Central Difference Method Third Order help

[abcdefgh]

How do I go about deriving the coefficients (-1/24, -27/24, 27/24,1/24) for a third-order polynomial approximation of a using the central difference method.

So far I have

ah^3-bh^2-ch+d=f(-h)
ah^3+bh^2+ch+d=f(h)

f(0)=d

f(-3/2)=a(-3/2)^3-b(-3/2)^2-c(-3/2)+d
f(-1/2) = a(-1/2)^3-b(-1/2)^2-c(-1/2)+d
f(1/2) = a(1/2)^3+b(1/2)^2+c(1/2)+d
f(3/2)=a((3/2)^3+b(3/2)^2+c(3/2)+d

I know that I am supposed to solve for c, the linear term but I don't know where to go from there
With the h=(-3/2,-1/2,0,1/2,3/2)

I know that the b and d terms are supposed to cancel but I don't know how

 

[Ishuda]

How do I go about deriving the coefficients (-1/24, -27/24, 27/24,1/24) for a third-order polynomial approximation of a using the central difference method.

So far I have

ah^3-bh^2-ch+d=f(-h)
ah^3+bh^2+ch+d=f(h)

f(0)=d

f(-3/2)=a(-3/2)^3-b(-3/2)^2-c(-3/2)+d
f(-1/2) = a(-1/2)^3-b(-1/2)^2-c(-1/2)+d
f(1/2) = a(1/2)^3+b(1/2)^2+c(1/2)+d
f(3/2)=a((3/2)^3+b(3/2)^2+c(3/2)+d

I know that I am supposed to solve for c, the linear term but I don't know where to go from there
With the h=(-3/2,-1/2,0,1/2,3/2)

I know that the b and d terms are supposed to cancel but I don't know how

Do you know the Taylor Series expansion?

f(x+h) = f(x) + f'(x) h + \(\displaystyle \frac{1}{2!}\) f''(x) h² + \(\displaystyle \frac{1}{3!}\) f'''(x) h³ + ...
and
f(x-h) = f(x) - f'(x) h + \(\displaystyle \frac{1}{2!}\) f''(x) h² - \(\displaystyle \frac{1}{3!}\) f'''(x) h³ + ...

If so, you can subtract f(x-h) from f(x+h), rearrange, and get
f'(x) = \(\displaystyle \frac{f(x+h)-f(x-h)}{2h}\) - \(\displaystyle \frac{1}{3}\) f'''(x) h³ + ...
If you drop the order 3 and higher terms [those with h³ and higher] you get a third order central difference approximation to the derivative of f
f'(x) ~ \(\displaystyle \frac{f(x+h)-f(x-h)}{2h}\)

A similar kind of thing can be done for the second derivative to get a fourth order centered formula approximation. You might want to look at
http://mathfaculty.fullerton.edu/mathews/n2003/differentiation/numericaldiffproof.pdf