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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 $ 4 $-degree Taylor polynomial for function $ f(x)= \sqrt{25-x^2} $ in $ x_0=0$.

Function and value of the function

$ f(x)= \sqrt{25-x^2} $,      $ f(0)= 5 $

Derivatives, derivatives at $0$ and coefficients of Taylor polynomial

$ i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!}$
1$ -{{x}\over{\sqrt{25-x^2}}} $$ 0 $$ 0 $
2$ -{{25{}\sqrt{25-x^2}}\over{x^4-50{}x^2+625}} $$ -{{1}\over{5}} $$ - {{1}\over{10}} $
3$ {{75{}x{}\sqrt{25-x^2}}\over{\left(x-5\right)^3{} \left(x+5\right)^3}} $$ 0 $$ 0 $
4$ {{75{}\left(4{}x^2+25\right)}\over{ \left(x-5\right)^3{}\left(x+5\right)^3{}\sqrt{25-x^2}}} $$ -{{3 }\over{125}} $$ -{{1}\over{1000}} $

Polinomios de Taylor

$ T_{4}(x)= 5-{{1}\over{10}}{}x^2-{{1}\over{ 1000}}{}x^4 $