We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

Introduced are the basic ideas of interpolation and approximation theory through a combination of theory and exercises written for extramural education at the university level. Topics treated are spline methods, Lagrange interpolation, trigonometric approximation, Fourier series, and polynomial approximation. (MDH)

This note considers the four classes of orthogonal polynomials--Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same…

Presents ways complex physics concepts can be demonstrated by music. Demonstrations and appropriate comments are provided for: (a) diffraction and interference, (b) standing waves in cavities and solids, (c) wave mechanics, (d) Fourier series, (e) propagation of waves in periodic media, and (f) group velocity. (DS)

Presents an exercise suitable for beginning calculus students that may give insight into series representations and allow students to see some elementary application of these representations. The Fourier series is used to approximate by taking sums of trigonometric functions of the form sin(ns) and cos(nx) for n is greater than or = zero. (PVD)

Reports on efforts to develop interactive general purpose programs (for very inexpensive microcomputers) to be used in engineering systems courses. Programs discussed (Microsoft Basic) include: (1) polynomial root finder; (2) partial fraction expansion; (3) transfer function frequency response; (4) transfer function simulator; and (5) Fourier…

Infrared (IR) spectra of biologically derived materials display distinct absorption bands correlating to individual macromolecules: protein, polysaccharide, lipid, and nucleic acids. A series of experiments aimed at teaching qualitative bioorganic spectroscopy using attenuated total reflectance (ATR) Fourier transform infrared (IR) with biological…

Demonstrates the principal concept and the application of MAPLE in mathematical education in various examples. Discusses lengthy and abstract topics like the convergence of Fourier series to a given function, performs the visualization of the wave equation in the case of a vibrating string, and computes the oscillations of an idealized skyscraper…

Non-linear second-order differential equations whose solutions are the elliptic functions "sn"("t, k"), "cn"("t, k") and "dn"("t, k") are investigated. Using "Mathematica", high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are…

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when…

This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on…

Students in introductory physics courses encounter harmonics when they study standing waves on strings and in pipes. The Fourier spectrum, which plots amplitude against harmonic number, can describe all periodic tones imaginable. However, most sounds are aperiodic and therefore have additional spectral components. A real-world application that…

We present a theoretical approach to the problem of the transition from Calculus to Analysis within the undergraduate mathematics curriculum. First, we formulate this problem using the anthropological theory of the didactic, in particular the notion of praxeology, along with a possible solution related to Klein's "Plan B": here,…

This paper presents a new method for estimating a given latent trait variable by the least-squares approach. The beta weights are obtained recursively with the help of Fourier series and expressed as functions of item parameters of response curves. The values of the latent trait variable estimated by this method and by maximum likelihood method…

A research project for senior undergraduates of chemistry has been developed to introduce syntheses of a series of monodispersed semiconductor PbS quantum dots (QDs) and their characterization methodologies. In this paper, we report the preparation of monodispersed semiconductor PbS QDs with sizes smaller than the exciton Bohr radius using a…