Calculus II: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=176
en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 04 Mar 2021 18:16:54 GMTThu, 04 Mar 2021 18:16:54 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=176
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 BaroudySolving 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 CastilloMomentum 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 ZJohn OgilvieJohn 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 CastilloMeasuring 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 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="https://www.maplesoft.com/view.aspx?si=99587/thumb.jpg" alt="Two Bodies Revolving Around Their Center of Mass with ANIMATION" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=99587&ref=FeedMon, 29 Nov 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom 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="https://www.maplesoft.com/view.aspx?si=1644/thumb3.jpg" alt="Classroom Tips and Techniques: Fitting Circles in Space to 3-D Data" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=1644&ref=FeedMon, 17 May 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom 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="https://www.maplesoft.com/view.aspx?si=34062/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing Regions of Integration" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=34062&ref=FeedWed, 21 Oct 2009 04:00:00 ZDr. Robert LopezDr. Robert LopezStreamlines in 2-Dimensional Vector Fields
https://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="https://www.maplesoft.com/view.aspx?si=6665/thumb.gif" alt="Streamlines in 2-Dimensional Vector Fields" style="max-width: 25%;" align="left"/>This worksheet gives two examples of Maple's capabilities for calculating and displaying the streamlines in a 2-dimensional vector field.https://www.maplesoft.com/applications/view.aspx?SID=6665&ref=FeedTue, 16 Sep 2008 00:00:00 ZMaplesoftMaplesoftWork-Winding Cable
https://www.maplesoft.com/applications/view.aspx?SID=5162&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="https://www.maplesoft.com/view.aspx?si=5162/appviewer.aspx.jpg" alt="Work-Winding Cable" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5162&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftNumeric Integration - Simpson's Rule
https://www.maplesoft.com/applications/view.aspx?SID=5175&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="https://www.maplesoft.com/view.aspx?si=5175/thumb.gif" alt="Numeric Integration - Simpson's Rule" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5175&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftMean Value Theorem for Integrals
https://www.maplesoft.com/applications/view.aspx?SID=5167&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="https://www.maplesoft.com/view.aspx?si=5167/thumb.gif" alt="Mean Value Theorem for Integrals" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5167&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftSeparable Differential Equations
https://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="https://www.maplesoft.com/view.aspx?si=5176/thumb.gif" alt="Separable Differential Equations" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5176&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftAverage Value
https://www.maplesoft.com/applications/view.aspx?SID=5184&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="https://www.maplesoft.com/view.aspx?si=5184/appviewer.aspx.jpg" alt="Average Value" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5184&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoftNumeric Integration: Trapezoid Rule
https://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="https://www.maplesoft.com/view.aspx?si=5174/appviewer.aspx.jpg" alt="Numeric Integration: Trapezoid Rule" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=5174&ref=FeedWed, 05 Sep 2007 00:00:00 ZMaplesoftMaplesoft