人工智能教程 - 數學基礎課程1.1 - 數學分析（一）15-17 微分方程和分離變量,定積分及性質,微積分第一定理

微分方程 differential equation

Ex:

$\frac{dy}{dx} = f(x)$

$y= \int f(x) dx$

solved substitution

Ex2:

$(\frac{d}{dx}+x)$爲annihilation operator 湮沒算符 in quantum mechnics

$\frac{dy}{dx} = -xy$

$\frac{dy}{y} = -xdx$

$\int \frac{dy}{y} = -\int xdx$

$ln y = -x^2/2 +C$

$e^{lny} = e^{-x^2/2}+C$

$y = Ae^{-x^2/2} (A=e^c)$

Solution:

$y = ae^{-x^2/2} \ any \ a$

$\frac{dy}{dx} = a.\frac{d}{dx} . e^{-x^2/2}$

$= a.(-x) . e^{-x^2/2}$

$= -x.y$

by the way,the function is known as the normal distribution 正態分佈

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分離變量法

SEPARATION OF VARIABLES

$\frac{dy}{dx} = f(x).g(y)$

$\frac{dy}{g(y)} = f(x)dx$

$H(y) = \int \frac{dy}{g(y)}; F(x) = \int f(x)dx$

$H(y)=F(x)+C \rightarrow implicit$

$y = H^{-1}(F(x)+C)$

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定積分

Definite Integrals

Find Area under a curve =$\int_{a}^{b} f(x) dx$

累積和(cumulative sum)

To compute the area

1. divide into rectangles
2. add up areas
3. take the limit as rectangles get thin

簡寫 abbreviation

$\sum_{i=1}^{n} a_i = a_1+a_2+...+a_n$

$\sum$ is called sigma

Notation (Riemann Sums)

General Procedure for definite integrals:

$\Delta x = \frac{b-a}{n}$

Pick any height of f in each interval:

${\color{Red} \sum_{i=1}^{n} f(c_i) \Delta x\rightarrow \int_{a}^{b} f(x)dx}$

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微積分第一定理

Fundamental theorem of calculus(FTC1)

If F’(x) = f(x), then

${\color{Red} {\color{Red} \int_{a}^{b} f(x)dx = F(b)-F(a)}=F(x)|_{a}^{b}}$

$F=\int f(x)dx$

NOTATION

$F(b) - F(a) = F(x)|_{a}^{b} = F(x)|_{x=a}^{x=b}$

Ex1:

$F(x) = x^3/3$

$F'(x) = x^2(=f(x))$

$\int_{a}^{b}x^2dx = F(b)-F(a)=\frac{b^3}{3}-\frac{a^3}{3}$

$\int_{0}^{b}x^2dx = \frac{x^3}{3}|_{0}^{b}=\frac{b^3}{3}$

True geometric interp of definit integral is that area above x-axis minus the area below the x-axis

定積分的性質

Properties of integrals

$1.\int_{a}^{b}(f(x)+g(x))dx = \int_{a}^{b}f(x)dx +\int_{a}^{b}g(x)dx$

$2.\int_{a}^{b}Cf(x)dx = C\int_{a}^{b}f(x)dx$

$3.\int_{a}^{b}f(x)dx +\int_{b}^{c}f(x)dx = \int_{a}^{c}f(x)dx \ \ (a<b<c)$

$4.\int_{a}^{a}f(x)dx = 0$

$5.\int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx$

6. (Estimation)If f(x)<=g(x),then:$\int_{a}^{b}f(x)dx \leq \int_{a}^{b}g(x)dx$

change of variables: (=substituion)

${\color{Red} \int_{u_1}^{u_2}g(u)dx = \int_{x_1}^{x_2}g(u(x)).u'(x)dx}$

u = u(x)	u1 = u(x1)
du = u’(x)dx	u2 = u(x2)

Only works if u’ does not change sign