what is… calculus? two major ideas approximation linearization
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 CalculusIntegral Calculus
Combining two ideas…
sin 𝑥=¿? ¿
Try 𝑥−𝑥3
6
Taylor series for sin x
sin 𝑥=¿ 𝑥− 𝑥3
6+ 𝑥5
120−
𝑥7
5040+…¿
Taylor series were first discovered by…
Taylor series were first discovered by…
James GregoryBornNovember 1638Drumoak, Aberdeenshire, Scotland
DiedOctober 1675 (aged 36)Edinburgh, Scotland
Taylor series were further studied by…
Taylor series were further studied by…
Brook TaylorBorn18 August 1685Edmonton, Middlesex, England
Died29 December 1731 (aged 46)London, England
Taylor series were extensively used by…
Taylor series were extensively used by…
Taylor series were extensively used by…
Sir Isaac Newton
Born25 December 1642Woolsthorpe, Lincolnshire, England
Died20 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
Example: a sawtooth wave
...
4
4sin
3
3sin
2
2sinsin
2)(
xxxxxs
Example: a sawtooth wave
...
4
4sin
3
3sin
2
2sinsin
2)(
xxxxxs
Fourier series were first studied by…
Fourier series were first studied by…
Fourier series were first studied by…
Leonhard EulerBorn15 April 1707Basel, Switzerland
Died18 September 1783 (aged 76)Saint Petersburg, Russian Empire
Fourier series were named after…
Fourier series were named after…
Joseph FourierBorn21 March 1768Auxerre, Burgundy, Kingdom of France (now in Yonne, France)
Died16 May 1830 (aged 62)Paris, Kingdom of France
(Some) applications of calculusArea of application What is used
All areas of science Integration and differentiation to solve differential equations
Almost all areas of science
Integration and differentiation as tools for statistical applications
Scientific computations Taylor series, other methods of approximation
MP3 and other compression algorithms
Fourier series, Fourier transform
JPEG and similar formats Wavelet analysisImaging (e.g. medical imaging)
Methods of optimization (e.g. Lagrange multipliers)