The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Courses » ap-calculus

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: [No Subject]
Replies: 1   Last Post: Aug 29, 1997 8:29 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Howard Swann

Posts: 6
Registered: 12/6/04
[No Subject]
Posted: Aug 28, 1997 7:14 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

I have been following the current discussion of the difficulties of
teaching limits with considerable interest.

Deltas and epsilons and the foundations of differential calculus
do not attract much interest in the beginning calculus sequence these
days. Yet in the beginning of this century there was great excitement
(in scientific circles anyway...) over the extraordinary finesse
proposed by Cauchy, Weierstrass, Dedekind and Co. to resolve some
ancient and deep philosophical problems.

For example, Bertrand Russell was so astonished that he pronounced
"...all goes smoothly until we reach those studies in which the notion
of infinity is employed - the infinitesimal calculus and the whole of
higher mathematics. The solution of the difficulties which formerly
surrounded the mathematically infinite is probably the greatest
achievement of which our age has to boast."

Nowadays, using quantifiers, the basic definitions of differential
calculus can be expressed with (laudable) logical efficiency, yet this
masks the excitement of discovering the triumph of human intellect
celebrated by Russell.

I enclose excerpts below from a handout I use in my college
(multisemester calculus) classes that attempts to recapture this
experience; it has proved to be highly effective in class and makes the
whole discussion of limits into an entertaining adventure. It may be
useful to you all as well, for the high-school students I occasionally get
in my classes are often the best in the class....

I would be happy to send the entire essay; send a note to my
e-mail at

Howard Swann
San Jose State University

* * * * * *
The Theory of Differential Calculus

Calculus provides a basis for much scientific reasoning.
Mathematicians sought methods of proof for the validity of the
procedures of calculus that are as indisputably correct as the
demonstrations of Euclidean geometry, where, for example, each step of
reasoning about congruence of triangles has a justification. Yet
calculus tries to deal with concepts that are, in part,
incomprehensible to the human mind. Our concept of number is curiously
inadequate. We sense that there should be a number that represents the
length of any line segment. The Greek Pythagoreans (c. 550 B.C.)
discovered that the hypotenuse of a right triangle with legs of length
1 would necessarily have a length whose square was 2, yet there was no
rational number, a number represented by the ratio of two integers,
whose square was 2. This discovery was viewed with great alarm, for the
Pythagoreans, a religious society, based their understanding of
creation on the supposition that the integers and rationals described
all that was `good', yet here was a (perfectly good) length that could
not be described in such terms. According to legend, one of the
Pythagoreans revealed this discovery to the Pythagorean's enemies, and
was shipwrecked at sea as punishment. For the rest of us, this
demonstration showed that some new `irrational' numbers representing
such lengths were needed to extend the set of rational numbers.
Straightforward algorithms for adding, multiplying and dividing
rational numbers - fractions - are easy to justify; if we are to invent
new irrational numbers we must find unquestionably correct ways to add,
multiply and divide such new numbers.

Some 2400 years later in the 19th century, axioms were proposed
that provided the structure to represent any length by a possibly
infinite (non-terminating) decimal. The axioms also give compellingly
plausible ways to define addition, multiplication and so forth for such
infinite decimals. The set of infinite decimals together with the
methods for doing the arithmetic operations is optimistically called
the real numbers, although infinite decimals seem somewhat unreal,
requiring that we become comfortable with a "number" representing the
notion of adding an infinite number of rationals which, interpreted
literally, cannot be done. The fundamentally incomprehensible concept
of infinity keeps keeps coming up in one way or another.

* * * * * *

Our sense of a correspondence between numbers and locations along
the `number line' leads to the idea that a certain trichotomy must
be true; given any two numbers a and b, either they are equal, or a <
b or b < a. The real numbers have this property; one just compares
the digits in two "infinite decimals" to make the required
determination. But the trichotomy property leads to confusion. Given
any real number a, there can be no real number b next to a for the
following reason: if there were such a b not equal to a, then a < b
or b < a. In either case the new average number (a+b)/2 is closer
to a than b. Thus the real numbers do not seem to model the
`continuum'that we sense represents locations in space or the flow
of time; there still seem to be points missing. If we insist that
the real numbers possess the trichotomy property (and have the
conventional order) we are stuck with this problem.

In view of these concerns, the extension of the rational numbers
to the real numbers does not provide a completely satisfactory
structure for a mathematical model for motion. We represent the motion
of a point along a straight line by a function p(t), where t is a real
number representing time, with real number p(t) giving the location on
the number line at time t. But how is motion possible? If a point is
located precisely at real number p(t) at time represented by real
number t, by the argument above, there is no next instant later in
time, nor is there any possible location next to p(t). Does the point
momentarily disappear on its way to a later location p(t') at later
time t' > t? If so, it disappears infinitely often.

In spite of this puzzle, motion clearly does occur somehow, and we
go ahead and use a real valued function p(t) to represent motion, only
to encounter a similar problem when we try to define what we mean by
the instantaneous velocity of point p(t), which is the derivative of
function p(t). We turn to the equivalent problem of defining what
should be the slope of a line tangent to the graph of a function.

* * * * * *

Suppose we seek the derivative of y = 2^x at x = 0. Thus we seek a
limit for T(h) := [2^h - 1]/h as h --> 0. We tabulate some values for

h 1 -1 .1 -.11 0.01 -.011 0.001 -.001
T(h) 1 .5 .71 0.6696 0.69555 0.690750 0.693387 0.692907

It appears from the table that T(h) is approaching some number close
to .693???.... as h approaches zero through negative or positive
values, but we don't really know what that limiting number might be.

We can expand the table above of values for T(h), letting h take
smaller and smaller values and thus generate more and more digits of
the apparent limit, but eventually our calculators will no longer be
able to make accurate computations, or will produce numbers that no
longer agree with the apparent trend of the computations. The following
values were obtained from a CASIO fx-7500G:
h 10E-5 10E-6 10E-7 10E-8 10E-9 10E-10 10E-11
T(h) .6931496 .693147 .69315 .6931 .693 .69 .00

We are uneasy accepting this termination of our efforts, for
our answers should be independent of human notions of "small" (or
large.) We may consider .0001 cm. to be "small", yet in some culture
on some planet in some other galaxy it may be very "large". We have
encountered the classical problem of dealing with the "infinitesimally
small", a notion that defies attempts to give it a comfortable

* * * * * * *

From the examples above, graphically, the task of finding a limit
for a function T(h) as h approaches 0 (assuming that there is a limit
in some satisfactory sense) is the same as finding a number L so that,
if you place a dot at point (0,L), the parts of the graph of T(h) for h
< 0 and h > 0 are then joined up so that the resulting graph is smooth
or `continuous' across the vertical line through (0,0).

The problem with giving this pictorial interpretation a
formal definition is that our model of the real numbers does not
allow us to convert the intuitive ideas to clearly defined terms.
There are NO numbers on either side of 0; there is no way to "link"
the two parts of the graph of T(h) together with some point (0,L);
our model for the real numbers and the points on the graph of T(h) is
"all full of holes."

We have encountered the fundamental conceptual problem with the
real numbers once again. We do not seem to be able to reconcile two
intuitively "obvious" properties that should be aspects of our sense of
how numbers describe the continuum we call the number line. We cannot
simultaneously retain a notion of smooth `continuous' one-dimensional
motion of a particle and hold that the real numbers describing the
location of the particle are `points', where we can determine if one
point is less than another. The problem with the `holes' cannot be
avoided. What is needed is a way to proceed that avoids these
difficulties yet still is sure to get the right limiting numbers. The
following finesse was proposed in the 19th century:

We consider the more general problem of determining whether or
not there is a limit for some function g(x) as x approaches some number
(a) What exactly would such a limiting `number' be? Is it a
`real number'? If so, what is its value?
(b) Is there some way of being sure that you have the right number?

Mathematician Karl Weierstrass (1815-1897) contrived the following
way around the difficulties discussed above. Essentially, he shifted
the burden back onto the function by proposing that we only discuss
limits at points a for functions g(x) where the function g(x) must
(1) provide a guess, a `real number' L as a limiting number for g(x)
as x goes to a AND

(2) g(x) must provide an additional function that is guaranteed to have
certain properties. ( will (eventually) be the name
of this additional `function'...H.S.)

He was able to show that
(i) these requirements are reasonable, and lots of functions satisfy
these requirements and
(ii) in spite of any confusion, the guess for the limit number for
any function satisfying these requirements must be correct.

His arguments successfully avoided any reference to infinity
or "infinitesimals" and, as we shall see, the reasoning is also
unquestionably independent of any human perception of what was "small"
or "close."

* * * * * * *

The foundations of calculus and the many branches of mathematics
that spring from calculus are rooted in this finesse of Weierstrass. In
the proof of the (essential uniqueness) theorem (for limits) you will
see how we manage to turn the profound difficulties concerning our
concept of number to our advantage; as long as we stay within the class
of functions defined to have limits in the sense of Weierstrass, the
conclusions we make concerning the accuracy of our limits must be
correct, even though we ultimately do not understand ...

Date Subject Author
Read [No Subject]
Howard Swann
Read Re:
Geoff Hagopian

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.