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Polinomios de Taylor

$ n$-degree Taylor polynomial for the function $ f(x)$ around $ x=x_0$\[\begin{align*} T_n(x)&=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\cdots\\&+\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i+\cdots \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \end{align*}\]

Problema

Find $ 2 $-degree Taylor polynomial for function $ f(x)= x^{{{1}\over{2}} } $ in $ x_0=1$.

Function and value of the function

$ f(x)= x^{{{1}\over{2}} } $,      $ f(1)= 1 $

Derivatives, derivatives at $1$ and coefficients of Taylor polynomial

$ i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!}$
1$ {{{{1}\over{2}}}\over{\sqrt{x}}} $$ {{1 }\over{2}} $$ {{1}\over{2}} $
2$ -{{{{1}\over{4}}}\over{x^{{{3}\over{2}} }}} $$ -{{1}\over{4}} $$ -{{1}\over{8}} $

Polinomios de Taylor

$ T_{2}(x)= 1+{{1}\over{2}}{}\left(x-1 \right)-{{1}\over{8}}{}\left(x-1\right)^2 $