## Taylorův polynom

Taylorův polynom $n$-tého stupně pro funkci $f(x)$ se středem v bodě $x=x_0$\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*}

Najděte Taylorův polynom stupně $3$ pro funkci$f(x)= {{\ln \left(x+1\right) }\over{x^2}}$ se středem v bodě $x_0=2$.

#### Funkce a funkční hodnota

$f(x)= {{\ln \left(x+1\right) }\over{x^2}}$,      $f(2)= {{1}\over{4}}{}\ln \left(3\right)$

#### Derivace, derivace v bodě $2$ a koeficienty Taylorova polynomu

$i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!} 1 -{{\left(2{}x+ 2\right){}\ln \left(x+1\right)-x}\over{x^4+x^3}}$$ {{1}\over{12}}- {{1}\over{4}}{}\ln \left(3\right) $${{1}\over{12}}-{{1}\over{4}}{} \ln \left(3\right) 2 {{\left(6{}x^2+12{}x+6\right){}\ln \left(x+1 \right)-5{}x^2-4{}x}\over{x^6+2{}x^5+x^4}}$$ {{3}\over{8}}{}\ln \left(3\right)-{{7}\over{36}} $${{1}\over{2}}{}\left({{3}\over{8}}{} \ln \left(3\right)-{{7}\over{36}}\right) 3 -{{\left(24{}x^3+72{}x^2 +72{}x+24\right){}\ln \left(x+1\right)-26{}x^3-42{}x^2-18{}x}\over{ x^8+3{}x^7+3{}x^6+x^5}}$$ {{103}\over{216}}-{{3}\over{4}}{}\ln \left(3\right)$${{1}\over{6}}{}\left({{103}\over{216}}-{{3}\over{4 }}{}\ln \left(3\right)\right)$

#### Taylorův polynom

$T_{3}(x)= {{1}\over{4}}{}\ln \left(3\right)- {{1}\over{12}}{}\left(3{}\ln \left(3\right)-1\right){}\left(x-2 \right)+{{1}\over{144}}{}\left(27{}\ln \left(3\right)-14\right){} \left(x-2\right)^2-{{1}\over{1296}}{}\left(162{}\ln \left(3\right)- 103\right){}\left(x-2\right)^3$