Define the [math]n[/math]th [b]Taylor polynomial [math]T_n[/math] of [math]f[/math] centered at [math]x=a[/math][/b] in the following way:[br][br][math]T_n\left(x\right)=f\left(a\right)+\frac{f'\left(a\right)}{1!}\left(x-a\right)+\frac{f''\left(a\right)}{2!}\left(x-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(x-a\right)^n[/math].[br][br]
1. Let [math]f\left(x\right)[/math] be a polynomial of degree [math]n[/math]. What is the degree of [math]T_m\left(x\right)[/math] if [math]0\le m\le n[/math]?
2. Let [math]f\left(x\right)[/math] be a polynomial of degree 2. What do you expect the graphs of [math]T_0[/math], [math]T_1[/math], and [math]T_2[/math] to look like?
3. If [math]f\left(x\right)[/math] is a polynomial of degree 3, what behavior will the graphs of [math]T_0[/math], [math]T_1[/math], and [math]T_2[/math] have near the point [math]\left(a,f\left(a\right)\right)[/math]?