# 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 $\boxed{ (f\circ H)^{(n)} =\displaystyle\sum_{\sum_{j=1}^n j\,m_j=n} \frac{n!}{m_!\ldots m_n!}\, \bigl(f^{(m_1+\ldots+m_n)})\circ H\bigr)\, \prod_{j=1}^n \left(\frac{H^{(j)}}{j!}\right)^{m_j} }$

# The Error Function & The Integral of e^(-x^2) $\boxed{ \displaystyle erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} {e^{-{t}^2}dt}}$  The Bell Curve: $\boxedy = e^{-x^2}$

# Improper Integral vs. Infinite Series $\displaystyle \int_{2}^{\infty}{\frac{1}{x^{2} - 1}dx} =\boxed{ \ln {3}} = 1.09$ $\displaystyle \sum_{n=2}^{\infty}{\frac{1}{n^{2} - 1}} = \boxed{\frac{3 } {2 }} = 1.5$

To test if the infinite series $\displaystyle \sum_{n=2}^{\infty}{\frac{1}{n^{2} - 1}}$ converges,
provided the function $\displaystyle f(n) = {\frac{1}{n^{2} - 1}}$
in the interval [2, ∞[ is:

• positive &
• decreasing function

you can convert it to $\displaystyle \int_{2}^{\infty}{\frac{1}{x^{2} - 1}dx}$
(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: $\displaystyle \int {e^{x}}\sin {x} dx$ 