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 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)

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)

Presents problems to find the integrals of logarithmic and inverse trigonometric functions early in the calculus sequence by using the Fundamental Theorem of Calculus and the concept of area, and without the use of integration by parts. (Author/ASK)

Because one of the difficulties with the standard presentation of the Fundamental Theorem of Calculus (FTC) is that essentially all functions used to illustrate this theorem are taken from earlier material, many students never fully appreciate the essential role played by continuity in statement and proof of FTC. Introduces the sim x function that…

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)

Outlines an attempt at integrating web-based activities into a precalculus course at a large university in which discussion of the development of the activities is initially provided. Investigates the effects of the use these activities in four classrooms. Focuses on the use of the activities by two instructors, only one of whom received…

Offers an approach to the understanding and to the teaching of the fundamental theorem of calculus. Stresses teaching the relation between a function and its derivative and the functions themselves. (YP)

A simple way to introduce natural logarithms and e is presented. The standard approach is outlined, followed by the approach via differentiating the exponential functions that the student knows about. (MNS)

Presented is a simple method which may be used to determine points of intersection in graphs of functions if they do exist. Several examples are given with illustrations of the functions. (CW)