Suppose, $f(x)=2x^3+7x^2+x-1$ is a polynomial of degree 3. If I were to expand the function about $x=2$ and approximate that up to 5 decimal place accuracy, how do I approach? (This came in my exam, no interval was given)
I have tried the following approach:
Expanding up to 2nd degree: $45+\left(x-2\right)\cdot53+\frac{\left(x-2\right)^{2}}{2!}\cdot38$
Expanding up to 3rd degree: $45+\left(x-2\right)\cdot53+\frac{\left(x-2\right)^{2}}{2!}\cdot38+\frac{\left(x-2\right)^{3}}{3!}\cdot12$
The last one is the exact same equation given in the question. But the second one does not ensure 5 decimal place accuracy. How do I find the region where the expansion is correct up to the given accuracy?