1. Short history of Calculus

2. 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)...

3. 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

4. AB

5. AB B1B1

6. AB B1B1 B2B2

7. AB B1B1 B2B2 B3B3

8. AB B1B1 B2B2 B3B3 B4B4

9. 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

10. Archimedes 287 BC – 212 BC

11. Area of segment of parabola

12. 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

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

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

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

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

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

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

19. Calculation of areas again 16th century Kepler: the area of sectors of an ellipse Method: the area is sum of lines Johannes Kepler 1571-1630

20. Method of indivisibles Cavalieri : the area/volume is made up from summing up infinite many „indivisible” lines/plan figures Bonaventura Francesco Cavalieri 1598-1647

21. 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

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

23. 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 Did Leibniz steal ideas from the unpublished notes (Newton shared them with some people)?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

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

25. 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.