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By use of a computer and Mathematica the elementary function cos x is visualized as a limit of the graphs of Taylor polynomials. A slight modification of the Taylor polynomials for cos x leads to a computer visualization of the graphs of the Bessel function Jo(x) . By similar modifications the student can develop and study the student's own analogues of trigonometric functions and of Bessel function. A computer model of a vibrating elastic string, which can be animated, is produced by use of Fourier series and Mathematica. Similarly, a model of a vibrating elastic membrane is developed by use of Jo(x). The student can perform numerous computer experiments involving vibrating membranes with various initial conditions and involving study of the period of the motion. In addition, the student can study the motion of 'pseudo-membranes' by use of the student's own analogues of Bessel functions.
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