sinx-0.5x=0

2006-11-12 6:43 am
sinx-0.5x=0
find x
更新1:

Navy gi︰ 0.5x =(sin30)x《=這些情況要用radian =(sin30x)《=沒什麼事嗎?無端端放進去 還有,sin a=sin b並不表示a=b 雖然的確是一個根,但你做出來的每一步都是錯的。

更新2:

myisland8132︰ 做法是Newton Method,方法用對了。但是計算時將一個四捨五入了很多的數字代進下一步,搞得做出來的答案不夠準確。一個會考可用的計數機都能比你的答案多四個位。 還有,我們可以證明如果f(x)=sinx-0.5x,f(x)=-f(-x),這樣只需要做一個根出來就可證明另一個根是他的負值,不需要做兩次。 而且最後答案沒有寫0這根進去。

更新3:

nancypoon02︰ 相信中學二年級的人也可以一眼看出其中一個根是0。更加不需要畫圖。

回答 (3)

2006-11-12 7:52 am
✔ 最佳答案
draw the graph
http://in.geocities.com/myisland8132/Book1.pub
we see that
x=0 is a solution by inspection
there is another root about [-2,-1] and [1,2] use netwon's method to find the roots
let f(x)=sinx-0.5x
df(x)/dx=cosx-0.5
x(n+1)=x(n)-f(x(n))/f'(x(n))
x(n+1)=x(n)-[sinx(n)-0.5x(n)]/[cosx(n)-0.5]
let x(0)=-2






interation
x(n)
x(n+1)

0
-2
-1.901

1
-1.901
-1.89551

2
-1.89551
-1.89549

3
-1.89549
-1.89549

4
-1.89549
-1.89549

5
-1.89549
-1.89549

6
-1.89549
-1.89549

7
-1.89549
-1.89549

8
-1.89549
-1.89549

9
-1.89549
-1.89549

10
-1.89549
-1.89549
let x(0)=1
then






interation
x(n)
x(n+1)

0
1
-7.47274

1
-7.47274
14.47852

2
14.47852
6.935115

3
6.935115
16.63568

4
16.63568
8.343938

5
8.343938
4.954633

6
4.954633
-8.30132

7
-8.30132
-4.81732

8
-4.81732
3.792574

9
3.792574
1.861061

10
1.861061
1.896214

11
1.896214
1.895495

12
1.895495
1.895494

13
1.895494
1.895494

14
1.895494
1.895494

15
1.895494
1.895494

16
1.895494
1.895494

17
1.895494
1.895494

18
1.895494
1.895494

19
1.895494
1.895494

20
1.895494
1.895494

21
1.895494
1.895494

22
1.895494
1.895494

23
1.895494
1.895494

24
1.895494
1.895494

25
1.895494
1.895494
so the roots of sinx=0.5x are -1.895494 0 and 1.895494
2006-11-12 7:57 am
sinx - 0.5x = 0
sinx = 0.5x

So, there are two Simultaneous Equations where y = sinx and y = 0.5x

Solve about these,
we have x = 0
參考: Myself
2006-11-12 7:46 am
sinx-0.5x=0

sin x = sin 30 x

x=30x

x=0


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