微积分发明的前夜
Tag Archives: Calculus
Feymann Calculus Trick
理查德·费曼非常聪明的求导 differentiate 技巧
Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.


Learn Calculus With These Four Online Courses
https://lifehacker.com/learn-calculus-with-these-four-online-courses-1836377912/amp
Four Online Free Calculus Courses:
1. http://www-math.mit.edu/~djk/calculus_beginners/
2. “Calculus Made Easy” (1910 )
3. “Essence of Calculus” (12 short videos, with Chinese subtitle)
4. [Best👍] “Calculus” (MIT Prof Gilbert Strang)
https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/
Walter Bradley Center Fellow Discovers Longstanding Flaw in Elementary Calculus
… The paper on the proof of the second derivative (Quotient rule) :
https://www.dropbox.com/s/mv6npb3tpe76xjl/4.pdf?dl=0
Reference:
Faà di Bruno Fomula
The universe’s secrets are at your fingertips – just learn calculus | Cornell Chronicle
Hacker Calculus
The Error Function & The Integral of e^(-x^2)
The Bell Curve:
Improper Integral vs. Infinite Series
To test if the infinite series converges,
provided the function
in the interval [2, ∞[ is:
- positive &
- decreasing function
you can convert it to
(although they converge to different values.)
Integration by parts, DI method
(Traditional Method learnt at O-Level) Integration by Parts:
NEW Technique taught by this UC-Berkley Chinese Mathematician: “DI” Method
First Stop: “D column at 0 “
2nd Stop:
3rd Stop:
WiFi Password = Integral Answer
China 南京航空航天大学 Nanjing University of Aeronautics and Astronautics set the WiFi password as the answer of this integral (first 6 digits).
Can you solve it?
(If can’t, please revise GCE “A-level” / Baccalaureate / 高考 Calculus 微积分)
Answer : Break the integral (I) into 2 parts:
I = A(x) + B(x)
A(x) = – A(-x) => Odd function
=> A(x) = 0 since its area canceled out over [-2, 2]
B(x) = B(-x) => Even function
Let x = 2 sin t => dx = 2 cos t. dt
x = 2 = 2 sin t => sin t = 1 => t = π / 2
x = 0 = 2 sin t => sin t = 0 => t = 0
A smarter method using Analytic Geometry: A circle of radius 2 is
What is math?
What is Math ? Interesting article below:
https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/
- Mathematics = “that which is learned“ –(Pythagoras)
Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…
What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.
Difference between Riemann Integral & Lebesgue Integral ?
Calculus = 微积分
Calculus = 微积分
出自: 荀况《荀子.大略》300 BCE [same time as Euclid]
“尽小者大, 积微者著”
=> 見微知著
Calculus Fundamental Technique
This is just a simple but powerful application of Calculus, behind which lies the philosophy of Leibniz:
1. D (=dy/dx) is the inverse function of
2. Calculus Fundamental Technique:
E.g. Sherlock Holmes example:
1. D first:
dT/dt = k(T-Ts)
=> can’t solve directly
2. Take D’s inverse:
=> can solve now !
Solution: Sherlock Holmes
Apply Newton’s Law of Cooling:
(Room Temperature)
At t=0, (normal body temperature)
=> A=16
Let t =x hour 1st temperature taken
…[1]
t = x+1 hour later
k = ln(3/4)
[1]: kx = ln(1/2)
x = ln(1/2) / ln(3/4)
x=2.41 hr = 2 hr 25m
Murder Time= 2am -2h25m =11:35 p.m. [QED]
Newtonian Calculus not rigorous !
Why Newton’s Calculus Not Rigorous?
…[1]
cancel x (≠0)from upper and below =>
…[2]
In [1]: we assume x ≠ 0, so cancel upper & lower x
But In [2]: assume x=0 to get L=5
[1] (x ≠ 0) contradicts with [2] (x = 0)
This is the weakness of Newtonian Calculus, made rigorous later by Cauchy’s ε-δ ‘Analysis’.