Robert J. Lopez

Vector formulation for interferogram surface fitting

Interferometry is an optical testing technique that quantifies the optical path difference (OPD) between a reference wave front and a test wave front based on the interference of light. Fringes are formed when the OPD is an integral multiple of the illuminating wavelength. The resultant two-dimensional pattern is called an interferogram. The function of any interferogram analysis program is to extract this OPD and to produce a representation of the test wave front (or surface). This is accomplished through a three-step process of sampling, ordering, and fitting. We develop a generalized linear-algebra vectornotation model of the interferogram sampling and fitting process.

L’Hôpital’s Rule

Even though Maple has a very capable limit command, L’Hopital’s Rule is still a conceptual necessity in the calculus. Here is an investigation that combines the insight of local linearity with the operational role of the derivative in understanding what L’Hopital’s Rule actually does.

Partial Fraction Decomposition

The partial fraction decomposition typically arises as a technique of integration. Some students see the technique again when inverting Laplace transforms. Nevertheless, the manipulations are algebraic, not analytic. Hence, we present partial fraction decomposition as an algebraic operation, divorced from its use in the applications.

A Separable Differential Equation

The following problem comes from the section Separable Differential Equations in a traditional calculus text. It requires no more than a separation of variables and an “integration of both sides” to produce the solution of the differential equation in an implicit form. However, we show that additional insight is possible if we ask questions like “How does the solution behave?” and “What is the nature of the implicit function describing the solution?” The role of Maple in answering these questions is essential.

Integration by Parts

The companion of Integration by Trigonometric Substitution is Integration by Parts. Let’s explore one way Maple can be used to learn about integration by parts. This approach is predicated on the belief that real integrals are never actually done “by parts” because in real life such integrals are found in tables of integrals.

Integration by Trigonometric Substitution

One of the most contentious issues in the introduction of computer algebra into the calculus classroom is that of “methods of integration.” For some, the unit on methods of integration is a sacred rite of passage from the ignorance of the laity into the exhalted state of high-priesthood. For others, the concept of a change of variables in an integral is sufficient, as long as there are good tables of integrals or powerful computer algebra systems available.

Newton’s Law of Cooling

One of the applications of integration arising in Calculus II is the separable ordinary differential equation. Separating variables and integrating both sides of the differential equation is an excuse for practicing integration. However, the assigned exercises usually involve skills far different than just the skills of integration.

Deriving Simpson’s Rule

Maple’s student package contains a built-in command for Simpson’s Rule for approximate numeric integration and an exploration of the companion built-in Trapezoidal Rule appears in Unit 14. Here, we explore a derivation of Simpson’s Rule.

Teaching the Definite Integral

Maple contains features both useful and effective for seeing the definite integral as area under a curve. We begin by posing the question: How can we find the area bounded by the graph of some function f(x) and the x-axis? We are then led to an approximate answer based on rectangles drawn under the graph of f(x).

An Implicit Function

Let’s see how we might use technology to explore the concept of the implicit function. Given the following quadratic expression q and the equation q = 0, an initial attempt at investigating its meaning and behavior might be a call to Maple’s plot command.