WHAT IS CALCULUS?
© 2001, G. William Moore, MD, PhD.

WHAT IS CALCULUS?

      INTEGRAL AND DIFFERENTIAL CALCULUS, typically known simply as CALCULUS, is a set of mathematical methods used to approximate values of slopes and areas under curves (mathematical functions), such as parabolas, circles, ellipses, etc. Co-invented in the seventeenth century by an English physicist, SIR ISAAC NEWTON, and a German philosopher, GOTTFRIED LEIBNIZ, calculus has largely been applied to physics and engineering problems (Courant et al, 1996). However, calculus is also useful tool for a range of curve-fitting and statistics problems, and for this reason should be studied by biomedical specialists as well. In the sample problem shown herein, calculus is used to estimate the second derivative for a series of measurements taken from the vertebral column, and from this the curvature (kyphosis, lordosis, etc.).

      DIFFERENTIAL CALCULUS comprises the mathematical methods used to approximate the values of SLOPES of mathematical functions. Originally known as FLUXIONS, the study of differential calculus was already well-developed before the time of Newton and Leibniz. Differential calculus is, by far, the less difficult of the two great branches of calculus, and is employed in many statistical applications.

      INTEGRAL CALCULUS comprises the mathematical methods used to approximate the values of AREAS under mathematical functions. Originally known as QUADRATURES, the study of integral calculus preceded Newton and Leibniz. The great contribution of Newton and Leibniz was the clear understanding that the two operations are the inverse of one another. This understanding allowed formulas to be exchanged between these two great branches of calculus, and thus enriched our knowledge of both. The present document covers possible biomedical applications of elementary calculus.


BASIC CONCEPTS.

      The basic ideas of elementary calculus apply to a CURVE, or MATHEMATICAL FUNCTION, lying on the xy plane. We employ the ordinary Cartesian plane of analytic geometry, with the angles, distances, etc., obeying the usual rules of Euclidean geometry. More advanced ideas in calculus involve multiple dimensions, and different rules of angles and distances.

      A MATHEMATICAL FUNCTION, is defined as a relationship between x and y on the xy plane, in which every x corresponds to ONE AND ONLY ONE VALUE OF y. A LIMIT in calculus is the destination-value for a sequence of values. A DERIVATIVE, or DIFFERENTIAL, for a mathematical function (curve) in calculus is the instantaneous slope of that curve. An INTEGRAL, or ANTIDERIVATIVE, for a mathematical function in calculus is the area under that function. The FUNDAMENTAL THEOREM OF INTEGRAL AND DIFFERENTIAL CALCULUS is the assertion derivative and integral are inverse processes. In this minitutorial, we shall prove the fundamental theorem for so-called well-behaved functions (continuous and everywhere differentiable).

      Finally, we shall develop the notion of LEAST SQUARES as a method in statistics and curve-fitting, for certain classes of biomedical prolems.


WHAT IS A LIMIT?

      A LIMIT in calculus is the destination-value for a sequence of values. We write:
Limx -> c f(x) = L.
to denote the
L = LIMIT OF f(x) AS x APPROACHES c.
This means that the function, f(x), gets closer-and-closer to L, as x gets closer-and-closer to c. As long as x-getting-closer-and-closer-to-c doesn't involve dividing by zero, then typically L = f(c). The limit process gets much tricker if a division by zero (or division by a very small number) is involved in calculating L. Unfortunately, taking a derivative involves just such a process.

      In an extension of the original definition of limits, we can speak of f(x) getting closer-and-closer to L as x gallops off to infinity:
Limx -> infinity f(x) = L.
denotes the
L = LIMIT OF f(x) AS x APPROACHES INFINITY.


      The vagaries of DIVISION BY ZERO befuddled our eighteenth century predecessors, and eventually led to the WEIERSTRASS DEFINITION OF LIMIT, in which the expression:
L = LIMIT OF f(x) AS x APPROACHES c
means that for every epsilon>0, and for every |x-c|<epsilon, there exists a delta>0 such that |f(x)-L|<delta. The analogous Weierstrass definition for x approaching infinity is:
Limx -> infinity f(x) = LIMIT OF f(x) AS x APPROACHES INFINITY.
means that for every N>0, and for every x>N, there exists a delta>0 such that |f(x)-L|<delta. Proofs in calculus are then reduced tofinding a formula for calculating delta, given a value of epsilon or N.

      In some circles of mathematicians, the traditional definition of limit was considered DISREPUTABLE (Courant, 1996). This disrepute of the traditional definition of limit in calculus has been somewhat mitigated by the work of Abraham Robinson (Tymoczko, 1998).

      The ancient Greek mathematician, ARCHIMEDES, came very close to the idea of limits in his book entitled, THE SAND RECKONER, which includes an attempt to calculate the circumference of a circle, namely, pi × diameter, using a series of N-sided, regular polygons, as N gets very large (Archimedes, 1939).

      DIVISION-BY-ZERO. Everybody knows that you shouldn't divide by zero, but do you know why? The answer is that DIVISION IS DEFINED IN TERMS OF MULTIPLICATION. That is, when you write x = z / y, then you are actually asking what x is it, for which
x × y = z.
For y=0, the answer is in two parts: IF y=0 AND z~=0, THEN THERE IS NO SUCH x. On the other hand, IF y=0 AND z=0, THEN EVERY POSSIBLE x SATISFIES THE MULTIPLICATION (Seife, 2000). Thus, z/0 is NONEXISTENT for z~=0; whereas 0/0 is ANYTHING. Unfortunately, taking a derivative in elementary calculus involves getting perilously close to division by zero (vide infra). This single fact is why reasoning in calculus is fundamentally more difficult than reasoning in algebra. In calculus, you must ALWAYS PAY CLOSE ATTENTION!

      There are several fool's proofs in the popular mathematics literature, purporting to demonstrate that 1 = 2 or that 2+2 = 5. In both cases, the proof works by setting x=0, and distracting the reader not to pay attention when one divides by x (Asimov, Realm of Numbers; Singh, Fermat's Enigma).


WHAT IS A DERIVATIVE?

      A DERIVATIVE, or DIFFERENTIAL for a mathematical function (curve) in calculus is the instantaneous slope of that curve. If the slope is going upward, then the derivative is positive; if the slope is going downward, then the derivative is negative; if the slope is exactly horizontal, then the derivative is zero. For a function, f(x), its FIRST DERIVATIVE may be defined as
Limh->0(f(x+h)-f(x))/h = L.
Think of a curve, f(x), and nudge x forward by a small amount, h. Then (f(x+h)-f(x)) is the amount that f() moves upward, while x advances by h. As h approaches zero, the (f(x+h)-f(x))/h becomes the slope of f(x).

      Trouble is, obtaining L by substituting 0 for h would involve dividing by zero, and yielding the answer 0/0. The way to avoid this paradox is to get h out of the denominator before completing the calculation. For example, let f(x) = x2. Then:
L = Limh->0((x+h)2-x2)/h.
or:
L = Limh->0(x2+2xh+h2-x2)/h.
or:
L = Limh->0(2x+h).
Now, let h=0 and complete the calculation, so that L=2x.

      In general, the first derivative, L, for xk is kxk-1


METHOD OF LEAST SQUARES.

Suppose that you have a collection of n points, (x1, y1), (x2, y2), ... (xn, yn), and suppose that you wish to represent values of y1, y2, ..., yn, optimally. In the METHOD OF LEAST SQUARES, one calculates a BEST a, such that
(y1 - a)2 + (y2 - a)2 ... + (yn - a)2 is minimized.
That is:
(y12 -2ay1 + a2) + (y22 -2ay2 + a2) ... + (yn2 -2ayn + a2) is minimized.
The first derivative with respect to a is:
(2y1a - 2a) + (2y2a - 2a) ... + (2yna - 2a) = 0.
Dividing by two:
(y1a - a) + (y2a - a) ... + (yna - a) = 0.
Reorganizing:
y1a + y2a ... + yna - na = 0.
Then:
i=1§n y1 = na.
So that:
a = i=1§nyi/n
where: i=1§n denotes the sum from i=1 through i=n, usually denoted by the upper case Greek sigma, but not available in most Internet fonts.

      Why difference-squared and not difference-cubed or difference-absolute-value? Because the derivative comes out cleaner this way. If there is such a big distinction in the results obtained with difference-cubed or difference-absolute-value, then your curve-fitting equation is probably too crude.

      One of the valuable features of a PARABOLA (QUADRATIC FUNCTION) is that it achieves its MINIMUM (or MAXIMUM) at a single point, where its derivative equals zero. This means that we can use differential calculus to find the point at which a function's derivative equals zero, and that point minimizes the value of the function. This trick underlies the METHOD OF LEAST SQUARES, in which the function to be minimized is a sum of parabolas.














REFERENCES.

      1. Courant R, Robbins H, Stewart I.
What is Mathematics? An Elementary Approach to Ideas and Methods. Second Edition.
Oxford: Oxford University Press. 1996.

      2. Asimov I.
Realm of Numbers.
Out of Print.

      3. Agnew RP.
Calculus and Analytic Geometry.
1962.

      4. Tymoczko T.
New Directions in the Philosophy of Mathematics.
Princeton, NJ: Princeton University Press. 1998.

      5. Greek Mathematics.
Goold GP, ed. Thomas I transl. Loeb Classical Library. #335.
Cambridge, MA: Harvard University Press. 1939.
ISBN 0-674-99369-1, 511 pages.

      6. Singh S.
Fermat's Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Division of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.

      7. Seife C.
Zero. The Biography of a Dangerous Idea.
London: Penguin Books. 2000.
ISBN: 0-670-88457-X, 248 pages.


Last modified, November 18, 2001, by G. William Moore, MD, PhD.