U. of Wisconsin-Milwaukee Dept of Mathematics: New Applications
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en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 18 May 2021 13:25:08 GMTTue, 18 May 2021 13:25:08 GMTNew applications published by U. of Wisconsin-Milwaukee Dept of Mathematicshttps://www.maplesoft.com/images/Application_center_hp.jpgU. of Wisconsin-Milwaukee Dept of Mathematics: New Applications
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Definition of the derivative
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In this worksheet, you will see how to solve two problems: the mathematical problem of finding the tangent line to the graph of a function, and the physical problem of defining the velocity of a moving object. Although these problems might seem unrelated at first, it will turn out that they both lead to the same mathematical concept: the derivative of a function. <img src="https://www.maplesoft.com/view.aspx?si=4031//applications/images/app_image_blank_lg.jpg" alt="Definition of the derivative" style="max-width: 25%;" align="left"/>In this worksheet, you will see how to solve two problems: the mathematical problem of finding the tangent line to the graph of a function, and the physical problem of defining the velocity of a moving object. Although these problems might seem unrelated at first, it will turn out that they both lead to the same mathematical concept: the derivative of a function. https://www.maplesoft.com/applications/view.aspx?SID=4031&ref=FeedFri, 03 Aug 2001 09:31:05 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsParametric arc length
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In this worksheet, we will use the process of integration to compute the lengths of plane parametric curves. The same approach will find the lengths of 3-dimensional curves, but we will not consider that extension<img src="https://www.maplesoft.com/view.aspx?si=3990//applications/images/app_image_blank_lg.jpg" alt="Parametric arc length" style="max-width: 25%;" align="left"/>In this worksheet, we will use the process of integration to compute the lengths of plane parametric curves. The same approach will find the lengths of 3-dimensional curves, but we will not consider that extensionhttps://www.maplesoft.com/applications/view.aspx?SID=3990&ref=FeedWed, 01 Aug 2001 13:38:51 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsParametric curves
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Some simple curves in a plane may be described algebraically by setting up a suitable co-ordinate system in the plane and describing the curve as the graph of a function. We have seen already how useful such a description is; for example, we used it to obtain a formula for the arclength of such a curve. The description fails, however, for even so simple a curve as a circle: there is no choice of co-ordinate system for which a circle is the graph of a function<img src="https://www.maplesoft.com/view.aspx?si=3989//applications/images/app_image_blank_lg.jpg" alt="Parametric curves" style="max-width: 25%;" align="left"/>Some simple curves in a plane may be described algebraically by setting up a suitable co-ordinate system in the plane and describing the curve as the graph of a function. We have seen already how useful such a description is; for example, we used it to obtain a formula for the arclength of such a curve. The description fails, however, for even so simple a curve as a circle: there is no choice of co-ordinate system for which a circle is the graph of a functionhttps://www.maplesoft.com/applications/view.aspx?SID=3989&ref=FeedWed, 01 Aug 2001 13:37:48 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsConic sections
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We start with the geometric definition of conic sections. Later in this lesson, we'll plot conic sections in the plane using their analytic representations. <img src="https://www.maplesoft.com/view.aspx?si=3988//applications/images/app_image_blank_lg.jpg" alt="Conic sections " style="max-width: 25%;" align="left"/>We start with the geometric definition of conic sections. Later in this lesson, we'll plot conic sections in the plane using their analytic representations. https://www.maplesoft.com/applications/view.aspx?SID=3988&ref=FeedWed, 01 Aug 2001 13:36:43 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsPower series expansions
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In this lesson, we explore methods of expanding functions into power series. The basic idea hinges on the geometric series expansion of 1/(1-x) . However, using differentiation and integration we can expand many more functions into power series also. In addition, we will examine the interval of convergence and how it is affected by the location of the expansion and features of the function such as vertical asymptotes. In general, this module will reinforce methods one might use by hand and not rely on the automated expansions Maple can generate.
<img src="https://www.maplesoft.com/view.aspx?si=3987//applications/images/app_image_blank_lg.jpg" alt="Power series expansions" style="max-width: 25%;" align="left"/> In this lesson, we explore methods of expanding functions into power series. The basic idea hinges on the geometric series expansion of 1/(1-x) . However, using differentiation and integration we can expand many more functions into power series also. In addition, we will examine the interval of convergence and how it is affected by the location of the expansion and features of the function such as vertical asymptotes. In general, this module will reinforce methods one might use by hand and not rely on the automated expansions Maple can generate.
https://www.maplesoft.com/applications/view.aspx?SID=3987&ref=FeedWed, 01 Aug 2001 11:42:54 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsApproximating general functions
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In the previous lesson, we saw how to approximate the sine and cosine functions arbitrarily accurately near x=0 by using polynomials. In this worksheet, we will see that the same approach can be used to approximate other functions, and that it can be used to get approximations near other points<img src="https://www.maplesoft.com/view.aspx?si=3986//applications/images/app_image_blank_lg.jpg" alt="Approximating general functions " style="max-width: 25%;" align="left"/>In the previous lesson, we saw how to approximate the sine and cosine functions arbitrarily accurately near x=0 by using polynomials. In this worksheet, we will see that the same approach can be used to approximate other functions, and that it can be used to get approximations near other pointshttps://www.maplesoft.com/applications/view.aspx?SID=3986&ref=FeedWed, 01 Aug 2001 11:41:38 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsApproximating sine and cosine
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Calculus lesson in showing how to approximate trig functions using Taylor series<img src="https://www.maplesoft.com/view.aspx?si=3985//applications/images/app_image_blank_lg.jpg" alt="Approximating sine and cosine " style="max-width: 25%;" align="left"/>Calculus lesson in showing how to approximate trig functions using Taylor serieshttps://www.maplesoft.com/applications/view.aspx?SID=3985&ref=FeedWed, 01 Aug 2001 11:40:32 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsTaylor series
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Suppose the nth derivative of y=f(x) is defined at x=a . We illustrate examples of the nth Taylor polynomial for f at x=a.
<img src="https://www.maplesoft.com/view.aspx?si=3984//applications/images/app_image_blank_lg.jpg" alt="Taylor series" style="max-width: 25%;" align="left"/>Suppose the nth derivative of y=f(x) is defined at x=a . We illustrate examples of the nth Taylor polynomial for f at x=a.
https://www.maplesoft.com/applications/view.aspx?SID=3984&ref=FeedWed, 01 Aug 2001 11:39:24 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsConvergence tests for infinite series
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We have seen what is meant by saying that an infnite series converges, with sum s. Unfortunately, there are very few series to which the definition can be applied directly; the most important is certainly the Geometric Series <img src="https://www.maplesoft.com/view.aspx?si=3983//applications/images/app_image_blank_lg.jpg" alt="Convergence tests for infinite series" style="max-width: 25%;" align="left"/>We have seen what is meant by saying that an infnite series converges, with sum s. Unfortunately, there are very few series to which the definition can be applied directly; the most important is certainly the Geometric Series https://www.maplesoft.com/applications/view.aspx?SID=3983&ref=FeedWed, 01 Aug 2001 11:38:16 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsGeometric curves
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The Snowflake Curve was initially described by Koch as an affirmative solution to the problem of whether there is a continuous curve that has no tangent line at any point on the curve.<img src="https://www.maplesoft.com/view.aspx?si=3982/snowflake.gif" alt="Geometric curves" style="max-width: 25%;" align="left"/>The Snowflake Curve was initially described by Koch as an affirmative solution to the problem of whether there is a continuous curve that has no tangent line at any point on the curve.https://www.maplesoft.com/applications/view.aspx?SID=3982&ref=FeedWed, 01 Aug 2001 11:36:17 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsConvergence of series
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Suppose you have established somehow, either directly or by some test, that a series converges to a number L. How do we calculate this number to any specified accuracy?
<img src="https://www.maplesoft.com/view.aspx?si=3981//applications/images/app_image_blank_lg.jpg" alt="Convergence of series" style="max-width: 25%;" align="left"/>Suppose you have established somehow, either directly or by some test, that a series converges to a number L. How do we calculate this number to any specified accuracy?
https://www.maplesoft.com/applications/view.aspx?SID=3981&ref=FeedWed, 01 Aug 2001 11:35:21 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntroduction to infinite series
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What does it mean to add up a sequence of numbers? (Remember that sequences are always infinite, so this is a question about adding up an infinite set of numbers.) This is a Maple lesson introducing convergence of series.<img src="https://www.maplesoft.com/view.aspx?si=3980//applications/images/app_image_blank_lg.jpg" alt="Introduction to infinite series" style="max-width: 25%;" align="left"/>What does it mean to add up a sequence of numbers? (Remember that sequences are always infinite, so this is a question about adding up an infinite set of numbers.) This is a Maple lesson introducing convergence of series.https://www.maplesoft.com/applications/view.aspx?SID=3980&ref=FeedWed, 01 Aug 2001 11:34:11 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsConvergence of sequences
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We have used sequences lots of times before. The sequence of estimates to the solution of an equation generated by Newton's Method is one. The sequence of estimates to the integral of a function over an interval obtained by subdividing the interval into more and more subintervals is another. These are examples of potentially infinite sequences. These are sequences we hope converge to the answer we seek, whether it be the solution of an equation or the value of an integral<img src="https://www.maplesoft.com/view.aspx?si=3979//applications/images/app_image_blank_lg.jpg" alt="Convergence of sequences " style="max-width: 25%;" align="left"/>We have used sequences lots of times before. The sequence of estimates to the solution of an equation generated by Newton's Method is one. The sequence of estimates to the integral of a function over an interval obtained by subdividing the interval into more and more subintervals is another. These are examples of potentially infinite sequences. These are sequences we hope converge to the answer we seek, whether it be the solution of an equation or the value of an integralhttps://www.maplesoft.com/applications/view.aspx?SID=3979&ref=FeedWed, 01 Aug 2001 11:33:15 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntroduction to sequences
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A sequence is simply an infinite list; that is an infinite collection of objects arranged in some order. In this course, we will deal almost exclusively with sequences of numbers<img src="https://www.maplesoft.com/view.aspx?si=3978//applications/images/app_image_blank_lg.jpg" alt="Introduction to sequences " style="max-width: 25%;" align="left"/>A sequence is simply an infinite list; that is an infinite collection of objects arranged in some order. In this course, we will deal almost exclusively with sequences of numbershttps://www.maplesoft.com/applications/view.aspx?SID=3978&ref=FeedWed, 01 Aug 2001 11:31:27 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsImproper integrals
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Calculus lesson on integration of functions over unbounded intervals<img src="https://www.maplesoft.com/view.aspx?si=3977//applications/images/app_image_blank_lg.jpg" alt="Improper integrals" style="max-width: 25%;" align="left"/>Calculus lesson on integration of functions over unbounded intervalshttps://www.maplesoft.com/applications/view.aspx?SID=3977&ref=FeedWed, 01 Aug 2001 11:30:22 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntegration techniques summary and review
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A review of the previous lessons on integration techniques, including substitution, integration of rational functions, integration by parts and integration of trig powers.
<img src="https://www.maplesoft.com/view.aspx?si=3976//applications/images/app_image_blank_lg.jpg" alt="Integration techniques summary and review " style="max-width: 25%;" align="left"/>A review of the previous lessons on integration techniques, including substitution, integration of rational functions, integration by parts and integration of trig powers.
https://www.maplesoft.com/applications/view.aspx?SID=3976&ref=FeedWed, 01 Aug 2001 11:28:27 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntegration of trigonometric powers
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One of the most important uses of reduction formulas is to evaluate anti-derivatives of powers of trigonometric functions.<img src="https://www.maplesoft.com/view.aspx?si=3975//applications/images/app_image_blank_lg.jpg" alt="Integration of trigonometric powers" style="max-width: 25%;" align="left"/>One of the most important uses of reduction formulas is to evaluate anti-derivatives of powers of trigonometric functions.https://www.maplesoft.com/applications/view.aspx?SID=3975&ref=FeedWed, 01 Aug 2001 11:27:15 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntegration of rational functions
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You know from the Fundamental Theorem of Calculus that every continuous function has an anti-derivative. On the other hand, not every (continuous) function has an elementary anti-derivative, which we could define loosely as an anti-derivative built up from functions discussed in first semester of Calculus. We have seen some examples already.
<img src="https://www.maplesoft.com/view.aspx?si=3974//applications/images/app_image_blank_lg.jpg" alt="Integration of rational functions" style="max-width: 25%;" align="left"/>You know from the Fundamental Theorem of Calculus that every continuous function has an anti-derivative. On the other hand, not every (continuous) function has an elementary anti-derivative, which we could define loosely as an anti-derivative built up from functions discussed in first semester of Calculus. We have seen some examples already.
https://www.maplesoft.com/applications/view.aspx?SID=3974&ref=FeedWed, 01 Aug 2001 11:26:15 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntegration by parts
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A calculus lesson in the study of anti-differentiation by parts . <img src="https://www.maplesoft.com/view.aspx?si=3973//applications/images/app_image_blank_lg.jpg" alt="Integration by parts " style="max-width: 25%;" align="left"/>A calculus lesson in the study of anti-differentiation by parts . https://www.maplesoft.com/applications/view.aspx?SID=3973&ref=FeedWed, 01 Aug 2001 11:24:35 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of MathematicsIntegration by substitution: 20 worked examples
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Here are 20 anti-derivatives. We reduce each of them to a simpler form by means of a substitution. Here are the rules for today's game:
Use the unevaluated Int command throughout the worksheet.
You may consider that you have "solved" the problem when you have reduced it to one of the standard forms or to a problem which can be done with a known reduction formula, or to a sum of problems of these types.
Try to find the simplest substitution that will work in each case.
<img src="https://www.maplesoft.com/view.aspx?si=3972//applications/images/app_image_blank_lg.jpg" alt="Integration by substitution: 20 worked examples " style="max-width: 25%;" align="left"/>Here are 20 anti-derivatives. We reduce each of them to a simpler form by means of a substitution. Here are the rules for today's game:
Use the unevaluated Int command throughout the worksheet.
You may consider that you have "solved" the problem when you have reduced it to one of the standard forms or to a problem which can be done with a known reduction formula, or to a sum of problems of these types.
Try to find the simplest substitution that will work in each case.
https://www.maplesoft.com/applications/view.aspx?SID=3972&ref=FeedWed, 01 Aug 2001 11:22:59 ZU. of Wisconsin-Milwaukee Dept of MathematicsU. of Wisconsin-Milwaukee Dept of Mathematics