Calculus II: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 01 Dec 2015 23:47:00 GMTTue, 01 Dec 2015 23:47:00 GMTNew applications in the Calculus II categoryhttp://www.mapleprimes.com/images/mapleapps.gifCalculus II: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=176
Measuring Water Flow of Rivers
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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
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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
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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 LopezTwo Bodies Revolving Around Their Center of Mass with ANIMATION
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<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 BaroudyClassroom Tips and Techniques: Fitting Circles in Space to 3-D Data
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<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 LopezClassroom Tips and Techniques: Visualizing Regions of Integration
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<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 - Simpson's Rule
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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=5175/thumb.gif" alt="Numeric Integration - Simpson's 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.5175Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftMean Value Theorem for Integrals
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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=5167/thumb.gif" alt="Mean Value Theorem for Integrals" 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.5167Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftSeparable Differential Equations
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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 ZMaplesoftMaplesoftWork: Spring of Unknown Length
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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=5163/appviewer.aspx.jpg" alt="Work: Spring of Unknown Length" 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.5163Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftWork-Winding Cable
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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=5162/appviewer.aspx.jpg" alt="Work-Winding Cable" 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.5162Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftNumeric Integration: Trapezoid Rule
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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 ZMaplesoftMaplesoftAverage Value
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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=5184/appviewer.aspx.jpg" alt="Average Value" 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.5184Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftHydrostatic Force: Trapezoidal Dam
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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=5165/appviewer.aspx.jpg" alt="Hydrostatic Force: Trapezoidal Dam" 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.5165Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftVolume of a Solid of Revolution Rotating about x=2
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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=5171/appviewer.aspx.jpg" alt="Volume of a Solid of Revolution Rotating about x=2" 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.5171Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftSeries of Constants
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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=5179/thumb.gif" alt="Series of Constants" 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.5179Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftArc Length
http://www.maplesoft.com/applications/view.aspx?SID=5185&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=5185/appviewer.aspx.jpg" alt="Arc Length" 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.5185Wed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftHyperbolic Functions -2
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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=5178/appviewer.aspx.jpg" alt="Hyperbolic Functions -2" 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.5178Wed, 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 ZMaplesoftMaplesoft