Back Page: My Favorite Lesson: Bridging Algebra and Calculus

2014 ◽  
Vol 107 (8) ◽  
pp. 640
Author(s):  
Marla A. Sole
Keyword(s):  
Real World ◽  
Rate Of Change ◽  
Abstract Concept ◽  
Instantaneous Rate ◽  
Algebra Students

My favorite lesson introduces algebra students to a key concept from calculus: instantaneous rate of change. In this lesson, I help students develop an intuitive understanding of this abstract concept by framing questions within a real-world context.

scholarly journals SOLVING INVERSE PROBLEM OF CHEMICAL KINETICS WITH USE OF CUBIC SPLINES

2020 ◽  
Vol 63 (7) ◽  
pp. 61-66
Author(s):  
Nikolay I. Kol'tsov
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Chemical Kinetics ◽  
Good Accuracy ◽  
Cubic Splines ◽  
Rate Of Change ◽  
Reference Points ◽  
Additional Advantage ◽  
Instantaneous Rate ◽  
Relaxation Curves

A simple effective method for solving the inverse problem of chemical kinetics based on non-stationary experiments for multistage reactions occurring in an isothermal reactor of ideal mixing is described. The idea of the method is based on taking into account the distinctive features (informativeness) of different fragments of relaxation curves for chemical reactions with arbitrary (non-monotonic) kinetics and their as accurate approximation as possible. For this purpose, non-linear (cubic) splines are used to describe different informative fragments of relaxation curves, which allow to approximate and interpolate experimental data as accurately as possible. An additional advantage of cubic splines, from the point of view of the implementation of the described method, is their continuity at all given points up to and including second-order derivatives (smoothness). This allows us to calculate with good accuracy not only the concentration of reagents, but also the instantaneous rate of change at any time. The consequence of this is the possibility of a sufficiently accurate solution of the inverse problem based on the data of non-stationary experiments. The correctness of the mathematical model used and the stability of the method were tested using variations of the original data. An example of using the method for determining the intervals of physical values of the rate constants of the stages of a two-stage reaction is given. The influence of the method of selecting the reference points (structure) of the spline and measurement errors (noise) of experimental data on the error of determining the speed constants of the stages is estimated. The efficiency of application and good accuracy of the method for solving the inverse problem of chemical kinetics of multistage reactions occurring in non-gradient systems with taking into account of noise is shown.

scholarly journals A dynamical time operator in Dirac's relativistic quantum mechanics

2014 ◽  
Vol 29 (06) ◽  
pp. 1450036 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Phase Velocity ◽  
Relativistic Quantum ◽  
Rate Of Change ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Instantaneous Rate ◽  
Expectation Value ◽  
Dynamical Time

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

scholarly journals Global patterns of kelp forest change over the past half-century

2016 ◽  
Vol 113 (48) ◽  
pp. 13785-13790 ◽  
Author(s):  
Kira A. Krumhansl ◽  
Daniel K. Okamoto ◽  
Andrew Rassweiler ◽  
Mark Novak ◽  
John J. Bolton ◽  
...  
Keyword(s):  
Multiple Scales ◽  
Global Analysis ◽  
Rate Of Change ◽  
Kelp Forests ◽  
Arctic Seas ◽  
Instantaneous Rate ◽  
Half Century ◽  
The Past ◽  
Past Half Century ◽  
Past Half

Kelp forests (Order Laminariales) form key biogenic habitats in coastal regions of temperate and Arctic seas worldwide, providing ecosystem services valued in the range of billions of dollars annually. Although local evidence suggests that kelp forests are increasingly threatened by a variety of stressors, no comprehensive global analysis of change in kelp abundances currently exists. Here, we build and analyze a global database of kelp time series spanning the past half-century to assess regional and global trends in kelp abundances. We detected a high degree of geographic variation in trends, with regional variability in the direction and magnitude of change far exceeding a small global average decline (instantaneous rate of change = −0.018 y−1). Our analysis identified declines in 38% of ecoregions for which there are data (−0.015 to −0.18 y−1), increases in 27% of ecoregions (0.015 to 0.11 y−1), and no detectable change in 35% of ecoregions. These spatially variable trajectories reflected regional differences in the drivers of change, uncertainty in some regions owing to poor spatial and temporal data coverage, and the dynamic nature of kelp populations. We conclude that although global drivers could be affecting kelp forests at multiple scales, local stressors and regional variation in the effects of these drivers dominate kelp dynamics, in contrast to many other marine and terrestrial foundation species.

scholarly journals Large herbivores surf waves of green-up during spring

2016 ◽  
Vol 283 (1833) ◽  
pp. 20160456 ◽  
Author(s):  
Jerod A. Merkle ◽  
Kevin L. Monteith ◽  
Ellen O. Aikens ◽  
Matthew M. Hayes ◽  
Kent R. Hersey ◽  
...  
Keyword(s):  
Vegetation Index ◽  
Normalized Difference Vegetation Index ◽  
Leading Edge ◽  
Rate Of Change ◽  
High Quality ◽  
Large Herbivores ◽  
Instantaneous Rate ◽  
Movement Data ◽  
Selection For ◽  
Step Selection

The green wave hypothesis (GWH) states that migrating animals should track or ‘surf’ high-quality forage at the leading edge of spring green-up. To index such high-quality forage, recent work proposed the instantaneous rate of green-up (IRG), i.e. rate of change in the normalized difference vegetation index over time. Despite this important advancement, no study has tested the assumption that herbivores select habitat patches at peak IRG. We evaluated this assumption using step selection functions parametrized with movement data during the green-up period from two populations each of bighorn sheep, mule deer, elk, moose and bison, totalling 463 individuals monitored 1–3 years from 2004 to 2014. Accounting for variables that typically influence habitat selection for each species, we found seven of 10 populations selected patches exhibiting high IRG—supporting the GWH. Nonetheless, large herbivores selected for the leading edge, trailing edge and crest of the IRG wave, indicating that other mechanisms (e.g. ruminant physiology) or measurement error inherent with satellite data affect selection for IRG. Our evaluation indicates that IRG is a useful tool for linking herbivore movement with plant phenology, paving the way for significant advancements in understanding how animals track resource quality that varies both spatially and temporally.

scholarly journals A nomenclature for the standard linguistic description of the kinematics of networks

2015 ◽  
Author(s):  
Jaderick P Pabico ◽  
Eliezer A Albacea
Keyword(s):  
Real World ◽  
Rate Of Change ◽  
Collaboration Network ◽  
Time Range ◽  
Linguistic Description ◽  
Friendship Relations ◽  
Computer Scientists ◽  
Twitter Users ◽  
Kinematic Properties ◽  
Over Time

The rate of change \(\partial M/\partial t\) of some metric \(M\) measured as one of the kinematic properties of a network described by a graph \(G\) transitioning from \(G(V_{t}, E_{t})\) to \(G(V_{t+\partial t}, E_{t+\partial t})\) over time range \(\partial t\) has been described in the literature with linguistic descriptions that often provide ambiguity. For example, one rate of change \((\partial M/\partial t)_{1}\) has been described as ``dynamic'' and another \((\partial M/\partial t)_{2}\) as ``highly dynamic'' but \((\partial M/\partial t)_{1}>(\partial M/\partial t)_{2}\). We propose in this paper a nomenclature for the standard linguistic description of the kinematics of networks in the hope that description in the literature will be standardized and understood with the corresponding quantitative meaning. We termed a network as ``static'' when \(\partial M/\partial t=0\), as ``non-volatile'' when \(0<\partial M/\partial t\le 1\), and as ``volatile'' when \(\partial M/\partial t>1\). In the development of the linguistic nomenclature, we borrowed heavily from the standard used in signal theory to provide linguistic descriptions to various ranges for \(\partial M/\partial t>1\). We described the kinematics of example real-world networks where the proposed nomenclature was used: (1) The collaboration network of Filipino Computer Scientists; (2) The network created from friendship relations among Batangas and Laguna Facebook users; and (3) The network created from the followed-follower relations among the top ten globally influential Twitter users.

scholarly journals A nomenclature for the standard linguistic description of the kinematics of networks

2015 ◽  
Author(s):  
Jaderick P Pabico ◽  
Eliezer A Albacea
Keyword(s):  
Real World ◽  
Rate Of Change ◽  
Collaboration Network ◽  
Time Range ◽  
Linguistic Description ◽  
Friendship Relations ◽  
Computer Scientists ◽  
Twitter Users ◽  
Kinematic Properties ◽  
Over Time

The rate of change \(\partial M/\partial t\) of some metric \(M\) measured as one of the kinematic properties of a network described by a graph \(G\) transitioning from \(G(V_{t}, E_{t})\) to \(G(V_{t+\partial t}, E_{t+\partial t})\) over time range \(\partial t\) has been described in the literature with linguistic descriptions that often provide ambiguity. For example, one rate of change \((\partial M/\partial t)_{1}\) has been described as ``dynamic'' and another \((\partial M/\partial t)_{2}\) as ``highly dynamic'' but \((\partial M/\partial t)_{1}>(\partial M/\partial t)_{2}\). We propose in this paper a nomenclature for the standard linguistic description of the kinematics of networks in the hope that description in the literature will be standardized and understood with the corresponding quantitative meaning. We termed a network as ``static'' when \(\partial M/\partial t=0\), as ``non-volatile'' when \(0<\partial M/\partial t\le 1\), and as ``volatile'' when \(\partial M/\partial t>1\). In the development of the linguistic nomenclature, we borrowed heavily from the standard used in signal theory to provide linguistic descriptions to various ranges for \(\partial M/\partial t>1\). We described the kinematics of example real-world networks where the proposed nomenclature was used: (1) The collaboration network of Filipino Computer Scientists; (2) The network created from friendship relations among Batangas and Laguna Facebook users; and (3) The network created from the followed-follower relations among the top ten globally influential Twitter users.

Optimal Windows for the Time-Frequency Analysis of Arbitrary Swept Frequency Signals

1995 ◽  
Author(s):  
James A. Mooney ◽  
Andres Soom
Keyword(s):  
Frequency Analysis ◽  
Wavelet Transforms ◽  
Fourier Transforms ◽  
Rate Of Change ◽  
Effective Bandwidth ◽  
Instantaneous Rate ◽  
Time Frequency Analysis ◽  
Time Frequency ◽  
Analysis Window ◽  
Constant Bandwidth

Abstract In noise and vibration analysis, as well as in many other engineering applications, it may be necessary to extract or analyze signals with time-varying frequency components. Examples include start-up and shut-down of rotating machinery, transient structural vibrations, vehicle passing noise, and speech analysis. Both Short-Time Fourier Transforms (STFT), representing a set of non-causal filters of constant bandwidth, and Wavelet Transforms, representing a set of non-causal filters of constant Q or constant percent bandwidth, have been used for such Joint Time Frequency Analysis (JTFA). In the present work, an arbitrary swept frequency signal is approximated locally, in time, by a linearized frequency sweep. We show that an optimal time window can be identified which, at a given frequency, is inversely proportional to the square root of the instantaneous rate of change of frequency. We find that the constant bandwidth of the STFT and the constant-Q of the Wavelet transform represent extreme cases which are each optimal for certain types of signals. In between the two extremes there lies a continuous range of variation of the effective bandwidth with frequency. Many important types of signals require analysis window variation in this range between STFT and Wavelet analysis. The paper concludes with some simple rules for optimizing the variation of the analysis window with frequency for various types of signals.

Review: Investigating the Development of Multiplicative Reasoning

1995 ◽  
Vol 26 (3) ◽  
pp. 282-288
Author(s):  
Martin A. Simon
Keyword(s):  
Mathematics Education ◽  
Education Reform ◽  
Real World ◽  
Rate Of Change ◽  
Nonlinear Functions ◽  
Multiplicative Reasoning ◽  
Ratio And Proportion ◽  
Geometric Figures ◽  
Linear And Nonlinear ◽  
Development Of Students

If the current mathematics education reform is to have significant impact on the mathematical development of students, the mathematical understandings encompassed by the multiplicative conceptual field are paramount. These mathematical understandings range from initial conceptions of multiplication and division through concepts of ratio and proportion. fractions, and linear and nonlinear functions. Not only is an understanding of multiplicative relationships important in the modeling of real world situations, but such relationships form the basis of the mathematics of probability, of similarity of geometric figures, and of rate of change.

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