The behavior of certain functions in advanced calculus is discussed, with the mathematics explained. (MNS)

The behavior of some functions near the point of origin is discussed. Each function oscillates, and as x approaches 0, the oscillations become increasingly more rapid; their behavior near the origin improves with increasing values of n. Examples for a calculus class to consider are given. (MNS)

Clarified is the assertion that the so-called complementary function is indeed the general solution of the homogeneous equation associated with a linear nth-order differential equation. Methods to obtain the particular integral, once the complementary function is determined, are illustrated for both cases of constant and of variable coefficients.…

A problem that most high school calculus students can explore is presented. It can help students understand such mathematical topics as functional notation, composition of functions, the solving of systems of equations, and the derivative. A computer program is included. (MNS)

Uses parametric equations and a graphing calculator to investigate the connections among the algebraic, numerical, and graphical representations of functions. (MKR)

Presents an activity to introduce the concepts of average rate change and instantaneous rate of change of a function and to explore the relationship between the value of the exponential function and its instantaneous rate of change. (ASK)

A definition of convexity with six conditions is discussed and illustrated. (MNS)

Two traditional presentations introducing the calculus of exponential functions are first presented. Then the suggested direct presentation using calculators is described. (MNS)

Illustrates how Lagrange's approach applies to the differential calculus of polynomial functions when approximations are obtained. Discusses how to obtain polynomial approximations in other cases. (YP)

Illustrates exploration of composition of functions, translations, and inverse functions on a graphing calculator. Includes reproducible student worksheets. (MKR)

Utilizing composition of elementary functions as prototype of dynamical system, notions of periodic points and their orbits in relation to concept of shift map are used to illustrate concept of continuity. A special case of Sarkovskii's theorem, dealing with period-3 point, is presented with proof relying solely upon Intermediate Value Theorem and…

Described is an example of a piecewise defined function developed naturally as a consequence of the solution to the given problem statement, thereby allowing calculus students the uncommon opportunity to generate such an otherwise, seemingly contrived function. (JJK)

Discusses a precalculus project in which students create a model United Nations to present and discuss the long-term prognosis for individual countries given data on population growth and food production. Students compare exponential and linear functions to determine whether starvation will occur and prepare oral and written presentations of their…

Three activities with Knuth functions are discussed and illustrated, with sample computer programs listed. (MNS)

Presents some examples from geometry: area of a circle; centroid of a sector; Buffon's needle problem; and expression for pi. Describes several roles of the trigonometric function in mathematics and applications, including Fourier analysis, spectral theory, approximation theory, and numerical analysis. (YP)