A TOUR OF THE CALCULUS From Pythagoras to Newton

TWO BASIC QUESTIONS ZENO STYLE How can you add an infinite number of things together and not get infinity as an answer? How can you divide something that is already infinitely small (nearly zero) and not get something else that is infinitely small (nearly zero)?

WHAT IS CALCULUS? Calculus is the arithmetic of infinites. What do you do in Calculus? The bulk of a Calculus I class is concerned with finding the slopes of curves and the areas underneath them. Why is Calculus so useful? It is the mathematics of change. The area of a rectangle is easy (e.g. 5x4). But the area under a parabola is very difficult to find. When you throw a stone or a cannonball it follows a parabolic arch. What if you wanted to know the area required to fill in an arch that large? Most real world problems with motion involve calculus.

Math Models

FROM WHENCE DOES THE NAME CALCULUS COME? Gottfried Leibniz “This was the first page of the first publication on calculus. In the October 1684 issue of the Acta eruditorum, Leibniz presented a ‘new method’ for finding maxima, minima, and tangents and, in the last line of the title, promised ‘a remarkable type of calculus for this.’ The name stuck.” (Dunham 4)

A STROLL DOWN MEMORY LANE

PHILOLAUS, THE PYTHAGOREAN Nature in the Cosmos is harmoniously composed of the limited and the unlimited, both the entire cosmos and everything in it. The Pythagoreans upheld this as the ancient headwaters for music understood as a liberal art.

NATURE IN THE COSMOS A Greek Lesson Nature in Greek is physis Gr. Kosmos means order

IS HARMONIOUSLY COMPOSED

OF THE LIMITED AND THE UNLIMITED Of the bounded and the unbounded (Gr. apeiron) A sequence of real numbers is called a Cauchy sequence, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > Npositiveinteger where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring absolute value to be infinitesimal for every pair of infinite m, n.infinitesimal Every Cauchy sequence of real (or complex) numbers is bounded (since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M is the largest absolute value of the terms up to and including the N-th, then no term of the sequence has absolute value greater than M+1).bounded

HARMONIES OF THE LIMITED AND UNLIMITED

BOTH THE ENTIRE COSMOS AND EVERYTHING IN IT

SO DID THE GREEKS HAVE CALCULUS? No!

WHY NOT? In the Fourth Century BC, Aristotle banned infinities from Greek mathematics. He did this as a knee jerk reaction to Zeno’s Paradoxes. Pythagoras, turned over in his grave…

ARCHIMEDES METHOD OF EXHAUSTION I’m so tired… Question: How do you measure the area under a parabola? Answer: Lots of….. Triangles!

NEWTON USED… Rectangles

THE ANCIENTS DID LITTLE WITH ALGEBRA

THE EARLY MODERN ALGEBRA REVIVAL In the 16 th century Italians Tartaglia and Cardano sought solutions to algebraic problems Francois Viete also revived the Diophantine tradition and extended it. In the 17 th century Pierre de Fermat began exploring ‘higher parabolas’ Fermat wrote his “Last Theorem” in the margins of Diophantus’ book Rene Descartes brought together Algebra and Geometry Descartes’ fusion is now called the Cartesian coordinate system

FERMAT AND PASCAL LEAD THE WAY TO CALCULUS

THESE TWO INSPIRED LEIBNIZ AND NEWTON The contribution of Newton and Leibniz is not the area under the curves of polynomials. That was already done. Nor is it the slopes of tangent lines to those curves. That was already done. Their great work is the Fundamental Theorem of Calculus. Newton and Leibniz each independently discovered that an inverse relationship holds between the slope and area of curves.

THE FUNDAMENTAL THEOREM OF CALCULUS Let an equation (curve B) describe the slope of a tangent line to curve A. In that case the area under curve B provides the values for curve A. Or Let an equation (curve A) describe the area of curve B. In that case the slope of curve A at any point is the same as the value of curve B at that location.

THE FUNDAMENTAL THEOREM OF CALCULUS

SOME EXAMPLES

So What’s Left after the Fundamental Theorem of Calculus?

THE MOST BEAUTIFUL EQUATION

Questions?