Question

Sa se determine polinoamele Taylor T1(x), T3(x) si T5(x) pentru functia f(x) = ln(2 +x) ın punctul x0= 1.

Answer 1

Răspuns:

Avem $f(1)=\ln 3$

$f'(x)=\dfrac{1}{x+2}, \ f'(1)=\dfrac{1}{3}$

$f''(x)=-\dfrac{1}{(x+2)^2}, \ f''(1)=-\dfrac{1}{9}$

$f'''(x)=\dfrac{2}{(x+2)^3}, \ f'''(1)=\dfrac{2}{27}$

$f^{(4)}(x)=-\dfrac{6}{(x+2)^4}, \ f^{(4)}(1)=-\dfrac{2}{27}$

$f^{(5)}(x)=\dfrac{24}{(x+2)^5}, \ f^{(5)}(1)=\dfrac{8}{81}$

Atunci

$T_1(x)=f(1)+\dfrac{f'(1)}{1!}(x-1)=\ln 3+\dfrac{1}{3}(x-1)$

$T_3(x)=f(1)+\dfrac{f'(1)}{1!}(x-1)+\dfrac{f''(1)}{2!}(x-1)^2+\dfrac{f'''(1)}{3!}(x-1)^3=\\=\ln 3+\dfrac{1}{3}(x-1)-\dfrac{1}{18}(x-1)^2+\dfrac{1}{81}(x-1)^3$

Analog se calculează $T_5(x)$.

Explicație pas cu pas: