2. What is… Calculus?

3. Two major ideas Approximation Linearization

4. Approximation - Examples Approximating the circle Approximating the pyramid

5. Linearization Replacing complicated function by a linear function

6. Limits In both cases, a sequence of intermediate steps leads us to the “ideal” object This can be mathematically described as finding the limit

7. This leads us to… Differential Calculus Integral Calculus

8. Combining two ideas… Try

9. Taylor series for sin x

10. Taylor series were first discovered by…

11. James Gregory Born November 1638 Drumoak, Aberdeenshire, Scotland Died October 1675 (aged 36) Edinburgh, Scotland

12. Taylor series were further studied by…

13. Brook Taylor Born 18 August 1685 Edmonton, Middlesex, England Died 29 December 1731 (aged 46) London, England

14. Taylor series were extensively used by…

16. Sir Isaac Newton Born 25 December 1642 Woolsthorpe, Lincolnshire, England Died 20 March 1726 (aged 84) Kensington, Middlesex, England

17. Problem Taylor series can approximate only “very nice” functions (so-called analytic functions which are infinitely many times differentiable)

18. New approach: Fourier series Use of trigonometric polynomials to approximate functions

19. Example: a sawtooth wave

22. Fourier series were first studied by…

24. Leonhard Euler Born 15 April 1707 Basel, Switzerland Died 18 September 1783 (aged 76) Saint Petersburg, Russian Empire

25. Fourier series were named after…

26. Joseph Fourier Born 21 March 1768 Auxerre, Burgundy, Kingdom of France (now in Yonne, France) Died 16 May 1830 (aged 62) Paris, Kingdom of France

27. (Some) applications of calculus Area of applicationWhat is used All areas of scienceIntegration and differentiation to solve differential equations Almost all areas of science Integration and differentiation as tools for statistical applications Scientific computationsTaylor series, other methods of approximation MP3 and other compression algorithms Fourier series, Fourier transform JPEG and similar formatsWavelet analysis Imaging (e.g. medical imaging) Methods of optimization (e.g. Lagrange multipliers)