice-versa.

Exponential growth corresponds to a linear dynamical system:

A(n) = (1 + R)A(n-1)

dynamical system

i.e., F(x) = (1 + R)x. Linear dynamical systems have closed form solutions. Very few others do!

It is non-linear dynamical systems that lead to remarkable new phenomena such as chaos.

Logistic Growth

Consider a bank that offers an annual interest rate R but also imposes a “small service fee E”

A(n) - A(n-1)

Now the rule for change is

= RA(n-1)

- one that is applied to the square of your balance.

- EA(n-1)

A(n) = (1 + R)A(n-1) - EA(n-1)2

When A > R/E, the fee exceeds the interest!

2

(Solve RA = EA2)

- one that is applied to the square of your balance.

When A = 0 or A = R/E.

Hmm, I’d better think about this

When does the fee equal the interest?

An important modification of logistic growth is one in which we focus on the "target" K = R/E.

A(n) - A(n-1) = RA(n-1) - EA(n-1)2

= A(n-1)[R - EA(n-1)]= A(n-1)E[R/E - A(n-1)] K

Given a target K = R/E, we can vary the parameter E to obtain a family of logistic curves approaching R/E.

In a banking context, K is the value of A(n) at which interest equals fee.

In an environmental context, K is the carrying capacity of the environment that supports a given population.

In Cosmos, students at the pre-calculus level see some of the remarkable phenomena exhibited by non-linear dynamical systems that some regard as a hallmark of 20th century mathematics. Our starting point is the solution of quadratic equations!

We supplement the litany "negative bee, plus or minus the square root of … " with a dynamical system --

"ay ex squared minus cee, all over two ay ex plus bee"

3. Applications and Modeling

In 1956 a prominent geologist named M. King Hubbert made a dramatic prediction.

He was right!

At a time when the US oil industry was booming , Hubbert predicted that "US oil production will peak in the early 1970s."

"Hubbert's method" has become the source of intense interest, mystique, and efforts at global replication.

Are Johnny and Mary prepared to address such issues?

Lets start with petro-numeracy:

On the average, Americans account for gallons of petroleum per day.

This corresponds to of petroleum per year. 25 barrels

There are 300+ million Americans.

US Oil Consumption: ≈ 7.5 billion barrels/year

Global Oil Consumption: ≈ 30 billion barrels/year≈ 1000 barrels/second

Uses of petroleum? cars

trucks&buses agriculture

militaryspace

heatingEtc, etc.

3

Given global equity, how long would 1000 billion barrels last ?

25 x

1000/167 ≈

6.7 billion ≈ 167 billion barrels/year

6 years

In the context of such numbers some "wrong questions" tend to be asked: Are we running out of oil?

Oil is a non-renewable resource. We have been running out of oil since 1859 when the first well was drilled in Titusville, Pennsylvania.

Of Course!

When will we run out of oil?

Never! Given other energy sources (e.g., natural gas) oil can be inefficiently recovered.

How can we arrive at a better question?

In "system dynamics" we make use of two basic icons:

reservoir flow

In modeling non-renewable resources such as petroleum, the earth is a reservoir.

"Life is a flow."

The values of A(n) describe a reservoir.

The values of a(n) describe a flow.

As someone who viewed the world through the laws of thermodynamics and had thought deeply about the role of energy in human affairs, Hubbert was sensitive to this distinction.

Hubbert addressed this question in 1956

This question is now being asked in a global context.

“When will oil production peak?” - as it applied

to “the lower 48 states.”

“In the early 1970s”His answer:

The question he posed was:

Peak Oil!

Hubbert's Peak

Was calculus necessary - or was it largely ornamental?

Hubbert’s data

When Did U.S. Oil Production Peak?

Hubbert's peak

Hubbert's model for non-renewable resource production is based on the following assumptions:

• Annual production data will be bell shaped.

• Annual production data will be symmetric.

• Cumulative production data will be logistic.

Some Informal Curve Fitting

Let us enter the EIA data for cumulative production into the spreadsheet used to model targeted logistic growth.

Given a value for the target K (total amount of petroleum to be recovered) we can fit the logistic curve generated by A(n) to this data.

The corresponding graph for A(n) - A(n-1) will indicate the peak in annual production.

Focusing on "the lower 48 states," Hubbert used 200 gigabarrels for K. Including Alaska and off-shore oil this corresponds roughly to K = 220.

An Old Joke:

In the middle of the night an inebriated man is under a lamppost on his hands and knees.

Another man comes by and asks, "What's the matter?" to which the inebriated man responds, "I've lost my keys."

The second man gets down on his hands and knees and, after some minutes asks, "Where did you lose them?"

"Oh, half a block down the street."

"So why are we looking here?"

"Because the light is better!"

Let's take another look at the data available to Hubbert in 1956:

Implicit in Hubbert's model is that the peak in annual production coincides with half the cumulative production,

200 ÷ 2 = 100

100 - 52.5 = 47.547.5 ÷ 2.5 = 191955 + 19 = 1974

It was Hubbert's estimate of the total amount of oil recoverable that was key to his "in the early 1970s" prediction. The fancy mathematics attributed to him was not required!

Was Hubbert a con artist?

Or a genius?

Or a little bit of both?

Who would take seriously a model based on 6th grade arithmetic?

Given that exponential growth has become the normal state of affairs, how do you get people's attention?

The Mystique of Calculus!

What is a mathematical model?

Like a political cartoon, it is a gross simplification of reality.

It can be used to distort reality,or to provide incisive insights that cannot be conveyed in a more reasoned format.

But there are bad models, and even corrupt models.

Hubbert's strength lay in his understanding of the world rather than in the model for which he is remembered.

“Human history can be divided into three distinct phases.

The second, based on the exploitation of fossil fuels and industrial metals, has been a period of spectacular exponential growth.

The first, prior to 1800, was characterized by a small human population, low levels of energy consumption, and very slow rates of change.

Because of the finite resources of the earth’s fossil fuels and metallic ores, the second phase can only be transitory.”

Epilog:

“The third phase must again become one of slow rates of growth,

Perhaps the foremost problem facing mankind is that of how to make a transition from the present exponential growth phase to the near steady state of the future

but initially at least with a large population

facing our

- M. King Hubbert

by as non-catastrophic a progression as possible.”

students

and a high rate of energy consumption.

Given global equity (at an American level of 25 barrels/year), how long would 1000 billion barrels last ?

25 x

1000/160 ≈

Hubbert’s Peak

6.4 billion ≈ 160 billion barrels/year

6.25 years

In the context of such numbers some "wrong questions" tend to be asked:

“Are we running out of oil?”

Oil is a non-renewable resource. We have been running out of oil since 1860.

Of Course!

“When will we run out of oil?”

As a resource becomes scarce, its price increases and substitutes take its place.

NEVER!

Note: Given other forms of energy, oil can be "produced"

(economists may tell us.)

- albeit inefficiently.