f(0)=2
f(1)=3
f(3)=4
f(7)=5
find f(x).
Please write it clearly
Thank ~
P.S. i don't know the answer.
[數學問題] polynomial
[數學問題] polynomial
由 E.T 於 星期六 五月 10, 2003 10:20 pm
Given f(x) is a 4th degree polynomial and
f(0)=2
f(1)=3
f(3)=4
f(7)=5
find f(x).
Please write it clearly Thank ~
P.S. i don't know the answer.
f(0)=2
f(1)=3
f(3)=4
f(7)=5
find f(x).
Please write it clearly
Thank ~
P.S. i don't know the answer.
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E.T - 文章: 1486
- 註冊時間: 2003-03-29
- 來自: 香港
Re: [數學問題] polynomial
由 --- 於 星期六 五月 10, 2003 10:58 pm
f(x) is a 4th degree polynomial and
f(0)=2
f(1)=3
f(3)=4
f(7)=5
f(x)=-2*(x-1)(x-3)(x-7)/3/7+3*x(x-3)(x-7)/2/6-4*x(x-1)(x-7)/3/2/4+5*x(x-1)(x-3)/4/6/7+ k *x(x-1)(x-3)(x-7)
k為任意數, k <>0
f(0)=2
f(1)=3
f(3)=4
f(7)=5
f(x)=-2*(x-1)(x-3)(x-7)/3/7+3*x(x-3)(x-7)/2/6-4*x(x-1)(x-7)/3/2/4+5*x(x-1)(x-3)/4/6/7+ k *x(x-1)(x-3)(x-7)
k為任意數, k <>0
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--- - 訪客
由 ---- 於 星期六 五月 10, 2003 11:02 pm
By interpolation formula,
f(x)
=0+3(x-0)(x-3)(x-7)(x-a)/(1-0)(1-3)(1-7)(1-a) + 4(x-0)(x-1)(x-7)(x-a)/(3-0)(3-1)(3-7)(3-a)+5(x-0)(x-1)(x-3)(x-a)/(7-0)(7-1)(7-3)(7-a) + b(x-0)(x-1)(x-3)(x-7)/(a-0)(a-1)(a-3)(a-7)
for f(a)=b
where a isn't 0,1,3,7
As not enough information is given, there are infinity polynomials satisfying.
f(x)
=0+3(x-0)(x-3)(x-7)(x-a)/(1-0)(1-3)(1-7)(1-a) + 4(x-0)(x-1)(x-7)(x-a)/(3-0)(3-1)(3-7)(3-a)+5(x-0)(x-1)(x-3)(x-a)/(7-0)(7-1)(7-3)(7-a) + b(x-0)(x-1)(x-3)(x-7)/(a-0)(a-1)(a-3)(a-7)
for f(a)=b
where a isn't 0,1,3,7
As not enough information is given, there are infinity polynomials satisfying.
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---- - 訪客
由 Raceleader 於 星期六 五月 10, 2003 11:15 pm
如果f(x)=ax4+bx3+cx2+dx+e
那麼e=2
a=(1/3528)(205-266d), b=(1/441)(329d-274), c=(1/3528)(5515-5894d)
那麼e=2
a=(1/3528)(205-266d), b=(1/441)(329d-274), c=(1/3528)(5515-5894d)
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Raceleader - 訪客
由 ---- 於 星期六 五月 10, 2003 11:25 pm
Meowth 寫到:if we know f'(0)=10, please decide f(x)
Stupid methodx2:
Let f(x)=ax^4+bx^3+cx^2+dx+e
f(0)=2
e=2
f'(0)=10
d=10
When x=1
a+b+c+12=3
a+b+c=-3
When x=3
81a+27b+9c+32=4
81a+27b+9c=-28
When x=7
2401a+343b+49c+72=5
2401a+343b+49c=-67
a=
- 代碼:
|-3 1 1|
|-28 27 9|
|-67 343 49|
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|1 1 1|
|81 27 9|
|2401 343 49|
如此類推
我太笨，不懂其他方法...[/code]
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