Calculus
Derivates, Calculus & ODE
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I. Derivates
Kepler planet rotator
– despite Taylor series looked simple – there is more under the hood:
II. Calculus
Intro
– one of the first practical use of Calculus came with Kepler in first half of 17th century as a form of areal velocity, but Kepler did not find general solution for particular derivates resp. integrals.. – see …
– first calculus theory was proposed by Isaac Newton in 50s of 17th century (who also used it on Kepler´s planets orbit..) and in 60s Gottfried Wilhelm Leibniz – from which notation is nowadays used.
– w/ Cauchy and Weierstrass 150 y. later calculus got its robust form..
Taylor series
– writing goniometric fnct w/ polynomials – derivates (Taylor Series – GeoGebra):
Lpic:
1. order: x
3rd order: x – x^3/ 3!
5th order: x – x^3/ 3! + x^5/ 5!
7th order: x – x^3/ 3! + x^5/ 5!
9 – 11 – 13 – 15 – 17 – nine orders give pretty exact shape.
…
As we can see between -0,5 and 0,5 tangent lines stays pretty the same.
Sin value of 45° 0,5 on x is root: 2/ 2 or 0,701 on y axis – that means, that in this quarter it rise 40 % faster, but tangent keeps same angle.
From 0,5 to 1 angle is changing much faster from 45 % to 90°, but speed decreasing to 0 (on y axis) for a while.
III. ODE
Kepler planet rotator
This instrument will use acceleration of planets firstly described by Kepler, but follows logic of rotating and object on the rope by the hand etc…
This instrument will use acceleration of planets firstly described by Kepler, but follows logic of rotating and object on the rope by the hand etc…
Intro
This instrument will use acceleration of planets firstly described by Kepler, but follows logic of rotating and object on the rope by the hand etc…
3Blue1brown
Kepler planet rotator
This instrument will use acceleration of planets firstly described by Kepler, but follows logic of rotating and object on the rope by the hand etc…
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