摘要:

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.

ISSN:0020-739X    

DOI:10.1080/0020739890200511

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

The problem of generation of surface waves in a liquid with an inertial surface due to an initial axially symmetric surface disturbance is discussed. The form of the inertial surface is obtained asymptotically for large values of time and distance.

ISSN:0020-739X    

DOI:10.1080/0020739890200512

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Following the common procedure of deriving the Euler method and the extended Euler method through the use of Taylor expansion, four terms of the expansion are considered and some cases are calculated numerically. It is shown that one can, as expected, obtain a rapid improvement of the results, and moreover this is obtained for a small number of sub‐intervals of the interval under consideration.

ISSN:0020-739X    

DOI:10.1080/0020739890200513

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

A first derivative test is formulated for determining the nature of extrema of a function f(x) of n real variables, subject to the constraints gi(x) = 0, i = 1,2.....r, where r<n. The test, which does not require the differentiability of f at the critical point, x = a, nor the existence of second order derivatives of gi, is compared with the second derivative test for such extrema. Examples are given which demonstrate advantages of the first derivative test, and it is shown that if the second derivative test classifies a strict local minimum or maximum for the constrained problem, then, in principle, so does the first derivative test.

ISSN:0020-739X    

DOI:10.1080/0020739890200514

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

The numerical evaluation of five Cauchy integrals frequently encountered in thin‐airfoil‐wing theory are presented based on the work of Stark. Of these, the quadrature rules for four of the integrals appear in a very simple form which make it particularly easy for coding on digital computers and pocket calculators. Finally, it is indicated how these quadrature rules may be used to solve an integral equation of the type

ISSN:0020-739X    

DOI:10.1080/0020739890200515

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

If f is a real‐valued function defined on an interval I, we find that the set {x0ϵI|f(x0‐) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) is countable. We are then able to conclude that the only set of discontinuities of f which is not necessarily countable is the set {x0ϵI|f(x0‐) does not exist (but is not infinite) and f(x0+)does not exist (but is not infinite)}.

ISSN:0020-739X    

DOI:10.1080/0020739890200516

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Energy conservation for a subway system can be achieved through the implementation of regenerative braking. This paper gives an energy regeneration model to mathematically relate the energy consumption under regeneration to the velocity profile of a subway train between a pair of stations and shows the optimality of a very simple linear velocity function.

ISSN:0020-739X    

DOI:10.1080/0020739890200517

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor