A method is presented which can be used to obtain analytical solutions for boundary value problems (BVPs) using the matrix exponential and Maple. Systems of second order, linear differential equations are expressed as two or more first order equations in matrix form, and their solutions are obtained using the matrix exponential, matrix…

Strategies are proposed that promote use of an Integrated Applied Mathematics (IAM) approach to enhance teaching of advanced problem-solving and analysis skills. Three scenarios of 1-dimensional transport processes are presented that support using Error Function analyses when considering short time/small penetration depths in finite geometries.…

The natural exponential and logarithm are typically introduced to undergraduate engineering students in a calculus course using the notion of limits. We here present an approach to introduce the natural exponential/logarithm through a novel interpretation of derivatives. This approach does not rely on limits, allowing an early and intuitive…

As part of a departmental reform project, a large University converted a "Precalculus College Mathematics" lecture-based course for 97 students into an active learning flipped classroom for first-year engineering students. The curriculum was designed specifically to provide applications of pre-calculus in various engineering subjects.…

In this note, some variants of Cauchy's mean value theorem are proved. The main tools to prove these results are some elementary auxiliary functions.

Mechanical engineering, mechanical engineering technology, and related educational programs are not addressing in a sufficient way the principles associated with applying analytical investigations in solving actual engineering problems. Because of this, graduates do not have the adequate skills required to use the methods of applied dynamics in…

We present the Active Learning tools introduced in basic mathematics courses in the Preparatory Curriculum for Engineering schools at Université Claude Bernard. Additions were introduced in the academic year 2018/2019 in Foundations of Mathematics 2, in the first year of bachelor study program. Its content are Linear Algebra (matrices, vector…

Simulation is an effective tool for analyzing probability models as well as for facilitating understanding of concepts in probability and statistics. Unfortunately, implementing a simulation from scratch often requires users to think about programming issues that are not relevant to the simulation itself. We have developed a Python package called…

We offer many specific detailed examples, several of which are new, that instructors can use (in lecture or as student projects) to revitalize the role of complex variables throughout the curriculum. We conclude with three primary recommendations: revise the syllabus of Calculus II to allow early introductions of complex numbers and linear…

This paper describes a simple experiment that lets first-year physics and engineering students discover an important physical property of a Slinky. The restoring force for the fundamental oscillation frequency is provided only by those coils between the support and the Slinky center of mass.

Our traditional model of calculus instruction at a large public research university emphasized factual knowledge and procedural fluency with few realistic applications, leaving a chasm between classroom mathematics and disciplinary practice. Amid an ongoing effort by our department to improve undergraduate learning outcomes, we replaced…

It is no easy feat to engage young people with abstract material as well as push them to greater depths of understanding. Add in the extra pressures of curriculum expectations and standards and the problem is exacerbated. Projects designed around standards and having multiple entry points clearly offer students the best opportunity to engage with…

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as…

Calculus courses often present a large number of difficulties to undergraduate students of scientific studies, especially in engineering degrees. These difficulties are sometimes related to teaching and assessment strategies. In this paper, a teaching innovation experience is presented within the framework of the Universitat Oberta de Catalunya.…

The interdisciplinary approach that science, technology, engineering and mathematics (STEM) projects inspire in both teachers and students "brings to light a larger picture that promotes real-world scientific applications, which has in turn been shown to increase undergraduate persistence in STEM." The high school students have been…