[left]This applet enables the user to explore Taylor polynomial approximations of three functions: f(x)=[math]e^x[/math], g(x)=cos(x), and h(x)=sin(x).[br][br]Switch between the three functions by toggling on/off the "Show [function]" checkbox. After toggling on a checkbox, another checkbox labeled "Show approximation of [function]" will appear. When you toggle on this checkbox, the graph of a Taylor polynomial approximation of the function will appear in the window above in blue and the algebraic representation of this Taylor polynomial approximation will appear on the right. Use the slider n to change the number of terms in the Taylor polynomial approximation. As you explore, consider the following questions. [/left][list][*]Under what conditions is a Taylor polynomial a good approximation of a function (e.g. the number of terms, for what interval, etc.)?[/*][/list][list][*]What do you think happens as the number of terms (n) in the Taylor polynomial approximation approaches infinity?[/*][/list]After exploring all three functions, compare and contrast the Taylor polynomial approximations of each function. Consider the following questions. [list][*]What do the Taylor polynomial approximations of cos(x) and sin(x) have in common with the Taylor polynomial approximation of [math]e^x[/math]?[/*][/list][list][*]How do the Taylor polynomial approximations of cos(x) and sin(x) differ from the Taylor polynomial approximation of [math]e^x[/math]?[/*][/list]Finally, consider how a person might provide an argument for Euler's formula, [math]e^{ix}=cos\left(x\right)+isin\left(x\right)[/math], using the Taylor series of these three functions.