[math]T_n(x)[/math] is the [math]n^{th}[/math] Taylor Polynomial of [math]f(x)[/math]. [math]T_n(x_0+h)[/math] is used to approximate [math]f(x_0+h)[/math]. Drag the point [math]x_0[/math] along the [math]x[/math]-axis to change the center of expansion. Slide [math]n[/math] to change the degree. Enter a new function in the text box to change the function. Drag the point [math]x_0+h[/math] along the [math]x[/math]-axis to change the point of approximation. The error in the approximation is shown below as [math]f(x_0+h)-T_n(x_0+h)[/math]. Its absolute value will remain between the theoretical bounds. The value [math]\xi[/math] guaranteed by Taylor's Theorem is calculated in the CAS view and shown in the Graphics view on the [math]x[/math]-axis between [math]x_0[/math] and [math]x_0+h[/math].
[b]Taylor's Theorem[/b] There exists [math]\xi[/math] between [math]x[/math] and [math]x_0[/math] such that [math]f(x)-T_n(x)=R_n(x)[/math] where [math]R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}[/math]