2019年5月30日星期四

陳育霖

https://www.youtube.com/watch?v=kQTM1i8QSA8

2019年5月29日星期三

等速/等加速直線運動

等加速直線運動

v=v0+at
s=v0 t+1/2a t^2
v^2=v0^2+2as

ggb code

n=6

t=Slider(0,n)

eq1:y=v0+ a t
eq2:x=t

Intersect(eq1,eq2)
setTrace(A,true)
StartAnimation(t)

eq3:y=v0 t+1/2 a t^2

Intersect(eq2,eq3)
setTrace(B,true)
StartAnimation(t)

============================
一次輸入ggb code

Execute{(
"n=6",
"t=Slider(0,n)",
"eq1:y=v0+ a t",
"eq2:x=t",
"Intersect(eq1,eq2)",
"setTrace(A,true)",
"setColor(A,blue)",
"StartAnimation(t)",
"eq3:y=v0 t+1/2 a t^2",
"Intersect(eq2,eq3)",
"setTrace(B,true)",
"setColor(B,red)",
"StartAnimation(t)")}

等速直線運動(a=0 等加速直線運動)

v=v0
s=v0 t

the same ggb code

a=0

斜拋運動

vx=v0 cos(th)
vy=v0 sin(th)

y=vy t-1/2 g t^2
x=vx t

v0最小值
y=v0 sin(th) t-1/2gt^2>0
v0sin(th)t>1/2gt^2
v0>1/(2 sin(th))gt
=1/2gt
=4.9t

落地時間
t=(2 v0 sin(th))/g

最大高度
h=(v0 sin(th))^2/g

水平位移
x=(v0 sin(2 th))/g

th1=pi/6
th2=pi/4
th3=pi/3
g=9.8

t=10
v0=10

nn(th)=2 v0 sin(th)/g
th=th2
n=nn(th)
t=Slider(0,n)

eq1:y=v0 sin(th1) t-1/2 g t^2
eq2:y=v0 sin(th2) t-1/2 g t^2
eq3:y=v0 sin(th3) t-1/2 g t^2
eq1x:x=v0 cos(th1) t
eq2x:x=v0 cos(th2) t
eq3x:x=v0 cos(th3) t

Intersect(eq1,eq1x)
Intersect(eq2,eq2x)
Intersect(eq3,eq3x)

setTrace(A,true)
setTrace(B,true)
setTrace(C,true)

StartAnimation(t)

==============================
一次輸入ggb code

Execute({"th1=pi/6",
"th2=pi/4",
"th3=pi/3",
"g=9.8",
"v0=10",
"nn(th2)=2 v0 sin(th2)/g",
"n=nn(th2)",
"t=Slider(0,n)",
"eq1:y=v0 sin(th1) t-1/2 g t^2",
"eq2:y=v0 sin(th2) t-1/2 g t^2",
"eq3:y=v0 sin(th3) t-1/2 g t^2 ",
"eq1x:x=v0 cos(th1) t",
"eq2x:x=v0 cos(th2) t",
"eq3x:x=v0 cos(th3) t",
"Intersect(eq1,eq1x)",
"Intersect(eq2,eq2x)",
"Intersect(eq3,eq3x)",
"setTrace(A,true)",
"setTrace(B,true)",
"setTrace(C,true) ",
"StartAnimation(t)"})

=================================

https://www.geogebra.org/m/ne6ug4jp

2019年5月20日星期一

group

unifrom expansin bounds for Cayley Graphs of SL2(Fp)

14. Semigroups and Monoids - Groups - Gate

Modern Algebra (Abstract Algebra) Made Easy - Part 6 - Cosets and Lagrange's Theorem

Ben1994

anlaysis and beyond

Analysis and Beyond

Celebrating Jean Bourgain's Work and Impacthttps://www.ias.edu/ideas/2016/bourgain-analysis-and-beyond

2019年5月19日星期日

machine learning

http://www.lps.ens.fr/~krzakala/

Kristian Roopnarine

https://medium.com/@kristian.roopnarine

KNN for Iris
https://medium.com/@kristian.roopnarine/building-a-k-nearest-neighbor-algorithm-with-the-iris-dataset-b7e76867f5d9

圓,二次曲線 circle equation

前言

數形合一
新課綱:數學學習重點與實施要點納入計算機工具與科技的使用與觀念。學習工具對於數學教學助益極大。除了傳統教具如圓規、三角板、方格紙等，資訊時代的計算機 (calculator) 、電腦 (computer) 、網路、多媒體、行動工具等都是有用的學習工具。數學是一種規律的科學，計算機及電腦可以協助落實探究活動。
姚鴻澤院士：教證明時, 若畫個圖能一目了然, 看到直覺的觀點, 那麼只畫一個圖就好了。這樣就非常簡單, 學生也很快可以學會。
吳建銘教授(東華應數系系主任)：20世紀的數學是加了電腦以後的數學

OUTLINE

圓方程式
圓方程式之判別式與圓之關係
橢圓方程式及定義
雙曲線方程式及定義
拋物線方程式及定義

ggb
https://www.geogebra.org/graphing
1.圓方程式
標準式：(x-h)^2+(y-k)^2=r^2
(畢氏定理)

ggb
Execute({"h=1","k=-1","O=(h,k)","r=5"})
(x-h)^2+(y-k)^2=r^2
Point(eq1)
(點圓上任意一點A)
x=h
y=k
Segment(A,O)
PerpendicularLine(A,y=k)
(or input OrthogonalLine(A,y=k)
(過A點做垂直線交y=k於B點)
setTrace(A,true)
StartAnimation(A)
(在A點點右鍵show trace, animation)
[r 動畫(A軌跡成心形線)]

Execute({"h=1","k=-1","O=(h,k)","r=5",
"eq1:(x-h)^2+(y-k)^2=r^2","A:Point(eq1)","x=h","f:y=k","Segment(A,O)",
"i:OrthogonalLine(A,f)","Intersect(f,i)","setTrace(A,true)","StartAnimation(A)"})

2.圓方程式
一般式：x^2+y^2+dx+ey+f=0
判別式: D=d^2+e^2-4f
D>0 <==>圓
D=0 <==>點
D<0 <==>虛圓

x^2+y^2+dx+ey+f=0
d=3
O=(d/-2,e/-2)
D=d^2+e^2-4f
r=1/2 sqrt(D)
Point(eq1)
Segment(A,O)
StartAnimation(d)

Execute({"d=3","eq1:x^2+y^2+dx+ey+f=0","O=(d/-2,e/-2)",
"D=d^2+e^2-4f","r=1/2 Sqrt(D)","A=Point(eq1)",
"g:Segment(A,O)","StartAnimation(d)"})

3.橢圓方程式
(x-h)^2/a^2+(y-k)^2/b^2=1
定義
PF1+PF2=2a

ggb

Execute({"a=5","b=4","c=sqrt(a^2-b^2)","h=1","k=-1","O=(h,k)"})
(x-h)^2/a^2+(y-k)^2/b^2=1
F1=(h-c,k)
F2=(h+c,k)
Point(eq1)
l1:Segment(A,F1)
l2:Segment(A,F2)
Length(l1)+length(l2)
setTrace(A,true)
StartAnimation(A)
(在A點點右鍵 show trace, animation(開始動畫))

由定義畫出軌跡
(Point A 動畫off)
eq2:Circle(F1,2 a)
Point(eq2)
l3:Segment(B,F1)
l4:Segment(B,F2)
l5:PerpendicularBisector( l4)
C:intersect(l5,l3)
setTrace(C,true)
StartAnimation(B)
(做BF2的中垂線交BF1,在交點上點右鍵 show trace. 圓上B 點右鍵animation)

4.雙曲線方程式
(x-h)^2/a^2-(y-k)^2/b^2=1
定義
|PF1-PF2|=2a

ggb

Execute({"a=3","b=4","c=sqrt(a^2+b^2)","h=1","k=-1","O=(h,k)"})
(x-h)^2/a^2-(y-k)^2/b^2=1
F1=(h-c,k)
F2=(h+c,k)
Point(eq1)
b(x-h)+a(y-k)=0
b(x-h)-a(y-k)=0

Segment(A,F1)

Segment(A,F2)
StartAnimation(A)
(在A點點右鍵 show trace, animation(開始動畫))

https://www.geogebra.org/m/eqxdczkb
=======================================
依次輸入ggb code

Execute({"(x-h)^2/a^2-(y-k)^2/b^2=1",
"F1=(h-c,k)",
"F2=(h+c,k)",
"Point(eq1)",
"b(x-h)+a(y-k)=0",
"b(x-h)-a(y-k)=0",
"Segment(A,F1)",
"Segment(A,F2)",
"StartAnimation(A)"})

=======================================
由定義畫出軌跡
(Point A 動畫off)
eq2:Circle(F1,2 a)
Point(eq2)
Line(B,F1)
Segment(B,F2)
(做BF2的中垂線交BF1,在交點上點右鍵 show trace. 圓上B 點右鍵animation)
StartAnimation(B)

5.拋物線
(y-k)^2=4c(x-h)
定義
d(P,L)=PF

Execute({"c=5","h=1","k=-1","V=(h,k)"})
(y-k)^2=4c(x-h)
F=(h+c,k)
L:x= h-c
A:Point(eq1)
l1:Segment(A,F)
l2:PerpendicularLine(A,L)
StartAnimation(A)

由定義畫出軌跡
(Point A 動畫off)

B:Point(L)
g:Segment(B,F)
l3:PerpendicularBisector(g)
C:Intersect(l2,l3 )
StartAnimation(C)