Approximating Functions with Polynomials

Complex functions can often be approximated over a region of their domain by a polynomial. This environment lets you explore this phenomenon for different types of function. Enter a function of your choice. Move the black dot on the x axis to the region of the domain where you would like the approximation to be good. Slider the slider the choose the order of the approximating polynomial. If you place the black dot on the origin - where is the polynomial approximation best for the exponential function? - why do the approximating polynomials for sin(x) contain no even powers of x?