The Eve Before Calculus : Fermat

微积分发明的前夜

https://m.toutiaocdn.com/i6648474300158312968/?app=news_article&timestamp=1614968855&use_new_style=1&req_id=202103060227340102120752062220EFB9&group_id=6648474300158312968&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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.

https://m.toutiaocdn.com/i6933218183247626755/?app=news_article&timestamp=1614316375&use_new_style=1&req_id=2021022613125401019410001308000E75&group_id=6933218183247626755&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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 )

http://calculusmadeeasy.org/

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/

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)

\displaystyle A(x) = \int_{-2 }^ {2} x^{3}. \cos \frac{x}{2}.\sqrt{4-x^2}dx

\displaystyle B(x) = \int_{-2 }^ {2} \frac{1}{2}\sqrt{4-x^2}dx

A(x) = – A(-x) => Odd function
=> A(x) = 0 since its area canceled out over [-2, 2]

B(x) = B(-x) => Even function
\displaystyle\implies B(x) = 2\int_{0 }^ {2} \frac{1}{2}\sqrt{4 - x^2}dx
\displaystyle\implies B(x) = \int_{0 }^ {2} \sqrt{4 - x^2}dx

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

\displaystyle B(x) = \int_{0 }^ {\pi/2} \sqrt{4 - 4.\sin^{2} {t} }. (2 \cos t. dt)

\displaystyle \implies B(x) = \int_{0 }^ {\pi/2} 2.\cos t. (2 \cos t. dt)

\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} 4 \cos^{2} t. dt

\displaystyle \cos ^{2} t = \frac {1 + \cos 2t} {2 }

\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} (2 + 2\cos 2t) . dt

\displaystyle\implies B(x) = (2 t) \Bigr|_{0 }^ {\pi/2} + (2. \frac{1}{2} \sin 2t) \Bigr|_{0 }^ {\pi/2}

\displaystyle\implies B(x) = (\pi ) + \sin \pi = \pi

\boxed{ I = \pi = 3.14159}

A smarter method using Analytic Geometry: A circle of radius 2 is

x^2 + y^2 = 4 \implies y = \sqrt {4 -x^2}

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 ?