[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

For each of the values of [math]n[/math] indicated, compute the Taylor polynomial [math]T_n[/math] which approximates the function [math]f[/math] near the number [math]a[/math].

• [a [math]f(x) = \frac1{x+1}[/math], [math]a=0[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].
• [math]f(x) = \frac1{1+x^2}[/math], [math]a=0[/math], [math]n=0[/math], [math]2[/math], and [math]4[/math].
• [math]f(x) = \frac1{1+x^2}[/math], [math]a=1[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].
• [math]f(x) = \sqrt{x+1}[/math], [math]a=3[/math], [math]n=1[/math], [math]2[/math], and [math]3[/math].
• [math]f(x) = \sin x[/math], [math]a=\frac{\pi}4[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].