## idku one year ago I am not going to approximate sqrt(9.3). Well, I will attempt.

1. idku

Tailor polynomial, of a function f(x) at x=a, of nth degree. $$\displaystyle\large\color{black}{\sum_{n=0}^{\infty}\frac{f^{n}(a)}{n!}(x-a)^n}$$ starting to work my f of a and nth order derivatives evaluated at x=a. (my a is 9) $$\displaystyle\large\color{black}{f(x)=\sqrt{x}~~~~~~~~\color{blue}{f(9)=3}}$$ $$\displaystyle\large\color{black}{f'(x)=\frac{1}{2\sqrt{x}}~~~~~~~~\color{blue}{f'(9)=1/6}}$$ $$\displaystyle\large\color{black}{f'(x)=-\frac{1}{4x\sqrt{x}}~~~~~~~~\color{blue}{f'(9)=1/216}}$$ so my first 3 terms I get $$\displaystyle\large\color{black}{\sum_{n=0}^{2}\frac{f^{n}(a)}{n!}(x-a)^n~~{\Huge|}_{{\LARGE a=9}}~~~=3+\frac{1}{6}(x-9)+\frac{1}{216}(x-9)^2}$$ $$\displaystyle\large\color{black}{\sqrt{9.3}\approx3+\frac{1}{6}(9.3-9)+\frac{1}{216}(9.3-9)^2}$$

2. idku

and then whatever that last line is going to give me that is an approximation of sqrt(9.3) using a Taylor polynomial of degree 3, at x=9.

3. idku

right ?

4. idku

the third row when I started to do my derivatives should be f''(x) and f''(9)