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I'm supposed to find a Taylor polynomal of the $n^{\text{th}}$ degree, where $x = a$, and estimate the error for the given interval. The problem I'm given is: $$f(x) = \sqrt{x}, a = 4, n = 2, 4 \leq x \leq 4.2$$

I've solved out the polynomial to be:

$$ P2(x) = 2 + \frac{x-4}{4} - \frac{(x-4)^2}{64} $$

I am completely lost after this point.

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You are doing well. The error term is $\frac {f^3(\xi )(x-4)^3}{3!}$. Since you don't know $\xi$, you can bound it by the maximum of the third derivative of $f$ in the interval of interest $4 \le x \le 4.2$ This will give you a bound on the error.

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