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It discusses graphical interpretations of convergence and divergence properties and uses tools including Dynamic System Geogebra (DSG) and CAS Maple to relate related function classes and their behavior in two- and three-dimensional spaces. Finally, the work argues that such visualization supports cognition in academic teaching and learning.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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\nFrancisco Regis Vieira Alves  \nAbstract: Admittedly, the notion of generalized integrals in one parameter has a fundamental role.  \nEn virtue that, in this paper, we discuss and characterize an approach for to promote the visualization of this scientific mathematical concept. We still indicate the possibilities of graphical interpretation of formal properties related to notion of convergence/divergence. Moreover, supported by technology, we show the relations between the class of functions z 􀀠 f (x, y) andy 􀀠 f (t, x) related to the two and three-dimensional spaces. At the end, we propose a visual criterion for the recognition and perception of graphical-dynamic properties provided by the Dynamic System Geogebra – DSG and the Computer Algebraic System – CAS Maple. Finally, our perspective indicates the visualization as an important cognitive component for the teaching and the learning at the academic level.  \nKey words: Visual criterion, Integrals in one parameter, Geogebra, CAS Maple.  \n1. Introduction  \nWe start our discussion by an illustrative example. With this attention, we consider the function f (x, y) 􀀠 (2x 􀀎 y3 )2 . From the Multivariable Calculus, we know that its graph lies in theIR3 (Tall, 1986) . Traditionally, we can consider two kinds of restrictions. In fact, we can take the following expression g(t, y) 􀀠 (2t 􀀎 y3 )2 . By the same argument, we can consider too h(x, t) 􀀠 (2x 􀀎 t3 )2 . These two restriction allow us to analyze its graphical behavior in the IR2 . Infact, we show some properties in the figure 1.  \nFigure 1. Descrption of the graph´s behavior in the two dimensional space and it´s restrictions  \nWhen we restrict our action to the analytical treatment, we can still  \nconsider 􀂳10f  \n(t, x)dx 􀀠 􀂳10 (2x 􀀎 t3 )dx 􀀠 􀂪􀂫􀂬 ~~ ~~(2x~~ ~~􀀎6~~ ~~t~~ ~~3~~ ~~)3~~ ~~ 􀂺􀂼􀂻 xx􀀠􀀠10 􀀠 43 􀀎 2t3 􀀎 t6 􀀠 h(t) . On the other  \nhand, en virtue that we considered only integrable functions, we can visualize the graphical-geometric behavior corresponding the two restriction aforementioned. In the figure 1, we can understand that under the restriction [0,1] 􀂍 IR we observe a limited image for both restrictions. We still indicate the numerical behavior of the corresponding areas provided by Fundamental Theorem of Calculus – FTA.  \nAdmittedly, the abstract character involved in this last analytical argumentation deserves a careful attention from the universities teachers (Tall, 1991) . In fact, if we carefully analyze these last results, we note that h(t) represents a function of t , which makes sense since the integrand depends on t . We must observe that we integrate over x and are left with something that depends only on t , and not depends of the variable x . We explore the technology with the aim of highlighting certain qualitative characters, such as: the convergence of generalized integrals; the behavior of the contributions of area. In the next section we discuss the notion of integrals that depend on a parameter t . We establish some formal properties related to the class of functions denoted by f (t, x) . Moreover, the Dynamic System Geogebra-DSG will provide to us the visualization.  \n2. Preliminary properties and some examples  \nLet us consider the class of the functions F (t) : t 􀂳ba f (t, x)dx or F (t) : t 􀂳􀀎􀁦a f (t, x)dx , where t 􀂏 I 􀂍 IR . We will indicate the two main theorems ofthis section. With these two theorems, we can conclude, from the formal point of view, something about the convergence behavior of each integral related to the function F(t) .  \nTheorem 1: Consider the function F(t) : t 􀂳ba f (t, x)dx , where  \nf  \n: I 􀁵 [a, b) 􀁯 IR (t, x) f(t,x)  \nand  \nexist 􀁍 such that: f (t, x) 􀁤􀁍(x) , for 􀀅t 􀂏 I, 􀀅x 􀂏 [a, b) . Moreover, we still have 􀂳ba􀁍(x)dx 􀀟􀁦 .  \nIn these terms, we conclude that F (t) : t 􀂳ba f (t, x)dx is continuous.  \nTheorem 2: For n􀂏IN , consider the function 􀁷~~ ~~nf􀁷t(tn,~~ ~~x) continue over I 􀁵 ","cbCaihT7Ftbodq41","https://ap.wps.com/l/cbCaihT7Ftbodq41","pdf",1415960,12,"English","# 1. Introduction\n## Illustrative examples and restrictions\n# 2. Preliminary properties and some examples","[{\"question\":\"What is the main goal of the paper regarding integrals with one parameter?\",\"answer\":\"To discuss and characterize an approach that promotes visualization of generalized integrals in one parameter and to connect this visualization with convergence/divergence behavior.\"},{\"question\":\"Which tools are used to support the visualization and graphical interpretation?\",\"answer\":\"The paper uses Dynamic System Geogebra (DSG) and the Computer Algebraic System Maple to visualize graphical-dynamic properties and related function relations.\"},{\"question\":\"How do the preliminary theorems help analyze convergence behavior?\",\"answer\":\"Theorem 1 establishes continuity for integrals of the form F(t)=∫_a^b f(t,x)dx under a set of conditions, while Theorem 2 provides differentiability and higher-derivative relations for F(t) when conditions on n-th derivatives hold.\"}]","VISUAL CRITERION FOR UNDERSTANDING THE NOTION OF CONVERGENCE IF INTEGRALS IN ONE PARAMETER - for teaching and learning | PDF",1792065288]