Estimating error using Taylor Polynomial I have searched and read quite a bit on this subject but I can't get this last bit straight. Reading the other answers did not help me unfortunately for me.
Anyway the problem: Suppose I have the function $g(x) = (8+3x)^{\frac{1}{3}}$ and I want to use Taylor expansion around $x = 0$ to find the third degree polynomial $P_3(x) = 2 + \frac{x}{4} - \frac{x^{2}}{32} + \frac{5x^{3}}{768}$. I shall then use this to find an approximation for $11^{\frac{1}{3}}$..
I found $g(x) = 11^{\frac{1}{3}}$ when $x = 1$. Then I'm supposed to find the remainder/error $E_3(x)$ and use this to find a small interval that will certainly contain the correct value $11^{\frac{1}{3}}$.
I want to use this formula $E_n(x) = \dfrac{g^{(n+1)}(c)(x-a)^{n+1}}{(n+1)!}$ 
to find $E_3(x)$ .. So I went ahead and found $g^{(4)}(x) = -\dfrac{80}{(8+3x)^{\frac{11}{3}}}$
Which means $E_3(x) = g^{(4)}(c)\dfrac{x^{4}}{4!}$
Now my problem is choosing $c$ and that's what I wanted to ask about. I don't know exactly what I'm doing here. Is it true that I must choose $c \in[0, 1]$ because I centered the series around $x = 0$, and $1$ because $g(1) = 11^{\frac{1}{3}}$ ? Also when choosing $c$, is it true that I want the one that returns the biggest value so the error won't be too small?
In that case I guess I must pick $c = 1$ ?
I hope someone can give an answer that is easy to understand, because im really not confident in this subject and dont't have the deep understanding(yet :)), also I'm quite new to calculus.

You shouldn't guess $c$, but you can estimate the error, i.e. you take the $c\in[x_0,x]$ ($f(x)=f(x_0)+...$) that give a worse case for $g^{(n)}$, i.e. biggest value,  and show that your error is less then something.
