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Hongwei Lou – International Journal of Mathematical Education in Science and Technology, 2025
In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.
Descriptors: Calculus, Mathematics, Mathematical Logic, Validity
Mehmet Pakdemirli – International Journal of Mathematical Education in Science and Technology, 2025
The hanging rope problem is considered. The rope is subject to rotation with the rotation axis being parallel to the rope. Using a continuum model and the basic principles of dynamics, the differential equation governing the motion is derived. The dynamic equilibrium case without vibrational motion is assumed in deriving the equation. The equation…
Descriptors: Mathematical Models, Calculus, Motion, College Mathematics
Morales, Juan C.; Arango, Carlos A. – International Journal of Mathematical Education in Science and Technology, 2023
An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The…
Descriptors: Physics, Calculus, Mathematical Applications, Methods
Thomas J. Pfaff – PRIMUS, 2024
The logistic differential equation is ubiquitous in calculus and differential equations textbooks. If the model is developed from first principles in these courses, it is usually done in an abstract mathematical way, rather than in one based in ecology. In this short note, we look at examples of how the model is derived in mathematical texts and…
Descriptors: Calculus, Mathematics Instruction, Textbooks, Ecology
Tracy Weyand – International Journal of Mathematical Education in Science and Technology, 2024
Spring-mass systems are presented as an application of second-order, constant coefficient differential equations in many differential equations textbooks. The phenomenon of resonance can then be analysed after nonhomogeneous differential equations are introduced. With some simplifying assumptions, the movement of the roof of a one-story building…
Descriptors: Theory Practice Relationship, Calculus, Mathematics Instruction, Seismology
Wha-Suck Lee – International Journal of Mathematical Education in Science and Technology, 2024
We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier…
Descriptors: Numbers, Algebra, Equations (Mathematics), Mathematical Concepts
Kristin M. Frank; Alexei Kolesnikov; Xiaoyin Wang – Journal of Postsecondary Student Success, 2025
We introduce a new metric--the classification power--to examine the effectiveness of postsecondary mathematics placement policies. This metric addresses the methodological challenges of contextualizing the effectiveness of a single placement policy and comparing the effectiveness of multiple placement policies across different student populations.…
Descriptors: Postsecondary Education, Mathematics Instruction, Student Placement, College Students
Adrianne L. Jenner; Pamela M. Burrage – International Journal of Mathematical Education in Science and Technology, 2024
Mathematics provides us with tools to capture and explain phenomena in everyday biology, even at the nanoscale. The most regularly applied technique to biology is differential equations. In this article, we seek to present how differential equation models of biological phenomena, particularly the flow through ion channels, can be used to motivate…
Descriptors: Cytology, Mathematical Models, Prediction, Equations (Mathematics)
Mark McCartney – International Journal of Mathematical Education in Science and Technology, 2024
Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of…
Descriptors: Maps, Mathematics Instruction, Mathematics Activities, Matrices
Yves Nievergelt – International Journal of Mathematical Education in Science and Technology, 2024
On 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near…
Descriptors: Air Transportation, Mathematical Models, Prevention, Calculus
Keith Brandt – PRIMUS, 2024
This paper describes a project assigned in a multivariable calculus course. The project showcases many fundamental concepts studied in a typical course, including the distance formula, equations of lines and planes, intersection of planes, Lagrange multipliers, integrals in both Cartesian and polar coordinates, parametric equations, and arc length.
Descriptors: Mathematics Instruction, Calculus, Equations (Mathematics), Design
Gabriel Gianni Cantanelli; Barbara A. Shipman – PRIMUS, 2024
Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to…
Descriptors: Mathematics Instruction, College Mathematics, Undergraduate Study, Visual Aids
Ribeiro, Ademir Alves; Barbosa, José Renato Ramos – International Journal of Mathematical Education in Science and Technology, 2022
This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient…
Descriptors: Calculus, Mathematics Instruction, Undergraduate Students, Teaching Methods
Shiv Krishna Madi Reddy; Meng Guo; Long Cai; Ralph E. White – Chemical Engineering Education, 2024
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…
Descriptors: Chemical Engineering, Engineering Education, Computer Software, Mathematics Instruction
Amenda N. Chow; Peter D. Harrington; Fok-Shuen Leung – Teaching Mathematics and Its Applications, 2024
Physical experiments in classrooms have many benefits for student learning, including increased student interest, participation and knowledge retention. While experiments are common in engineering and physics classes, they are seldom used in first-year calculus, where the focus is on solving problems analytically and, occasionally, numerically. In…
Descriptors: Mathematics Instruction, Calculus, Computer Software, Programming