Calculus II: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=157
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 27 Feb 2020 19:36:56 GMTThu, 27 Feb 2020 19:36:56 GMTNew applications in the Calculus II categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgCalculus II: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=157
Bee-Cell Structure.mw
https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=Feed
Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.<img src="https://www.maplesoft.com/view.aspx?si=154603/3b4238cc9e7aeb803a42872bde31a350.gif" alt="Bee-Cell Structure.mw" style="max-width: 25%;" align="left"/>Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=FeedThu, 13 Feb 2020 17:34:36 ZAhmed BaroudyAhmed BaroudyAtwood Machine
https://www.maplesoft.com/applications/view.aspx?SID=154598&ref=Feed
The following is a detailed study of the motion of an unconventional Atwood Machine where one mass is constrained to move along a fixed vertical axis.
The differences with the regular Atwood Machine are :
1- the tension T on the string on either side of the pulley though it is the same, however it is not constant in the present case because of the obliquity of the 2d part of the string.
2- the unique and constant acceleration (a) in the simple machine is replaced in here with two different and variable accelerations whose ratio is however constant.
3- In the simple machine the constant acceleration makes plotting and animation of the system a straightforward procedure according to
s = (1/2)*at^2.However in the modified Atwood machine that we present in here the accelerations being variable there is no way to get the displacement as a direct function of time. This seems to make plotting & animation an impossible task. However we were able to devise a trick to overcome this difficulty.<img src="https://www.maplesoft.com/view.aspx?si=154598/Modified_Atwood_Machine.jpg" alt="Atwood Machine" style="max-width: 25%;" align="left"/>The following is a detailed study of the motion of an unconventional Atwood Machine where one mass is constrained to move along a fixed vertical axis.
The differences with the regular Atwood Machine are :
1- the tension T on the string on either side of the pulley though it is the same, however it is not constant in the present case because of the obliquity of the 2d part of the string.
2- the unique and constant acceleration (a) in the simple machine is replaced in here with two different and variable accelerations whose ratio is however constant.
3- In the simple machine the constant acceleration makes plotting and animation of the system a straightforward procedure according to
s = (1/2)*at^2.However in the modified Atwood machine that we present in here the accelerations being variable there is no way to get the displacement as a direct function of time. This seems to make plotting & animation an impossible task. However we were able to devise a trick to overcome this difficulty.https://www.maplesoft.com/applications/view.aspx?SID=154598&ref=FeedSat, 25 Jan 2020 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyInterpretación geométrica del proceso de solución de una ecuación trigonométrica
https://www.maplesoft.com/applications/view.aspx?SID=154110&ref=Feed
Esta aplicación tiene como objetivo ayudar al estudiante a comprender el significado geométrico de resolver la ecuación trigonométrica sen(theta) = c en un intervalo de longitud 2Pi.
La barra deslizante de la aplicación permite variar el valor de c, mientras que los gráficos ayudan al estudiante a visualizar y comprender el proceso de búsqueda de soluciones de la ecuación trigonométrica de interés.<img src="https://www.maplesoft.com/view.aspx?si=154110/app.png" alt="Interpretación geométrica del proceso de solución de una ecuación trigonométrica" style="max-width: 25%;" align="left"/>Esta aplicación tiene como objetivo ayudar al estudiante a comprender el significado geométrico de resolver la ecuación trigonométrica sen(theta) = c en un intervalo de longitud 2Pi.
La barra deslizante de la aplicación permite variar el valor de c, mientras que los gráficos ayudan al estudiante a visualizar y comprender el proceso de búsqueda de soluciones de la ecuación trigonométrica de interés.https://www.maplesoft.com/applications/view.aspx?SID=154110&ref=FeedWed, 06 Jun 2018 04:00:00 ZRanferi GutierrezRanferi Gutierrezabe - Absolute Expression
https://www.maplesoft.com/applications/view.aspx?SID=154462&ref=Feed
The procedure abe, given an algebraic expression and expression x=a .. b, returns a new algebraic expression with the original expression x=a .. b.
If integrated the above will return the absolute area of the original input.
Using the algebraic expressions above you can create equations for the positive and the negative areas above which, when integrated, gives their values.<img src="https://www.maplesoft.com/view.aspx?si=154462/2018-01-31_173205.jpg" alt="abe - Absolute Expression" style="max-width: 25%;" align="left"/>The procedure abe, given an algebraic expression and expression x=a .. b, returns a new algebraic expression with the original expression x=a .. b.
If integrated the above will return the absolute area of the original input.
Using the algebraic expressions above you can create equations for the positive and the negative areas above which, when integrated, gives their values.https://www.maplesoft.com/applications/view.aspx?SID=154462&ref=FeedThu, 31 May 2018 04:00:00 ZJohn AllenJohn AllenSistemas de coordenadas polares: Graficas y areas
https://www.maplesoft.com/applications/view.aspx?SID=154460&ref=Feed
Esta hoja permite al estudiante comprender la forma en que se construye, en un sistema de
coordenadas polares, el grafico de una funcion r=r(theta) a partir de la grafica de la
misma en un sistema de coordenadas rectangulares.
<BR><BR>
Adicionalmente se muestran algunos ejemplos relacionados con el calculo de areas en
coordenadas polares.<img src="https://www.maplesoft.com/view.aspx?si=154460/polar.PNG" alt="Sistemas de coordenadas polares: Graficas y areas" style="max-width: 25%;" align="left"/>Esta hoja permite al estudiante comprender la forma en que se construye, en un sistema de
coordenadas polares, el grafico de una funcion r=r(theta) a partir de la grafica de la
misma en un sistema de coordenadas rectangulares.
<BR><BR>
Adicionalmente se muestran algunos ejemplos relacionados con el calculo de areas en
coordenadas polares.https://www.maplesoft.com/applications/view.aspx?SID=154460&ref=FeedFri, 25 May 2018 04:00:00 ZRanferi GutierrezRanferi GutierrezCompilacion de problemas sobre calculo de volumen de solidos de seccion transversal conocida
https://www.maplesoft.com/applications/view.aspx?SID=154458&ref=Feed
Una de las aplicaciones de la integral definida que mas dificultades presenta a los estudiantes es la del calculo
de volumenes de solidos de secciones transversales conocidas.
<BR><BR>
Entre las dificultades que encuentran los estudiantes se pueden mencionar el dibujar correctamente una seccion transversal
tipica del solido, asi como el poder imaginar la forma final que este tendra.
<BR><BR>
Esta hoja permite, a traves de las capacidades de Maple y el uso de componentes, ayudar al estudiante principiante a salvar esas dificultades y ayudarlo en su entrenamiento para resolver problemas sobre el calculo de volumenes de secciones transversales conocidas.
<BR><BR>
Cada uno de los problemas que se presentan en la hoja viene acompañado del solido en su totalidad asi como de una barra deslizante que permite al estudiante visualizar la seccion transversal en diferentes posiciones dentro del mismo. Se incluye la respuesta a cada uno de los ejercicios propuestos.<img src="https://www.maplesoft.com/view.aspx?si=154458/section.PNG" alt="Compilacion de problemas sobre calculo de volumen de solidos de seccion transversal conocida" style="max-width: 25%;" align="left"/>Una de las aplicaciones de la integral definida que mas dificultades presenta a los estudiantes es la del calculo
de volumenes de solidos de secciones transversales conocidas.
<BR><BR>
Entre las dificultades que encuentran los estudiantes se pueden mencionar el dibujar correctamente una seccion transversal
tipica del solido, asi como el poder imaginar la forma final que este tendra.
<BR><BR>
Esta hoja permite, a traves de las capacidades de Maple y el uso de componentes, ayudar al estudiante principiante a salvar esas dificultades y ayudarlo en su entrenamiento para resolver problemas sobre el calculo de volumenes de secciones transversales conocidas.
<BR><BR>
Cada uno de los problemas que se presentan en la hoja viene acompañado del solido en su totalidad asi como de una barra deslizante que permite al estudiante visualizar la seccion transversal en diferentes posiciones dentro del mismo. Se incluye la respuesta a cada uno de los ejercicios propuestos.https://www.maplesoft.com/applications/view.aspx?SID=154458&ref=FeedThu, 24 May 2018 04:00:00 ZRanferi GutierrezRanferi GutierrezSolving 2nd Order Differential Equations
https://www.maplesoft.com/applications/view.aspx?SID=154426&ref=Feed
This worksheet illustrates how to use Maple to solve examples of homogeneous and non-homogeneous second order differential equations, including several different methods for visualizing solutions.<img src="https://www.maplesoft.com/view.aspx?si=154426/2nd_order_des.PNG" alt="Solving 2nd Order Differential Equations" style="max-width: 25%;" align="left"/>This worksheet illustrates how to use Maple to solve examples of homogeneous and non-homogeneous second order differential equations, including several different methods for visualizing solutions.https://www.maplesoft.com/applications/view.aspx?SID=154426&ref=FeedMon, 26 Mar 2018 04:00:00 ZEmilee CarsonEmilee CarsonSolving ODEs using Maple: An Introduction
https://www.maplesoft.com/applications/view.aspx?SID=154422&ref=Feed
In Maple it is easy to solve a differential equation. In this worksheet, we show the basic syntax. With this you should be able to use the same basic commands to solve many second-order DEs.<img src="https://www.maplesoft.com/view.aspx?si=154422/ode.PNG" alt="Solving ODEs using Maple: An Introduction" style="max-width: 25%;" align="left"/>In Maple it is easy to solve a differential equation. In this worksheet, we show the basic syntax. With this you should be able to use the same basic commands to solve many second-order DEs.https://www.maplesoft.com/applications/view.aspx?SID=154422&ref=FeedFri, 23 Mar 2018 04:00:00 ZDr. Francis PoulinDr. Francis PoulinImplementation of Maple apps for the creation of mathematical exercises in engineering
https://www.maplesoft.com/applications/view.aspx?SID=154388&ref=Feed
In this research work has allowed to show the implementation of applications developed in the Maple software for the creation of mathematical exercises given the different levels of education whether basic or higher.
For the majority of teachers in this area, it seems very difficult to implement apps in Maple; that is why we show the creation of exercises easily and permanently. The purpose is to get teachers from our institutions to use applications ready to be evaluated in the classroom. The results of these apps (applications with components made in Maple) are supported on mobile devices such as tablets and / or laptops and taken to the cloud to be executed online from any computer. The generation of patterns is a very important alternative leaving aside random numbers, which would allow us to lose results
onscreen. With this; Our teachers in schools or universities would evaluate their students in parallel on the blackboard without losing the results of any student and thus achieve the competencies proposed in the learning sessions. In Spanish.<img src="https://www.maplesoft.com/view.aspx?si=154388/genexr.png" alt="Implementation of Maple apps for the creation of mathematical exercises in engineering" style="max-width: 25%;" align="left"/>In this research work has allowed to show the implementation of applications developed in the Maple software for the creation of mathematical exercises given the different levels of education whether basic or higher.
For the majority of teachers in this area, it seems very difficult to implement apps in Maple; that is why we show the creation of exercises easily and permanently. The purpose is to get teachers from our institutions to use applications ready to be evaluated in the classroom. The results of these apps (applications with components made in Maple) are supported on mobile devices such as tablets and / or laptops and taken to the cloud to be executed online from any computer. The generation of patterns is a very important alternative leaving aside random numbers, which would allow us to lose results
onscreen. With this; Our teachers in schools or universities would evaluate their students in parallel on the blackboard without losing the results of any student and thus achieve the competencies proposed in the learning sessions. In Spanish.https://www.maplesoft.com/applications/view.aspx?SID=154388&ref=FeedFri, 26 Jan 2018 05:00:00 ZProf. Lenin Araujo CastilloProf. Lenin Araujo CastilloTaylor and Maclaurin Series
https://www.maplesoft.com/applications/view.aspx?SID=154375&ref=Feed
Tutorial for Calculus students. Features evaluation of power series, formal series expansions, discussion of series versus taylor commands, and direct construction of taylor polynomials.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Taylor and Maclaurin Series" style="max-width: 25%;" align="left"/>Tutorial for Calculus students. Features evaluation of power series, formal series expansions, discussion of series versus taylor commands, and direct construction of taylor polynomials.https://www.maplesoft.com/applications/view.aspx?SID=154375&ref=FeedTue, 12 Dec 2017 05:00:00 ZJuergen GerlachJuergen GerlachParametric Curves and Polar Coordinates
https://www.maplesoft.com/applications/view.aspx?SID=154376&ref=Feed
Tutorial for Calculus students.
Discusses parametric curves in maple, construction and display of tangent lines, calculation of arc length and areas under parametric curves.
The second illustrates the construction of polar graphs, either directly, or with the aid of the plots package, it includes finding slopes and the display of tangent lines for polar curves, and concludes with arc length and area calculations.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Parametric Curves and Polar Coordinates" style="max-width: 25%;" align="left"/>Tutorial for Calculus students.
Discusses parametric curves in maple, construction and display of tangent lines, calculation of arc length and areas under parametric curves.
The second illustrates the construction of polar graphs, either directly, or with the aid of the plots package, it includes finding slopes and the display of tangent lines for polar curves, and concludes with arc length and area calculations.https://www.maplesoft.com/applications/view.aspx?SID=154376&ref=FeedTue, 12 Dec 2017 05:00:00 ZJuergen GerlachJuergen GerlachMomentum with two variable force
https://www.maplesoft.com/applications/view.aspx?SID=154273&ref=Feed
This app shows the calculation of the final velocity of a body after it made contact with a variable force taking as reference the initial velocity, mass and the graph of the variation of F as a function of time. Made with native maple syntax (use of promt) and embedded components.
In Spanish.<img src="https://www.maplesoft.com/view.aspx?si=154273/cmimp.png" alt="Momentum with two variable force" style="max-width: 25%;" align="left"/>This app shows the calculation of the final velocity of a body after it made contact with a variable force taking as reference the initial velocity, mass and the graph of the variation of F as a function of time. Made with native maple syntax (use of promt) and embedded components.
In Spanish.https://www.maplesoft.com/applications/view.aspx?SID=154273&ref=FeedTue, 04 Jul 2017 04:00:00 ZProf. Lenin Araujo CastilloProf. Lenin Araujo CastilloMathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
<BR><BR>
Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
<BR><BR>
Other chapters are in preparation and will be released in due course.
<BR><BR>
Last updated on March 19, 2019<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
<BR><BR>
Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
<BR><BR>
Other chapters are in preparation and will be released in due course.
<BR><BR>
Last updated on March 19, 2019https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieCentroid with defined integral
https://www.maplesoft.com/applications/view.aspx?SID=154064&ref=Feed
With this application and using the rules of calculation we can show that procedures embedded in Maple components can also be used for teaching purposes in engineering. <br/><br/> In Spanish.<img src="https://www.maplesoft.com/view.aspx?si=154064/as.png" alt="Centroid with defined integral" style="max-width: 25%;" align="left"/>With this application and using the rules of calculation we can show that procedures embedded in Maple components can also be used for teaching purposes in engineering. <br/><br/> In Spanish.https://www.maplesoft.com/applications/view.aspx?SID=154064&ref=FeedSun, 20 Mar 2016 04:00:00 ZProf. Lenin Araujo CastilloProf. Lenin Araujo CastilloGuia de estudio para integrales dobles
https://www.maplesoft.com/applications/view.aspx?SID=153595&ref=Feed
<p>Esta guía de estudio tiene como objetivo aprovechar las capacidades de Maple para generar gráficas interactivas y lograr con ellas que el estudiante comprenda el problema geométrico que da origen a la integral doble, la interpretación geométrica de una integral doble cuando el integrando es positivo, y la interpretación geométrica del cálculo de integrales iteradas en una integral doble.</p><img src="https://www.maplesoft.com/view.aspx?si=153595/Preview_figure.png" alt="Guia de estudio para integrales dobles" style="max-width: 25%;" align="left"/><p>Esta guía de estudio tiene como objetivo aprovechar las capacidades de Maple para generar gráficas interactivas y lograr con ellas que el estudiante comprenda el problema geométrico que da origen a la integral doble, la interpretación geométrica de una integral doble cuando el integrando es positivo, y la interpretación geométrica del cálculo de integrales iteradas en una integral doble.</p>https://www.maplesoft.com/applications/view.aspx?SID=153595&ref=FeedTue, 03 Jun 2014 04:00:00 ZDr. Ranferi GutierrezDr. Ranferi GutierrezMeasuring Water Flow of Rivers
https://www.maplesoft.com/applications/view.aspx?SID=153480&ref=Feed
In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.<img src="https://www.maplesoft.com/view.aspx?si=153480/thumb.jpg" alt="Measuring Water Flow of Rivers" style="max-width: 25%;" align="left"/>In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.https://www.maplesoft.com/applications/view.aspx?SID=153480&ref=FeedFri, 13 Dec 2013 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes
https://www.maplesoft.com/applications/view.aspx?SID=137299&ref=Feed
A May 13, 2012, post to MaplePrimes asked some interesting questions about the tractrix defined parametrically by <em>x(s)</em> = sech<em>(s), y(s)</em> = <em>s</em> - tanh(s), s ≥ 0. I answered these questions on May 14 in a worksheet that forms the basis for this month's article.</p>
<p>It behooves me to write this article because the solution given in the May 14 MaplePrimes reply wasn't completely correct, the error stemming from a confounding of the variables <em>x, y, </em>and <em>s</em>. Mea culpa.<img src="https://www.maplesoft.com/view.aspx?si=137299/thumb.jpg" alt="Classroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes" style="max-width: 25%;" align="left"/>A May 13, 2012, post to MaplePrimes asked some interesting questions about the tractrix defined parametrically by <em>x(s)</em> = sech<em>(s), y(s)</em> = <em>s</em> - tanh(s), s ≥ 0. I answered these questions on May 14 in a worksheet that forms the basis for this month's article.</p>
<p>It behooves me to write this article because the solution given in the May 14 MaplePrimes reply wasn't completely correct, the error stemming from a confounding of the variables <em>x, y, </em>and <em>s</em>. Mea culpa.https://www.maplesoft.com/applications/view.aspx?SID=137299&ref=FeedWed, 12 Sep 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
https://www.maplesoft.com/applications/view.aspx?SID=136471&ref=Feed
In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.<img src="https://www.maplesoft.com/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" style="max-width: 25%;" align="left"/>In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.https://www.maplesoft.com/applications/view.aspx?SID=136471&ref=FeedTue, 14 Aug 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Sliders for Parameter-Dependent Curves
https://www.maplesoft.com/applications/view.aspx?SID=130674&ref=Feed
Methods for building slider-controlled graphs are explored, and used to show the variations in the limaçon. Then, the conchoid of a cubic is explored with the same set of tools.<img src="https://www.maplesoft.com/view.aspx?si=130674/thumb.jpg" alt="Classroom Tips and Techniques: Sliders for Parameter-Dependent Curves" style="max-width: 25%;" align="left"/>Methods for building slider-controlled graphs are explored, and used to show the variations in the limaçon. Then, the conchoid of a cubic is explored with the same set of tools.https://www.maplesoft.com/applications/view.aspx?SID=130674&ref=FeedTue, 14 Feb 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic
https://www.maplesoft.com/applications/view.aspx?SID=101922&ref=Feed
Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.<img src="https://www.maplesoft.com/view.aspx?si=101922/thumb.jpg" alt="Classroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic" style="max-width: 25%;" align="left"/>Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.https://www.maplesoft.com/applications/view.aspx?SID=101922&ref=FeedTue, 22 Feb 2011 05:00:00 ZDr. Robert LopezDr. Robert Lopez