Question

How do I approximate `sqrt(128)` using a Taylor polynomial centered at 125?

Answer 1

To approximating `f(x)=sqrtx` by a Taylor polynomial centered at `125` , you'll need to know `sqrt125`.

The Taylor polynomial, of degree `n+1`, to approximate `f(x)` centered at `a` involves:

`f(a), f'(a), f''(a), f'''(a), . . . , f^((n+1))(a)`

and also involves `x-a`

Derivatives of `sqrtx` of every order will involve `sqrt(x)^k`, Thus, every term will require `sqrt125`

Re-read the question you were asked.

The exercise would make more sense if either:
(a) the question is to approximate `root(3)128`
(`f(x)=root(3)x` and its derivatives are not too hard to find when `x=125`

or

(b) `a` is not specified
(In which case, note that `sqrt128=sqrt(2^7)=2^(7/2)` so you can use `f(x)=x^(7/2)` centered at `1` to approximate `f(2)`)