Series ExpansionsRobert J. LopezEmeritus Professor, RHIT Maple Fellow, MaplesoftLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 post, we examine the relevant commands and interface devices that access these functionalities. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= In Maple, the output of the taylor command appears to be a polynomial to which a "big Oh" term has been attached. The series command produces a generalized series expansion that could be a Taylor or Laurent series, or a more general series. In particular, the results of these commands can be seen in Table 1. 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlRig=

JSFH 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

LCgqKCkiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiKUkieEc2IkYmRidGJyoqI0YnIiInRidGJEYnRihGKylGLSMiIiZGJUYnRistSSJPR0YqNiMqJClGLSMiIipGJUYnRic=

Table 1 Application of taylor and series commands to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ4RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== In Table 1, the output of the taylor command is not just a polynomial to which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjZGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0 has been added. It is a special series construct that can be converted to a polynomial with the convert command, as in 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

LChJInhHNiIiIiIqJiNGJSIiJ0YlKUYjIiIkRiUhIiIqJiNGJSIkPyJGJSlGIyIiJkYlRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The default value of the Order variable in Maple is 6. It can be changed by assigning it another integer value. However, it can also be temporarily (i.e., locally) changed inside either of the taylor or series commands by simply including an integer as a third argument. For example, Order can be set temporarily to 7 as follows. JSFH 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlIiIo

JSFH This output suggests that care should be taken in interpreting the "big Oh" term produced by these commands. The outputs in Table 1 are technically correct in that they indicate the remainder term is "big Oh" of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn to the power of Order. In fact, the remainder terms in Table 1 are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEib0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjZGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0, that is, "little oh" of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI2RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ==. Thus, the commands in Table 1 do not produce the tight bound LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjdGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0 that would apply if some form of the remainder were analytically examined for its true order. The situation is further compounded by the Maple "remember table," an internal table that stores certain calculations to reduce the time it takes to access the calculation again. We see the effect of this table in the following calculation. 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlIiIo

The second time the command is called with the default value of Order, namely, six, the remember table comes into play, and now, Maple indicates that the remainder term is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjdGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0. The contents of the remember table can be emptied with the forget command, as in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnZm9yZ2V0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEnc2VyaWVzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlRig=

Thus, it is useful to think in terms of "series order" as an integer that Maple uses to keep track of the truncation of a series, and it is not necessarily the precise analytic order of the remainder term. Moreover, it is clearly not the degree of the truncated polynomial obtained after converting to a polynomial. It is unfortunate that Maple uses "order" in many places where "degree" would be the more appropriate term or construct. When the output of the series command contains fractional exponents, it is not the special series construct, but rather, an expression in the form of a sum of products to which the "big Oh" term has been added. The "big Oh" term can be removed by the same convert option that works for the output of the taylor command. JSFH

JSFH The series command can be accessed directly, as in Table 1, or from the Context Menu. The taylor command can be accessed directly, as in Table 1, from several Task Templates, or from the TaylorApproximation command in the Student Calculus 1 package. JSFH Figure 1 shows the dialog that results from selecting Series\342\211\273Series\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn from the Context Menu applied to an expression such as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ4RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== or 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, the BesselJ function of index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiL0YnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkotRiw2JVEiMkYnRi9GMkYvRjI=. 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Figure 1 Series option in the Context Menu JSFH The term "Series order" is the value that will be assigned to the Order option or variable. If the "Truncate to polynomial" checkbox is checked, a polynomial of degree at least one less than "Series order" will result. JSFH Figure 2 shows the Taylor Approximation tutor with its default settings. The default function is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ4RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== and the default "series order" is four. The TaylorApproximation command shown at the bottom of the tutor provides a Taylor polynomial as an approximation to the input function, but Maple declares this polynomial approximation to be of "order," that is, of "series order," rather than of "degree." The TaylorApproximation command is merely an interface to the underlying taylor command. 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Figure 2 Taylor Approximation tutor accessing the TaylorApproximation command Table 2 provides the path to a Task Template that invokes the TaylorApproximation command and/or launches the Taylor Approximation tutor. The syntax for implementing the TaylorApproximation command is provided by this Task Template. Note that the local setting of the Order variable must be done in the form of an equation whose left-hand side is the word "order." As in Figure 2, "order" means the degree of the resulting polynomial, a usage that conflicts with that of the underlying taylor command. Student[Calculus1][TaylorApproximationTutor](); Tools\342\211\273Tasks\342\211\273Browse: Calculus\342\211\273 Taylor Approximation of a Univariate Function

Calculate the Taylor approximation of a specified degree for a univariate function. Enter the function as an expression. 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

LCYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kmX194X19HRigiIiItSSRzaW5HRiVGKUYr

Specify the order, and then calculate the Taylor approximation. 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

LC4iIiJGI0kmX194X19HNiJGIyomI0YjIiIjRiMpRiRGKEYjISIiKiYjRiMiIidGIylGJCIiJEYjRioqJiNGIyIjQ0YjKUYkIiIlRiNGIyomI0YjIiQ/IkYjKUYkIiImRiNGIw==

Alternatively, you can use the Taylor Approximation tutor, a point-and-click interface. There are two ways to launch this tutor.From the Tools menu, select Tutors, Calculus - Single Variable, and then Taylor Approximation.Enter the function in one variable to be approximated below, then click the following Taylor Approximation button. try TaylorApproximation_temp := Student:-Calculus1:-TaylorApproximationTutor(DocumentTools:-Do(%TextArea0)); if TaylorApproximation_temp <> NULL then DocumentTools:-SetProperty('Plot0','visible','true'); DocumentTools:-SetProperty('Plot0','value',TaylorApproximation_temp); fi; catch: error end try;

Table 2 The Task Template "Taylor Approximation of a Univariate Function JSFH Pressing the "Taylor Approximation" button will launch the Taylor Approximation tutor pre-loaded with the function in the box to the right of the button. Alternatively, if the Student Calculus 1 package is first loaded into Maple's memory, the Context Menu for an expression will contain the option "Tutors" from which the Taylor Approximation tutor can be selected. This would be more direct than accessing the tutor from the Task Template. JSFH Table 3 provides the path to a Task Template that invokes the taylor command and provides an interactive tool for obtaining Taylor expansions and polynomials. It is possible to access just this tool, which in turn provides access to the Taylor Approximation tutor. Tools\342\211\273Tasks\342\211\273Browse: Calculus\342\211\273Taylor Expansion and Polynomials

Calculate the Taylor expansion for a univariate function about a specified point and up to a specific order. Enter the function as an expression: 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

LCYqJiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZfX3hfX0dGK0YlRiVGJUYl

Specify an expansion point and an expansion order, and then calculate the Taylor series: 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

Ky9JJl9feF9fRzYiIiIkIiIhIiIjIiIiRihGJyNGKEYlRiUjRigiIzciIiUtSSJPRyUqcHJvdGVjdGVkRzYjRigiIiY=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1 Alternatively, you can use the following interface to calculate the Taylor expansion and Taylor polynomial:

Taylor Expansions and Polynomials Enter the function: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRjJGQA== 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2NDc-@=Ll#!#:7$$"1ommY^*o-$!#:$!2B$pYK*zRu#!#:7$$"1baauX?7J!#:$!2ER(e!)zPKG!#:7$$"1ihh@**R#>$!#:$!2D5gI!yC(*G!#:7$$"2(444p1#=F$!#;$!2:d%GHi/TH!#:7$$"2ljjjf/8N$!#;$!2DA**4zU8'H!#:7$$"29444<tBV$!#;$!1:/*[!pGaH!#97$$"1#===]/$=N!#:$!2(=*y@!)o>"H!#:7$$"2%edd<^4(f$!#;$!2a(H!G%**ePG!#:7$$"2DDDDdr[n$!#;$!22;\`%\yEF!#:7$$"2LGGGK8!eP!#;$!2%)4Hn#4liD!#:7$$"273333u4%Q!#;$!2VxS0&QqYB!#:7$$"1pnn(GOa"R!#:$!1;%HysmR5#!#97$$"#S!""$!2dmmmmmmw"!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%%VIEWG6$;$!#S!""$"#S!"";$!)Hp(>"!"($").z*>"!"(-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6%Q"x6"Q"y6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$"$I"!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%SP!""-%.BOUNDS_HEIGHTG6#$"%SV!""-%)CHILDRENG6"G6" use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, a, F; Do(f = %MathContainer0); if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))); end if; F := Student:-Calculus1:-TaylorApproximationTutor(f,a); if F <> NULL then DocumentTools:-SetProperty('Plot1','visible','true'); DocumentTools:-SetProperty('Plot1','value', F); end if; end module; end use; Degree: Base-Point: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, q, a, F, Q; Do(f = %MathContainer0); q := op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer1','value')))); if is(q,posint) then Q:=q else Q:=5; end if; if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))); end if; F := taylor(f, x=a, Q+1); SetProperty('MathContainer3', 'value', F); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, q, Q, a, F; Do(f = %MathContainer0); q := op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer1','value')))); if is(q,posint) then Q:=q else Q:=5; end if; if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))); end if; F := convert(taylor(f, x=a, Q+1),polynom); SetProperty('MathContainer4', 'value', F); end module; end use; LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() SetProperty('MathContainer0', 'value', NULL); SetProperty('MathContainer1', 'value', NULL); SetProperty('MathContainer3', 'value', NULL); SetProperty('MathContainer2', 'value', NULL); SetProperty('MathContainer4', 'value', NULL); Do(%Plot1=plots[display](plots[textplot]([0,0,""]),axes=none)); end module; end use;

Table 3 The "Taylor Expansion and Polynomials" Task Template For both the series expansion and the Taylor polynomial, the degree of the polynomial is governed by the "Degree" option. If no base-point is provided, the tool assumes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8. Launching the tutor from this tool loads the function and the base-point, but not the desired degree. Closing the tutor places its graph into the graphing window of the tool. JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Laurent series 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 can be obtained with the series command, or with the laurent command from the numapprox package. Usage is illustrated in Table 4. 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

KytJInhHNiIiIiIhIiMjRiUiIiciIiEjIiIoIiRnJCIiIy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEqbnVtYXBwcm94RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtRiw2JVEobGF1cmVudEYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY2NiQtRiM2Jy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGQC1GIzYqLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSJ+RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZvLUYsNiVRJHNpbkYnL0YwRj9GQC1GNjYkLUYjNidGVUYvRj0vJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGMkZARi9GPUZhcEYyLyUubGluZXRoaWNrbmVzc0dGUi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZocC8lKWJldmVsbGVkR0Y/LUZZNi1RIixGJ0ZARmZuL0ZpbkYxRmpuRlxvRl5vRmBvRmJvRmRvL0Zob1EsMC4zMzMzMzMzZW1GJ0ZVRj1GQEZARj1GQA==

KytJInhHNiIiIiIhIiMjRiUiIiciIiEjIiIoIiRnJCIiIy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 4 The Laurent series via the series and laurent commands LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Asymptotic power series, even series with fractional powers, can be obtained in any of the four ways listed in Table 5. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnc2VyaWVzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiwtRiw2JVEoQmVzc2VsSkYnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1GNjYkLUYjNictSSNtbkdGJDYkUSIwRidGPy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRi9GMi8lK2V4ZWN1dGFibGVHRj5GP0Y/RklGW28tRko2LVEiPUYnRj9GTS9GUEY+RlFGU0ZVRldGWS9GZm5RLDAuMjc3Nzc3OGVtRicvRmluRmVvLUYsNiVRKWluZmluaXR5RidGL0YyRkktRkY2JFEiMkYnRj9GXm9GP0Y/LUYsNiNRIUYnRl5vRj8=

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

Table 5 Asymptotic power series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= From Table 5 we see that the series, taylor, and asympt commands can all be used to generate asymptotic power series, even series with fractional powers. The asympt command assumes that the expansion point is the point at infinity. Both the taylor and series commands need to be told that the expansion point is at infinity. It is even possible to obtain an asymptotic expansion by changing variables so 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, expanding about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8, and restoring LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn via 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. An asymptotic power series, even one with fractional powers, is available through the Context Menu, under the option Series\342\211\273Asymptotic Series\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. The resulting dialog, shown in Figure 3, suggests it is possible to obtain an asymptotic power series interactively, either as a series construct or as a truncated expansion. 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3 Dialog for Context Menu option: Series\342\211\273Asymptotic Series Unfortunately, Maple does not have a function for generating an asymptotic series with respect to an arbitrary asymptotic basis. However, if 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 is an asymptotic sequence as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJnNyYXJyO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJhRidGL0YyRjVGMkY8, then the coefficients in the expansion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RKCZUaWxkZTtGJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUkrbXVuZGVyb3ZlckdGJDYnLUZDNi5RJiZTdW07RidGMkZAL0ZHUSZ1bnNldEYnL0ZJRmpuL0ZLRjEvRk1Gam4vRk9GMS9GUUYxL0ZTRmpuL0ZVUSYwLjBlbUYnL0ZYUSwwLjE2NjY2NjdlbUYnLUYjNictRiw2JlEia0YnRi9GMkY1LUZDNi5RIj1GJ0YyRkBGRkZIRkpGTEZORlBGUkZURlctSSNtbkdGJDYlUSIwRidGMkZARjJGQC1GIzYoLUYsNiZRIk5GJ0YvRjJGNUYvLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRidGMi8lLHBsYWNlaG9sZGVyR0YxRjVGUi8lLGFjY2VudHVuZGVyR0Y0LUklbXN1YkdGJDYlLUYsNiZRImNGJ0YvRjJGNS1GIzYmRmdvRi9GMkY1LyUvc3Vic2NyaXB0c2hpZnRHRmBwLUZecTYlLUYsNiZRJyYjOTY2O0YnL0YwRjRGMkZARmNxRmVxRjhGMkZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= can be obtained recursively via 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The mtaylor command is used to generate a Taylor polynomial for a multivariate function. For example, the Taylor expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEieEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEieUYnRi9GMkY1RjJGREYyRkRGMkZE, taken about 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, would be of the form 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 Table 6 contains the expansion of 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 at the point 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. 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

LC4jIiImIiIlISIiKiYjIiIiRiVGKUkieEc2IkYpRikqJkYoRilJInlHRitGKUYpKiYjRikiIilGKSksJkYqRilGKUYmIiIjRilGJiooRihGKSwmRi1GKUYzRiZGKUYyRilGJiomRi9GKSlGNUYzRilGJg==

Table 6 Multivariate Taylor expansion of 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Even though Maple does not have a series construct for the multivariate case, the notion of "series order" is used to determine the degree of the resulting polynomial. Unfortunately, this results in a polynomial of total degree one less than the "series order" used as a parameter in the mtaylor command. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Because there is no "series construct" for the multivariate case, Maple does not tack on an order term for expansions generated by the mtaylor command. Because of this, I can recall that colleagues at RHIT would teach their students only about the mtaylor command, even in single-variable calculus. This would avoid the need to follow up with the "convert-to-polynom" step, thereby lessening the syntax burden in the days when all Maple calculations were dependent on knowing the appropriate commands.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Figure 4 shows the Taylor Approximation tutor from the Student Multivariate Calculus package. 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Figure 4 Multivariate Calculus Taylor Approximation tutor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The first-degree Taylor polynomial for the function 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, taken at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJ5RidGNEY3RjpGN0ZBRjdGQS1GPjYuUSI9RidGN0ZBRkMvRkZGOUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGaW4tRiw2JS1GIzYnLUkjbW5HRiQ2JVEiMUYnRjdGQUY9LUZgbzYlUSIyRidGN0ZBRjdGQUY3RkFGN0ZB is obtained in Figure 4. In addition, a graph of the function and the Taylor polynomial is drawn. The graph of the first-degree Taylor polynomial is the graph of the tangent plane. At the bottom of the tutor, the TaylorApproximation command can be seen. The syntax of this command contrasts to that of the mtaylor command and to the similarly named command from the Calculus 1 package. Note in particular how the expansion point is expressed, and note that the explicit use of the equation "order = 1" is not required. In the absence of the "output = plot" parameter, the expression for the Taylor polynomial would be returned. Finally, note that both the command and the tutor use "Order" to mean "degree." Given that there is no "series construct" in the multivariate case, and given that both the command and the tutor specifically reference "Approximation," it's striking that the notion of "series order" is used, and used where "degree" would be more appropriate. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 7 shows the path to a Task Template that provides access to the Taylor Approximation command and tutor. Tools\342\211\273Tasks\342\211\273Browse: Multivariate Calculus\342\211\273Taylor Approximation of a Multivariate Function

Calculate the Taylor approximation of a specified degree for a multivariate function. Enter the function. LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Li1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZSLUY8Ni1RIzo9RidGP0ZBRkRGRkZIRkpGTEZORlBGU0Y7LUkobWZlbmNlZEdGJDYkLUYwNigtRiw2JVEieEYnRjVGOC1GPDYtUSIsRidGP0ZBRkRGRkZIRkpGTEZORlBGU0Y7LUYsNiVRInlGJ0Y1RjgvJStleGVjdXRhYmxlR0ZDRj9GP0Y7LUY8Ni1RLSZSaWdodEFycm93O0YnRj9GQUZERkZGSEZKRkxGTkZQRlNGOy1GWTYkLUYwNidGKy1JKG1hY3Rpb25HRiQ2JC1GMDYoRistRjA2J0YrLUYjNixGKy1JJW1zdXBHRiQ2JS1GLDYoRmluRjUvJStmb3JlZ3JvdW5kR1ErWzk1LDAsMjE3XUYnLyUnb3BhcXVlR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1E4TWFwbGV+SW5wdXR+UGxhY2Vob2xkZXJGJ0Y4LUkjbW5HRiQ2J1EiM0YnRmdwRmpwRlxxRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjstRjw2MFEnJnBsdXM7RidGZ3BGanBGXHFGP0ZBRkRGRkZIRkpGTEZORlBGU0Y7LUYwNiktRiw2KFEkc2luRidGNUZncEZqcEZccUY4LUY8Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj9GQUZERkZGSEZKRkxGTkZQRlMtRlk2JS1GMDYmLUYsNihGX29GNUZncEZqcEZccUY4RmpwRmBvRj9GanBGP0YrRmpwRmBvRj9GK0ZqcEZgb0Y/RitGYG9GP0YrRmBvLyUscGxhY2Vob2xkZXJHRjdGPy8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGYG9GP0Y/RitGYG9GP0YrRmBvRj8=

Zio2JEkmX194X19HNiJJJl9feV9fR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiokKUYkIiIkIiIiRi4tSSRzaW5HRiU2I0YmRi5GJUYlRiU=

Specify an expansion point and order, and then calculate the Taylor approximation. 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

LDAqJiMiIiIiJD8iRiUpSSZfX3lfX0c2IiIiJkYlRiUqJiNGJSIiJ0YlKUYoIiIkRiUhIiIqJkYvRiVJJl9feF9fR0YpRiVGJUYoRiUiIiNGMComRi9GJSksJkYyRiVGJUYwRjNGJUYlKiQpRjZGL0YlRiU=

Alternatively, you can use the Taylor Series tutor, a point-and-click interface. There are two ways to launch this tutor.From the Tools menu, select Tutors, Calculus - Multi-Variable and then Taylor Series.Enter a multivariate function below, then click the following Taylor Series button. try TaylorSeries_temp := Student:-MultivariateCalculus:-TaylorApproximationTutor(DocumentTools:-Do(%TextArea1)); if TaylorSeries_temp <> NULL then DocumentTools:-SetProperty('Plot2','visible','true'); DocumentTools:-SetProperty('Plot2','value',TaylorSeries_temp); fi; catch: error end try;

Table 7 The Task Template "Taylor Approximation of a Multivariate Function" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Again, if the Student Multivariate Calculus package is loaded, the Context Menu for an expression will contain the option "Tutors," from which the Taylor Approximation tutor can be selected. This proves to be a more efficient connection to the tutor than the Task Template. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple has two separate processes for producing the complete formal power series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=. Each is accessed as an option to the convert command, but even though it looks like there is a common interface, the underlying approaches to obtaining the formal power series are different. In most cases, either process will work, but there are functions for which just one will succeed in finding the desired series. Table 8 illustrates the two relevant functionalities. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJImtHRiciIiItSSpmYWN0b3JpYWxHRiU2IywmKiYiIiNGLUYsRi1GLUYtRi1GKylJInhHRidGMUYtL0YsOyIiIUkpaW5maW5pdHlHRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRJHNpbkYnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1GNjYkLUYjNiUtRiw2JVEieEYnRi9GMi8lK2V4ZWN1dGFibGVHRj5GP0Y/LUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSRTdW1GJ0YvRjJGSEY/Rj9GSEY/

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJJF9rMUdGJyIiIilJInhHRicsJiomIiIjRi1GLEYtRi1GLUYtRi0tSSpmYWN0b3JpYWxHRiU2I0YwRisvRiw7IiIhSSlpbmZpbml0eUdGJQ==

Table 8 Obtaining complete formal power series in Maple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= It should be obvious from Table 8 that two different functionalities have been activated. In the result on the right, the summation index can be declared by the user, as we show in the following calculation. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJImtHRiciIiItSSpmYWN0b3JpYWxHRiU2IywmKiYiIiNGLUYsRi1GLUYtRi1GKylJInhHRidGMUYtL0YsOyIiIUkpaW5maW5pdHlHRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Context Menu provides the option Series\342\211\273Formal Power Series. The dialog that appears in shown in Figure 5. 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5 Dialog for the Context Menu option: Series\342\211\273Formal Power Series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= From Figure 5, it should be clear that even from the Context Menu it is possible to declare the summation index in the formal power series returned. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Finally, note that one can obtain a complete formal asymptotic power series, as illustrated in Table 9. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJikhIiJJImtHRiciIiIpKiRJInhHRidGKywmRixGLUYtRi1GLS9GLDsiIiFJKWluZmluaXR5R0Yl

Table 9 Complete formal asymptotic power series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH JSFH JSFH JSFH JSFH Expans\303\265es de s\303\251rieRobert J. LopezEm\303\251rito Professor, RHIT Maple Fellow, MaplesoftJSFH

JSFH O Maple tem a capacidade de fornecer v\303\241rias expans\303\265es em s\303\251rie e seus truncamentos, bem como s\303\251ries formais completas para uma variedade de fun\303\247\303\265es elementares e especiais. Nesta postagem, examinamos os comandos e dispositivos de interface relevantes que acessam essas funcionalidades. JSFH

JSFH No Maple, a sa\303\255da do comando taylor parece ser um polin\303\264mio ao qual um termo "Big oh" foi anexado. O comando series produz uma expans\303\243o de s\303\251rie generalizada que poderia ser uma s\303\251rie de Taylor ou Laurent ou uma s\303\251rie mais geral. Em particular, os resultados desses comandos podem ser vistos na Tabela 1. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LCgqKCkiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiKUkieEc2IkYmRidGJyoqI0YnIiInRidGJEYnRihGKylGLSMiIiZGJUYnRistSSJPR0YqNiMqJClGLSMiIipGJUYnRic=

Tabela 1 Aplica\303\247\303\243o de taylor e series comandos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ4RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== Na Tabela 1, a sa\303\255da do comando taylor n\303\243o \303\251 apenas um polin\303\264mio para o qual LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjZGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0 foi adicionado. \303\211 uma constru\303\247\303\243o de s\303\251rie especial que pode ser convertida em um polin\303\264mio com o comando convert, como em 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

LChJInhHNiIiIiIqJiNGJSIiJ0YlKUYjIiIkRiUhIiIqJiNGJSIkPyJGJSlGIyIiJkYlRiU=

JSFH O valor padr\303\243o da vari\303\241vel Ordem no Maple \303\251 6. Ele pode ser alterado atribuindo-lhe outro valor inteiro. No entanto, tamb\303\251m pode ser temporariamente (i.e., localmente) alterado dentro dos comandos taylor ou series simplesmente incluindo um inteiro como um terceiro argumento. Por exemplo, Ordem pode ser definido temporariamente para 7 da seguinte maneira. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlIiIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Esta sa\303\255da sugere que deve-se tomar cuidado ao interpretar o termo "Big Oh" produzido por esses comandos. As sa\303\255das na Tabela 1 s\303\243o tecnicamente corretas, pois indicam que o termo restante \303\251 "big Oh" x elevado \303\240 pot\303\252ncia do pedido. Na verdade, os demais termos da Tabela 1 s\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEib0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjZGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0, isto \303\251, "little oh" de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI2RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ==. Assim, os comandos na Tabela 1 n\303\243o produzem o limite estreito LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjdGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0 isso se aplicaria se alguma forma do restante fosse examinada analiticamente para determinar sua verdadeira ordem. A situa\303\247\303\243o \303\251 ainda agravada pela "tabela de recorda\303\247\303\243o" do Maple, uma tabela interna que armazena certos c\303\241lculos para reduzir o tempo que leva para acessar o c\303\241lculo novamente. Vemos o efeito desta tabela no c\303\241lculo a seguir. 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlIiIo

Na segunda vez que o comando \303\251 chamado com o valor padr\303\243o de Ordem, a saber, seis, a tabela de lembrar entra em jogo e, agora, o Maple indica que o termo restante \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3VwR0YkNiUtRiw2JlEieEYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRictRiM2Ji1JI21uR0YkNiVRIjdGJ0YyRjRGQkYyRkQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNEYyRjRGMkY0. O conte\303\272do da tabela lembrar pode ser esvaziado com o comando LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EnZm9yZ2V0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0YuLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJw==, como em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnZm9yZ2V0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEnc2VyaWVzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA 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

KytJInhHNiIiIiJGJSMhIiIiIiciIiQjRiUiJD8iIiImLUkiT0clKnByb3RlY3RlZEc2I0YlRig=

Assim, \303\251 \303\272til pensar em termos de "series ordem" como um inteiro que o Maple usa para rastrear o truncamento de uma s\303\251rie, e n\303\243o \303\251 necessariamente a ordem anal\303\255tica precisa do termo restante. Al\303\251m disso, claramente n\303\243o \303\251 o grau do polin\303\264mio truncado obtido ap\303\263s a convers\303\243o em um polin\303\264mio. \303\211 lament\303\241vel que o Maple use "ordem" em muitos lugares onde "grau" seria o termo ou constru\303\247\303\243o mais apropriado. Quando a sa\303\255da do comando da s\303\251rie cont\303\251m expoentes fracion\303\241rios, n\303\243o \303\251 a constru\303\247\303\243o da s\303\251rie especial, mas sim uma express\303\243o na forma de uma soma de produtos para a qual o "big Oh" termo adicionado. O termo "big Oh" pode ser removido pela mesma op\303\247\303\243o convert que funciona para a sa\303\255da do comando taylor. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= O comando da s\303\251rie pode ser acessado diretamente, como na Tabela 1, ou a partir do Menu de Contexto. O comando taylor pode ser acessado diretamente, como na Tabela 1, de v\303\241rios modelos de tarefas ou do comando TaylorApproximation no pacote Student Calculus 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A Figura 1 mostra a caixa de di\303\241logo que resulta da sele\303\247\303\243o Series\342\211\273Series\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn do Menu de Contexto aplicado a uma express\303\243o como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ4RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRictSSNtb0dGJDYuUSIvRidGNUZBLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGVy1GPjYlUSIyRidGNUZBRjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLzYmUSJ4RidGMkY1RjhGNUZBRjVGQUY1RkE=, a fun\303\247\303\243o BesselJ de \303\255ndiceLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiL0YnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkotRiw2JVEiMkYnRi9GMkYvRjI=. 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Figura 1 Op\303\247\303\243o de s\303\251rie no menu de contexto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= O termo "Pedido em s\303\251rie" \303\251 o valor que ser\303\241 atribu\303\255do \303\240 op\303\247\303\243o ou vari\303\241vel do pedido. Se a caixa de sele\303\247\303\243o "Truncar para polin\303\264mio" estiver marcada, um polin\303\264mio de grau pelo menos um menor do que "Ordem da s\303\251rie" resultar\303\241. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR0Y3LyUncnNwYWNlR0Y3Ri8tRjA2JkYyRjVGOC9GO1ElYXV0b0YnRisvJStleGVjdXRhYmxlR0ZGRkE=A Figura 2 mostra o tutor Aproxima\303\247\303\243o de Taylo com suas configura\303\247\303\265es padr\303\243o. A fun\303\247\303\243o padr\303\243o \303\251 "sin (x)" e a "ordem das s\303\251ries" padr\303\243o \303\251 quatro. O comando TaylorApproximation mostrado na parte inferior do tutor fornece um polin\303\264mio de Taylor como uma aproxima\303\247\303\243o para a fun\303\247\303\243o de entrada, mas Maple declara esta aproxima\303\247\303\243o polinomial como de "ordem", isto \303\251, de "ordem da s\303\251rie", ao inv\303\251s de "grau. " O comando TaylorApproximation \303\251 apenas uma interface para o comando taylor subjacente. 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Figura 2 Tutor do Aproxima\303\247\303\243o Taylor acessando o comando TaylorApproximation A Tabela 2 fornece o caminho para um modelo de tarefa que invoca o comando TaylorApproximation e / ou inicia o tutor Aproxima\303\247\303\243o Taylor . A sintaxe para implementar o comando TaylorApproximation \303\251 fornecida por este modelo de tarefa. Observe que a configura\303\247\303\243o local da vari\303\241vel Ordem deve ser feita na forma de uma equa\303\247\303\243o cujo lado esquerdo \303\251 a palavra "ordem". Como na Figura 2, "ordem" significa o grau do polin\303\264mio resultante, um uso que entra em conflito com o do comando taylor subjacente. Student[Calculus1][TaylorApproximationTutor](); Ferramenta\342\211\273Tarefas\342\211\273Navegar: C\303\241lculo\342\211\273 Aproxima\303\247\303\243o de Taylor de uma fun\303\247\303\243o univariada

Calcule a aproxima\303\247\303\243o de Taylor de um grau especificado para uma fun\303\247\303\243o univariada. Insira a fun\303\247\303\243o como uma express\303\243o. 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

LCYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kmX194X19HRigiIiItSSRzaW5HRiVGKUYr

Especifique a ordem e, em seguida, calcule a aproxima\303\247\303\243o de Taylor. 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

LC4iIiJGI0kmX194X19HNiJGIyomI0YjIiIjRiMpRiRGKEYjISIiKiYjRiMiIidGIylGJCIiJEYjRioqJiNGIyIjQ0YjKUYkIiIlRiNGIyomI0YjIiQ/IkYjKUYkIiImRiNGIw==

Alternativamente, voc\303\252 pode usar o tutor Taylor Approximation, uma interface de apontar e clicar. Existem duas maneiras de iniciar este tutor.No menu Ferramenta, selecione Tutoriais, C\303\241lculo - Vari\303\241vel \303\232nica,e depois Aproxima\303\247\303\243o de Taylor.Insira a fun\303\247\303\243o em uma vari\303\241vel a ser aproximada abaixo e clique no seguinte bot\303\243o Aproxima\303\247\303\243o de Taylor . try TaylorApproximation_temp := Student:-Calculus1:-TaylorApproximationTutor(DocumentTools:-Do(%TextArea2)); if TaylorApproximation_temp <> NULL then DocumentTools:-SetProperty('Plot3','visible','true'); DocumentTools:-SetProperty('Plot3','value',TaylorApproximation_temp); fi; catch: error end try;

Tabela 2 O modelo de tarefa "Aproxima\303\247\303\243o de Taylor de uma fun\303\247\303\243o univariada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Pressionar o bot\303\243o "Aproxima\303\247\303\243o de Taylor" iniciar\303\241 o tutor Taylor Approximation pr\303\251-carregado com a fun\303\247\303\243o na caixa \303\240 direita do bot\303\243o. Alternativamente, se o pacote Student Calculus 1 for carregado pela primeira vez na mem\303\263ria do Maple, o Menu de Contexto para uma express\303\243o conter\303\241 a op\303\247\303\243o "Tutors" a partir da qual o tutor Aproxima\303\247\303\243o de Taylor pode ser selecionado. Isso seria mais direto do que acessar o tutor a partir do modelo de tarefa. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A Tabela 3 fornece o caminho para um modelo de tarefa que invoca o comando taylor e fornece uma ferramenta interativa para obter expans\303\265es Taylor e polin\303\264mios. \303\211 poss\303\255vel acessar apenas esta ferramenta, que por sua vez fornece acesso ao tutorAproxima\303\247\303\243o de Taylor. Ferramenta\342\211\273Tarefas\342\211\273Navegar: C\303\241lculo\342\211\273Expans\303\243o de Taylor e Polin\303\264mios

Calcule a expans\303\243o Taylor para uma fun\303\247\303\243o univariada sobre um ponto especificado e at\303\251 um order. Insira a fun\303\247\303\243o como uma express\303\243o: 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

LCYqJiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZfX3hfX0dGK0YlRiVGJUYl

Especifique um ponto de expans\303\243o e uma ordem de expans\303\243o e, a seguir, calcule a s\303\251rie de Taylor: 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

Ky9JJl9feF9fRzYiIiIkIiIhIiIjIiIiRihGJyNGKEYlRiUjRigiIzciIiUtSSJPRyUqcHJvdGVjdGVkRzYjRigiIiY=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1 Como alternativa, voc\303\252 pode usar a seguinte interface para calcular a expans\303\243o de Taylor e o polin\303\264mio de Taylor:

Expans\303\265es e polin\303\264mios de Taylor Digite a fun\303\247\303\243o: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRjJGQA== 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2NDc-@=Ll#!#:7$$"1ommY^*o-$!#:$!2B$pYK*zRu#!#:7$$"1baauX?7J!#:$!2ER(e!)zPKG!#:7$$"1ihh@**R#>$!#:$!2D5gI!yC(*G!#:7$$"2(444p1#=F$!#;$!2:d%GHi/TH!#:7$$"2ljjjf/8N$!#;$!2DA**4zU8'H!#:7$$"29444<tBV$!#;$!1:/*[!pGaH!#97$$"1#===]/$=N!#:$!2(=*y@!)o>"H!#:7$$"2%edd<^4(f$!#;$!2a(H!G%**ePG!#:7$$"2DDDDdr[n$!#;$!22;\`%\yEF!#:7$$"2LGGGK8!eP!#;$!2%)4Hn#4liD!#:7$$"273333u4%Q!#;$!2VxS0&QqYB!#:7$$"1pnn(GOa"R!#:$!1;%HysmR5#!#97$$"#S!""$!2dmmmmmmw"!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%%VIEWG6$;$!#S!""$"#S!"";$!)Hp(>"!"($").z*>"!"(-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6%Q"x6"Q"y6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$"$I"!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%SP!""-%.BOUNDS_HEIGHTG6#$"%SV!""-%)CHILDRENG6"G6" use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, a, F; Do(f = %MathContainer5); if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))); end if; F := Student:-Calculus1:-TaylorApproximationTutor(f,a); if F <> NULL then DocumentTools:-SetProperty('Plot4','visible','true'); DocumentTools:-SetProperty('Plot4','value', F); end if; end module; end use; Grau: Ponto Base: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, q, a, F, Q; Do(f = %MathContainer5); q := op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer6','value')))); if is(q,posint) then Q:=q else Q:=5; end if; if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))); end if; F := taylor(f, x=a, Q+1); SetProperty('MathContainer8', 'value', F); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local f, q, Q, a, F; Do(f = %MathContainer5); q := op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer6','value')))); if is(q,posint) then Q:=q else Q:=5; end if; if nops(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))=1 then a:=0 else a := eval(op(2,Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer7','value'))))); end if; F := convert(taylor(f, x=a, Q+1),polynom); SetProperty('MathContainer9', 'value', F); end module; end use; LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() SetProperty('MathContainer5', 'value', NULL); SetProperty('MathContainer6', 'value', NULL); SetProperty('MathContainer8', 'value', NULL); SetProperty('MathContainer7', 'value', NULL); SetProperty('MathContainer9', 'value', NULL); Do(%Plot4=plots[display](plots[textplot]([0,0,""]),axes=none)); end module; end use;

Tabela 3 O modelo de tarefa "Expans\303\243o de Taylor e polin\303\264mios" Para a expans\303\243o da s\303\251rie e o polin\303\264mio de Taylor, o grau do polin\303\264mio \303\251 governado pela op\303\247\303\243o "Grau". Se nenhum ponto de base for fornecido, a ferramenta assume "x = 0". Iniciar o tutor a partir desta ferramenta carrega a fun\303\247\303\243o e o ponto base, mas n\303\243o o grau desejado. Fechar o tutor coloca seu gr\303\241fico na janela de gr\303\241ficos da ferramenta. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH A s\303\251rie de Laurent 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 pode ser obtido com o comando series, ou com o comando laurent do pacote numapprox. O uso \303\251 ilustrado na Tabela 4. 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

KytJInhHNiIiIiIhIiMjRiUiIiciIiEjIiIoIiRnJCIiIy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJA==

JSFH 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

KytJInhHNiIiIiIhIiMjRiUiIiciIiEjIiIoIiRnJCIiIy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJA==

JSFH Table 4 A s\303\251rie Laurent atrav\303\251s dos comandos series e laurent JSFH

JSFH As s\303\251ries de pot\303\252ncias assint\303\263ticas, mesmo as s\303\251ries com pot\303\252ncias fracion\303\241rias, podem ser obtidas em qualquer uma das quatro maneiras listadas na Tabela 5. 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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

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

LCgqKikiIiMjIiIiRiVGJylJI1BpRyUqcHJvdGVjdGVkR0YmISIiLUkkc2luRzYkRipJKF9zeXNsaWJHNiI2IywmSSJ4R0YwRicqJiNGJyIiJUYnRilGJ0YnRicpKiRGM0YrRiZGJ0YnKiwjRiciIilGJ0YkRidGKEYrLUkkY29zR0YuRjFGJylGOCMiIiRGJUYnRistSSJPR0YqNiMqJClGOCMiIiZGJUYnRic=

Tabela 5 S\303\251rie de pot\303\252ncia assint\303\263tica JSFH Na Tabela 5, vemos que os comandos series, taylor e asympt podem ser usados \342\200\213\342\200\213para gerar s\303\251ries de pot\303\252ncias assint\303\263ticas, mesmo s\303\251ries com pot\303\252ncias fracion\303\241rias. O comando asympt assume que o ponto de expans\303\243o \303\251 o ponto no infinito. Tanto o comando taylor quanto o series precisam ser informados de que o ponto de expans\303\243o est\303\241 no infinito. \303\211 ainda poss\303\255vel obter uma expans\303\243o assint\303\263tica alterando as vari\303\241veis \342\200\213\342\200\213de modo 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, expanding about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8, e restaurando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn quando 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. Uma s\303\251rie de pot\303\252ncias assint\303\263ticas, mesmo com pot\303\252ncias fracion\303\241rias, est\303\241 dispon\303\255vel no Menu de Contexto, na op\303\247\303\243o Series\342\211\273Asymptotic Series\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. O di\303\241logo resultante, mostrado na Figura 3, sugere que \303\251 poss\303\255vel obter uma s\303\251rie de pot\303\252ncias assint\303\263ticas interativamente, seja como uma constru\303\247\303\243o de s\303\251rie ou como uma expans\303\243o truncada. 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3 Op\303\247\303\243o de di\303\241logo para menu de contexto: Series\342\211\273Asymptotic Series Infelizmente, o Maple n\303\243o tem uma fun\303\247\303\243o para gerar uma s\303\251rie assint\303\263tica em rela\303\247\303\243o a uma base assint\303\263tica arbitr\303\241ria. No entanto, se 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 \303\251 uma sequ\303\252ncia assint\303\263tica quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJnNyYXJyO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJhRidGL0YyRjVGMkY8, ent\303\243o os coeficientes na expans\303\243o 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 JSFH pode ser obtido recursivamente via 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 JSFH

JSFH O comando mtaylor \303\251 usado para gerar um polin\303\264mio de Taylor para uma fun\303\247\303\243o multivariada. Por exemplo, a expans\303\243o de Taylor de f (x, y), tomada sobre "(x, y) = (a, b)", seria da forma 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 JSFH 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 A Tabela 6 cont\303\251m a expans\303\243o de 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 no ponto 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. 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

LC4jIiImIiIlISIiKiYjIiIiRiVGKUkieEc2IkYpRikqJkYoRilJInlHRitGKUYpKiYjRikiIilGKSksJkYqRilGKUYmIiIjRilGJiooRihGKSwmRi1GKUYzRiZGKUYyRilGJiomRi9GKSlGNUYzRilGJg==

Tabela 6 Expans\303\243o Taylor multivariada de 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 JSFH Mesmo que o Maple n\303\243o tenha uma constru\303\247\303\243o de s\303\251rie para o caso multivariado, a no\303\247\303\243o de "ordem das s\303\251ries" \303\251 usada para determinar o grau do polin\303\264mio resultante. Infelizmente, isso resulta em um polin\303\264mio de grau total um a menos que a "ordem da s\303\251rie" usada como par\303\242metro no comando mtaylor. JSFH Como n\303\243o h\303\241 nenhuma "constru\303\247\303\243o em s\303\251rie" para o caso multivariado, o Maple n\303\243o adiciona um termo de pedido para expans\303\265es geradas pelo comando mtaylor. Por causa disso, posso lembrar que colegas do RHIT ensinavam a seus alunos apenas o comando mtaylor, mesmo no c\303\241lculo de vari\303\241vel \303\272nica. Isso evitaria a necessidade de seguir com a etapa "converter para polin\303\264mio", diminuindo assim a carga de sintaxe nos dias em que todos os c\303\241lculos do Maple dependiam do conhecimento dos comandos apropriados.

JSFH A Figura 4 mostra o tutor Aproxima\303\247\303\243o de Taylor do pacote Student Multivariate Calculus. 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Figura 4 Tutor de c\303\241lculo multivariado de aproxima\303\247\303\243o de Taylor JSFH O polin\303\264mio de Taylor de primeiro grau para a fun\303\247\303\243o 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, tirado 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 \303\251 obtido na Figura 4. Al\303\251m disso, um gr\303\241fico da fun\303\247\303\243o e do polin\303\264mio de Taylor \303\251 desenhado. O gr\303\241fico do polin\303\264mio de Taylor de primeiro grau \303\251 o gr\303\241fico do plano tangente. Na parte inferior do tutor, o comando TaylorApproximation pode ser visto. TA sintaxe deste comando contrasta com a do comando mtaylor e com o comando com nome semelhante do pacote Calculus 1. Observe em particular como o ponto de expans\303\243o \303\251 expresso e observe que o uso expl\303\255cito da equa\303\247\303\243o "ordem = 1" n\303\243o \303\251 necess\303\241rio. Na aus\303\252ncia do par\303\242metro "output = plot", a express\303\243o para o polin\303\264mio de Taylor seria retornada. A Tabela 7 mostra o caminho para um modelo de tarefa que fornece acesso ao comando Taylor Approximation e ao tutor. Ferramenta\342\211\273Tarefas\342\211\273Navegar: C\303\241lculo Multivarieis\342\211\273Aproxima\303\247\303\243o de Taylor de uma fun\303\247\303\243o multivariada

Calcule a aproxima\303\247\303\243o de Taylor de um grau especificado para uma fun\303\247\303\243o multivariada. De entrada na fun\303\247\303\243o. 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

Zio2JEkmX194X19HNiJJJl9feV9fR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiokKUYkIiIkIiIiRi4tSSRzaW5HRiU2I0YmRi5GJUYlRiU=

Especifique um ponto de expans\303\243o e ordem e, a seguir, calcule a aproxima\303\247\303\243o de Taylor. 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

LDAqJiMiIiIiJD8iRiUpSSZfX3lfX0c2IiIiJkYlRiUqJiNGJSIiJ0YlKUYoIiIkRiUhIiIqJkYvRiVJJl9feF9fR0YpRiVGJUYoRiUiIiNGMComRi9GJSksJkYyRiVGJUYwRjNGJUYlKiQpRjZGL0YlRiU=

Alternativamente, voc\303\252 pode usar o tutor da s\303\251rie de Taylor, uma interface de apontar e clicar. Existem duas maneiras de iniciar este tutor.No menu Ferramentas, selecione Tutores, C\303\241lculo - Multi-vari\303\241vel e, em seguida, S\303\251rie de Taylor.nsira uma fun\303\247\303\243o multivari\303\241vel abaixo e clique no seguinte bot\303\243o S\303\251rie de Taylor . try TaylorSeries_temp := Student:-MultivariateCalculus:-TaylorApproximationTutor(DocumentTools:-Do(%TextArea3)); if TaylorSeries_temp <> NULL then DocumentTools:-SetProperty('Plot5','visible','true'); DocumentTools:-SetProperty('Plot5','value',TaylorSeries_temp); fi; catch: error end try;

Tabela 7 O modelo de tarefa "Aproxima\303\247\303\243o de Taylor de uma fun\303\247\303\243o multivariada" JSFH

JSFH Maple tem dois processos separados para produzir a s\303\251rie completa de pot\303\252ncia formal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=. Cada um \303\251 acessado como uma op\303\247\303\243o para o comando convert, mas embora pare\303\247a que haja uma interface comum, as abordagens subjacentes para obter as s\303\251ries de pot\303\252ncias formais s\303\243o diferentes. Na maioria dos casos, qualquer um dos processos funcionar\303\241, mas existem fun\303\247\303\265es para as quais apenas um conseguir\303\241 encontrar a s\303\251rie desejada. A Tabela 8 ilustra as duas funcionalidades relevantes. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJImtHRiciIiItSSpmYWN0b3JpYWxHRiU2IywmKiYiIiNGLUYsRi1GLUYtRi1GKylJInhHRidGMUYtL0YsOyIiIUkpaW5maW5pdHlHRiU=

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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJJF9rMUdGJyIiIilJInhHRicsJiomIiIjRi1GLEYtRi1GLUYtRi0tSSpmYWN0b3JpYWxHRiU2I0YwRisvRiw7IiIhSSlpbmZpbml0eUdGJQ==

Table 8 Obtendo s\303\251ries de pot\303\252ncia formais completas em Maple JSFH Deve ser \303\263bvio pela Tabela 8 que duas funcionalidades diferentes foram ativadas. No resultado \303\240 direita, o \303\255ndice de soma pode ser declarado pelo usu\303\241rio, como mostramos no c\303\241lculo a seguir. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCkhIiJJImtHRiciIiItSSpmYWN0b3JpYWxHRiU2IywmKiYiIiNGLUYsRi1GLUYtRi1GKylJInhHRidGMUYtL0YsOyIiIUkpaW5maW5pdHlHRiU=

JSFH O Menu de Contexto fornece a op\303\247\303\243o S\303\251rie\342\211\273S\303\251rie de Pot\303\252ncia Formal. A caixa de di\303\241logo que aparece \303\251 mostrada na Figura 5. 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5 Caixa de di\303\241logo para a op\303\247\303\243o do menu de contexto: S\303\251rie\342\211\273S\303\251rie de Pot\303\252ncia Formal JSFH Finalmente, observe que pode-se obter uma s\303\251rie de pot\303\252ncias assint\303\263ticas formais completas, conforme ilustrado na Tabela 9. 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJikhIiJJImtHRiciIiIpKiRJInhHRidGKywmRixGLUYtRi1GLS9GLDsiIiFJKWluZmluaXR5R0Yl

Table 9 S\303\251rie de pot\303\252ncia assint\303\263tica formal completa JSFH

JSFH JSFH JSFH JSFH JSFH JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=