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 Eleaabout 490 BC - about 425 BC

A B

A BB1

A BB1B2

A BB1B2B3

A BB1B2B3B4

Method of exhaustion

- expanding areas measures (not only polygons)

- to be able to account for more and more of the required area

Eudoxus410 or 408 BC – 355 or 347

BC

Archimedes287 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

Δx

f

Δy

Derivative

Δy/Δx = tan(α) – the slope of the secant line

Δx

f

Δyα

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)

x

y

x

yx

0lim

d

d

Calculation of areas again

16th century

Kepler: the area of sectors of an ellipse

Method: the area is sum of lines

Johannes Kepler1571-1630

Method of indivisibles

Cavalieri : the area/volume is made up from summing up infinite many „indivisible” lines/plan figures

Bonaventura Francesco Cavalieri 1598-1647

Fundamental theorem of Calculus

Newton: applying calculus to physics

Leibniz: notations which is used in calculus today

Gottfried Wilhelm Leibniz 1646-1716

Sir Isaac Newton 1642-1727

Fundamental theorem of CalculusNewton and Leibniz

- laws of differentiation and integration

- second and higher derivatives

- notations

Fundamental theorem of CalculusNewton 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Did Leibniz steal ideas from the unpublished notes (Newton shared them with some people)?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 footingCauchyReimannWeierstrass

- generalization of the integral Lebesgue

- generalization of the differentiationSchwarz

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.