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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

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