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’.