Newton 362, proof of the Newton-Leibniz formula
Newton-Leibniz (cí) formula (Baidu Encyclopedia): ...
... Newton-Leibniz (cí) formula: see Newton 358, 359...
Derivation
... Derivation, Derivation: See Euclid 7...
(...Euclid: Novel title... ）
Define a variable upper integral function Φ(x) = ∫[a,x]f(t)dt, so that the function Φ(x) gets the increment △x, then the corresponding function increment is:
△Φ(x)=Φ(x+△x)-F(x)
=∫[a，x+△x]f（t）dt-∫[a，x]f（t）dt
=∫[x,x+△x]f(t)dt (adductability of definite integrals to intervals)
... Definition, Righteousness, Definition: See Euclid 28...
... Functions, numbers, functions: see Euclid 52...
... Φ: The 21st Greek alphabet. Pronunciation: fài... See Newton 359...
... ∫: integral symbol, elongated for the letter s... See Newton 338...
... d: differential initial...
[differential): n. (noun) difference; difference; difference; wage scale difference (especially for different types of work in the same industry).
adj. (adjective) differentiated; differentiated; differentiated.
- Newton 321
What does dx mean? ——Netizens ask questions
2019-09-07, cats who want to play games: d(x) represents differential to x.
d in dy/dx means "tiny increment", which means a tiny increment y divided by a tiny increment x. In functions, differentiation means.
dx is the differentiation of x, is to trivialize the increment, dx is a very small x.
- Newton 3]
... Variable upper-limit integral function Φ(x) = ∫[a,x]f(t)dt: see Newton 359~361...
... △: The pronunciation is "Delta". The phonetic alphabet is /deltə/.
In physics, △ is often used as a prefix to a variable to indicate the amount of change in the variable, such as: △t (time change), △T (temperature change), △X (displacement change), △v (speed change) and so on... See Newton 8...
... Properties: 1. The properties of the substance; the properties of the substance due to the inclusion of a certain ingredient: viscosity. Bullet ~. Medicine ~. Alkali~. Oil~. 2. Suffixes, which are added after nouns, verbs or adjectives to form abstract nouns or attribute words, indicate a certain nature or performance of things: party ~. Discipline ~. Create~. Adaptation ~. Excellent ~. Universal ~. Congenital ~. Pop ~... See Euclid 10...
……
Derivation one
According to the theorem of the median integral: △Φ(x) = ∫[x,x+△x]f(t)dt=f(ξ)△x, (ξ is between x and x+△x)
... ξ: uppercase Ξ, lowercase ξ, is the fourteenth Greek letter, transliteration of Chinese: Kesi.
Lowercase ξ for: mathematical random variables...
{The theorem for the median of the integral: if the function f(x) is continuous over the closed interval [a,b], then at least one point ξ exists on the integral interval [a,b], so that the ∫ [a,b]f(x)dx=f(ξ)(b-a). （a≤ξ≤b）
See Newton 351 for proof. }
According to the derivative definition, to:
F'(x)=lim(△x→0)△Φ(x)/△x
=lim（△x→0）f（ξ）△x/△x
=lim（△x→0）f（ξ）
=lim(o→x)f(o)
=f（x）
(''△x→0' and 'ξ→x' are different expressions of the same mathematical fact and are therefore equivalent.) Modern scholars say. ）
... Derivatives, numbers, derivatives: see Newton 288~294...
… lim：limit…
[... limit: n. limit; limit; limit; limit; limit; limit; limit; (regional or local) realm, boundary, range.
v. Limit; Limit; Limit; Reduce...]
即:F'(x)=f(x)
From the definition of indefinite integrals, it is obtained:
∴ Φ(x)=F(x)+C
∵ Φ（a）=F（a）+C=0
∴ C=-F（a）
∴ Φ(x)=F(x)+C=F(x)-F(a)
Φ（x）=F（x）-F（a）
∴ Φ(b)=F(b)-F(a)
即：∫[a，b]f（x）dx=F（b）-F（a）
Proof.
In 1670, the English mathematician Isaac Barrow expressed in geometric form the tangent problem in the inverse proposition of the area problem in his book Lecture Notes on Geometry, which is actually a geometric expression of the Newton-Leibniz formula.
See the next episode, "A Brief History of the Development of Newton 363 and the Newton-Leibniz Formula"
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