Write the Taylor series for the function $f(x)= x^2-3x+1$ using $x=2$ as the point of expansion , i.e. write a formula for $f(2+h)$
Even though I know what Taylor Expansion is I am really confused what the point of expansion is and how to find $f(2+h)$
Best Answer
You're probably accustomed to Taylor series about $x=0$. In general, the Taylor series about $x=a$ is $$f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots.$$
So, we compute the derivatives $f'(x)$, $f''(x)$, and so on (since this is a polynomial, it'll have a finite number of non-zero derivatives). We use them to find the values $f'(a)$, $f''(a)$, and so on, then substitute them into the above equation. In this case $a=2$.