Calculus II: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=157
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 20 Apr 2014 10:48:04 GMTSun, 20 Apr 2014 10:48:04 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
Measuring Water Flow of Rivers
http://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
http://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
http://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
http://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
http://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
http://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
http://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
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<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
http://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
http://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
http://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
http://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
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<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
http://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
http://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 LopezStreamlines in 2-Dimensional Vector Fields
http://www.maplesoft.com/applications/view.aspx?SID=6665&ref=Feed
This worksheet gives two examples of Maple's capabilities for calculating and displaying the streamlines in a 2-dimensional vector field.<img src="/view.aspx?si=6665/thumb.gif" alt="Streamlines in 2-Dimensional Vector Fields" align="left"/>This worksheet gives two examples of Maple's capabilities for calculating and displaying the streamlines in a 2-dimensional vector field.6665Tue, 16 Sep 2008 00:00:00 ZMaplesoftMaplesoftNumeric Integration: Trapezoid Rule
http://www.maplesoft.com/applications/view.aspx?SID=5174&ref=Feed
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
The steps in the document can be repeated to solve similar problems.<img src="/view.aspx?si=5174/appviewer.aspx.jpg" alt="Numeric Integration: Trapezoid Rule" align="left"/>This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
The steps in the document can be repeated to solve similar problems.5174Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftWork: Pumping a Cylindrical Tank
http://www.maplesoft.com/applications/view.aspx?SID=5164&ref=Feed
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.<img src="/view.aspx?si=5164/appviewer.aspx.jpg" alt="Work: Pumping a Cylindrical Tank" align="left"/>This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.5164Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftPower Series
http://www.maplesoft.com/applications/view.aspx?SID=5180&ref=Feed
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
The steps in the document can be repeated to solve similar problems.<img src="/view.aspx?si=5180/appviewer.aspx.jpg" alt="Power Series" align="left"/>This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
The steps in the document can be repeated to solve similar problems.5180Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftSeparable Differential Equations
http://www.maplesoft.com/applications/view.aspx?SID=5176&ref=Feed
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.<img src="/view.aspx?si=5176/thumb.gif" alt="Separable Differential Equations" align="left"/>This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus™ methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.5176Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoft