The possibility of approximating a function with a linear combination of exponential functions of the form e[superscript x], e[superscript 2x], ... is considered as a parallel development to the notion of Taylor polynomials which approximate a function with a linear combination of power function terms. The sinusoidal functions sin "x" and cos "x" are used as targets to assess how well the various approximations fit a given function. Some of the particularly nice properties of Taylor polynomials are shown not to apply to exponential approximations, a good lesson for students who can thereby gain a deeper appreciation of the power of Taylor approximations. (Contains 11 figures.)