What is… Calculus? Two major ideas Approximation Linearization.

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Presentation transcript:

What is… Calculus?

Two major ideas Approximation Linearization

Approximation - Examples Approximating the circle Approximating the pyramid

Linearization Replacing complicated function by a linear function

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

This leads us to… Differential Calculus Integral Calculus

Combining two ideas… Try

Taylor series for sin x

Taylor series were first discovered by…

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

Taylor series were further studied by…

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

Taylor series were extensively used by…

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

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

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

Example: a sawtooth wave

Fourier series were first studied by…

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

Fourier series were named after…

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

(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)