The Taylor Polynomial of a function is a polynomial that approximates the function at a point [math]c[/math]. The nth Taylor polynomial (or approximation) is given by [br][math]P_n\left(x\right)=f\left(c\right)+f'\left(c\right)\left(x-c\right)+\frac{f''\left(c\right)}{2!}\left(x-c\right)^2+...+\frac{f^{\left(n\right)}\left(c\right)}{n!}\left(x-c\right)^n[/math][br]This is very useful for approximating values of transcendental functions such as [math]e^x,sin(x),cos(x)[/math], etc...[br][br]The graph below shows the first four Taylor polynomial approximations of the function [math]f(x)=Asin(x)[/math] centered at zero, where [math]A[/math] is a constant. Why is each approximation more accurate than the previous one? How does the value of [math]A[/math] affect the graphs of the approximations?