Calculus II: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=157
en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 21 Jul 2017 20:35:35 GMTFri, 21 Jul 2017 20:35:35 GMTNew applications in the Calculus II categoryhttp://www.mapleprimes.com/images/mapleapps.gifCalculus II: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=157
Momentum 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="/view.aspx?si=154273/cmimp.png" alt="Momentum with two variable force" 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.154273Tue, 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.
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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.
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Other chapters are in preparation and will be released in due course.<img src="/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" 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.154267Tue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieInterpretación 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="/view.aspx?si=154110/232a3a3435a381a76ee84170be3fcee2.gif" alt="Interpretación geométrica del proceso de solución de una ecuación trigonométrica" 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.154110Tue, 24 May 2016 04:00:00 ZRanferi GutierrezRanferi GutierrezCentroid 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="/view.aspx?si=154064/as.png" alt="Centroid with defined integral" 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.154064Sun, 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="/view.aspx?si=153595/Preview_figure.png" alt="Guia de estudio para integrales dobles" 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>153595Tue, 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="/view.aspx?si=153480/thumb.jpg" alt="Measuring Water Flow of Rivers" 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.153480Fri, 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="/view.aspx?si=137299/thumb.jpg" alt="Classroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes" 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.137299Wed, 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="/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" 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.136471Tue, 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="/view.aspx?si=130674/thumb.jpg" alt="Classroom Tips and Techniques: Sliders for Parameter-Dependent Curves" 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.130674Tue, 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="/view.aspx?si=101922/thumb.jpg" alt="Classroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic" 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.101922Tue, 22 Feb 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems from the Little Red Book of Maple Magic
https://www.maplesoft.com/applications/view.aspx?SID=100897&ref=Feed
Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.<img src="/view.aspx?si=100897/thumb.jpg" alt="Classroom Tips and Techniques: Gems from the Little Red Book of Maple Magic" align="left"/>Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.100897Fri, 14 Jan 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezTwo Bodies Revolving Around Their Center of Mass with ANIMATION
https://www.maplesoft.com/applications/view.aspx?SID=99587&ref=Feed
<p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p><img src="/view.aspx?si=99587/thumb.jpg" alt="Two Bodies Revolving Around Their Center of Mass with ANIMATION" align="left"/><p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p>99587Mon, 29 Nov 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyDetermine Integrals by Monte-Carlo method
https://www.maplesoft.com/applications/view.aspx?SID=96010&ref=Feed
<p>We build two procedures to determine approximately single variable and two variable integrals by Monte-Carlo method.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Determine Integrals by Monte-Carlo method" align="left"/><p>We build two procedures to determine approximately single variable and two variable integrals by Monte-Carlo method.</p>96010Sat, 14 Aug 2010 04:00:00 ZDuong Ngoc HaoDuong Ngoc HaoClassroom Tips and Techniques: Visualizing Regions of Integration
https://www.maplesoft.com/applications/view.aspx?SID=94845&ref=Feed
<p>Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path</p>
<p>Tools ≻ Tasks ≻ Browse: Calculus - Multivariate ≻ Integration ≻ Visualizing Regions of Integration</p>
<p>Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template is provided.</p><img src="/view.aspx?si=94845/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing Regions of Integration" align="left"/><p>Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path</p>
<p>Tools ≻ Tasks ≻ Browse: Calculus - Multivariate ≻ Integration ≻ Visualizing Regions of Integration</p>
<p>Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template is provided.</p>94845Tue, 06 Jul 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Fitting Circles in Space to 3-D Data
https://www.maplesoft.com/applications/view.aspx?SID=1644&ref=Feed
<p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p><img src="/view.aspx?si=1644/thumb3.jpg" alt="Classroom Tips and Techniques: Fitting Circles in Space to 3-D Data" align="left"/><p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p>1644Mon, 17 May 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezSéries de puissances et séries de Fourier
https://www.maplesoft.com/applications/view.aspx?SID=87622&ref=Feed
<p>Cette application maplets permet d'obtenir le développement<br />
en série de puissances (Taylor ou Maclaurin) d'une fonction<br />
indéfiniment dérivable au voisinage d'un point ainsi que le<br />
développement en série de Fourier d'une fonction périodique<br />
continue ayant éventuellement un nombre fini de points de<br />
discontinuité de première espèce sur [-p,p].</p><img src="/view.aspx?si=87622/0\tf.png" alt="Séries de puissances et séries de Fourier" align="left"/><p>Cette application maplets permet d'obtenir le développement<br />
en série de puissances (Taylor ou Maclaurin) d'une fonction<br />
indéfiniment dérivable au voisinage d'un point ainsi que le<br />
développement en série de Fourier d'une fonction périodique<br />
continue ayant éventuellement un nombre fini de points de<br />
discontinuité de première espèce sur [-p,p].</p>87622Mon, 10 May 2010 04:00:00 ZAndre LevesqueAndre LevesqueClassroom Tips and Techniques: Stepwise Solutions in Maple - Part 1
https://www.maplesoft.com/applications/view.aspx?SID=35165&ref=Feed
<p>In Maple, there are commands, Assistants, Tutors, and Task Templates that show stepwise calculations in algebra, calculus (single-variable, multivariable, vector), and linear algebra. In this article we discuss Maple's functionality for providing these stepwise solutions to mathematical problems in algebra and calculus (both of one and several variables).</p><img src="/view.aspx?si=35165/thumb2.jpg" alt="Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 1" align="left"/><p>In Maple, there are commands, Assistants, Tutors, and Task Templates that show stepwise calculations in algebra, calculus (single-variable, multivariable, vector), and linear algebra. In this article we discuss Maple's functionality for providing these stepwise solutions to mathematical problems in algebra and calculus (both of one and several variables).</p>35165Wed, 10 Feb 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Series Expansions
https://www.maplesoft.com/applications/view.aspx?SID=34685&ref=Feed
<p>Maple has the ability to provide various series expansions and their truncations, as well as complete formal series for a variety of elementary and special functions. In this month's article, we examine the relevant commands and interface devices that access these functionalities.</p><img src="/view.aspx?si=34685/thumb.jpg" alt="Classroom Tips and Techniques: Series Expansions" align="left"/><p>Maple has the ability to provide various series expansions and their truncations, as well as complete formal series for a variety of elementary and special functions. In this month's article, we examine the relevant commands and interface devices that access these functionalities.</p>34685Thu, 05 Nov 2009 05:00:00 ZDr. Robert LopezDr. Robert LopezVelociraptor Math
https://www.maplesoft.com/applications/view.aspx?SID=34103&ref=Feed
<p>This worksheet solves the popular velociraptor math problem, posed by the webcomic xkcd.com:</p>
<p>How long does it take a raptor to catch and eat you, assuming it starts from a position 40 meters away, accelerating at 4 meters/second-squared to its top speed of 25 meters/second?</p><img src="/view.aspx?si=34103/thumb.jpg" alt="Velociraptor Math" align="left"/><p>This worksheet solves the popular velociraptor math problem, posed by the webcomic xkcd.com:</p>
<p>How long does it take a raptor to catch and eat you, assuming it starts from a position 40 meters away, accelerating at 4 meters/second-squared to its top speed of 25 meters/second?</p>34103Mon, 26 Oct 2009 04:00:00 ZStephanie RozekStephanie RozekClassroom Tips and Techniques: Visualizing Regions of Integration
https://www.maplesoft.com/applications/view.aspx?SID=34062&ref=Feed
<p>In this month's article, the synergy between the visual and the analytic is demonstrated with a learning tool built with Maple's embedded components.</p><img src="/view.aspx?si=34062/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing Regions of Integration" align="left"/><p>In this month's article, the synergy between the visual and the analytic is demonstrated with a learning tool built with Maple's embedded components.</p>34062Wed, 21 Oct 2009 04:00:00 ZDr. Robert LopezDr. Robert Lopez