Short history of Calculus

Greeks Numbers: - Integers (1, 2, 3,...) - Ratios of integers (1/2, 8/7,...) The “number line” contains “holes” - e.g. they hadn't got such a numbers: “pi”, “e”, sqrt(2)...

Problems about infinite Zeno's paradoxes - Achilles and the tortoise - The dichotomy paradox* - The arrow paradox Zeno of Elea about 490 BC - about 425 BC

AB

AB B1B1

AB B1B1 B2B2

AB B1B1 B2B2 B3B3

AB B1B1 B2B2 B3B3 B4B4

Method of exhaustion - expanding areas measures (not only polygons) - to be able to account for more and more of the required area Eudoxus 410 or 408 BC – 355 or 347 BC

Archimedes 287 BC – 212 BC

Area of segment of parabola

Theorem: The area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex. Idea of proof: A, A+A/4, A+A/4+A/16, A+A/4+A/16+A/64 … are more and more closer to the area of triangle (T), so T = A*(1 + 1/4 + 1/16 + 1/64 +…) = A*4/3 - first known example of the summation an infinite series. Area of segment of parabola

Area of a circle - using the method of exhaustion, too - „secondary product”: approximate value of „pi”

New problems 12th century India – Bhaskara II. An early version of derivative Persia – Sharaf al-Din al-Tusi Derivative of cubic polynomials

Derivative a measurement: how a function changes when its input changes: Δy/Δx ΔxΔx f ΔyΔy

Derivative Δy/Δx = tan(α) – the slope of the secant line ΔxΔx f ΔyΔy α

Derivative the value of Δy/Δx when Δx is increasingly smaller

Differentiation a method: the process of computing the derivative of a function. Today’s form from Leibnitz and Newton (17th century)

Calculation of areas again 16th century Kepler: the area of sectors of an ellipse Method: the area is sum of lines Johannes Kepler

Method of indivisibles Cavalieri : the area/volume is made up from summing up infinite many „indivisible” lines/plan figures Bonaventura Francesco Cavalieri

Fundamental theorem of Calculus Newton: applying calculus to physics Leibniz: notations which is used in calculus today Gottfried Wilhelm Leibniz Sir Isaac Newton

Fundamental theorem of Calculus Newton and Leibniz - laws of differentiation and integration - second and higher derivatives - notations

Fundamental theorem of Calculus Newton contra Leibniz Newton derived his result first. Leibniz published his result first. Did Leibniz steal ideas from the unpublished notes (Newton shared them with some people)? examination of the papers they got their results independently

Further development - from the 19th century - more and more rigorous footing Cauchy Reimann Weierstrass - generalization of the integral Lebesgue - generalization of the differentiation Schwarz

Summary Calculus: calculation with infinite/infinitesimal Two different parts:- integral - differentiation It was almost complete in the 17th century that we use nowadays in the business mathematics.