[math]T_n(x)[/math] is the [math]n^{th}[/math] Taylor Polynomial of [math]f(x)[/math].[br][br]Activity:[br][br]1. Place the point [math]x_0[/math] at the origin[br]2.Slide [math]n[/math] slowly to change the degree.[br]3.Observe the rule and the graph of [math]T_n\left(x\right)[/math] when sliding [math]n[/math] [br]4.List two things you find here[br][br][br]Enter a new function in the text box to change [math]f\left(x\right)[/math] and start all over again. [br]

[b]Taylor's Formula[/b]:[br][br]The Taylor series of  [math]f\left(x\right)[/math] that is [url=https://en.wikipedia.org/wiki/Infinitely_differentiable_function]infinitely differentiable[/url] at  [math]a[/math] is the [url=https://en.wikipedia.org/wiki/Power_series]power series[/url][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(a\right)}{3!}\left(x-a\right)^3+...[/math]