Stepwise Solutions in Maple An Overview JSFH Robert J. Lopez Emeritus Professor of Mathematics, RHIT Maple Fellow, Maplesoft Three recent articles in the Tips & Techniques series addressed the question of stepwise solutions in Maple. Just what is it that Maple provides by way of stepwise solutions for standard calculations in the mathematical curricula? There are commands, assistants, tutors, and task templates that provide stepwise calculations in precalculus, calculus, linear algebra, and vector calculus. In addition, since Maple can implement nearly any mathematical operation, any stepwise calculation can be reproduced in Maple by assembling the appropriate intermediate steps, just as they would be assembled when working with pencil and paper. This worksheet summarizes the available stepwise tools. Full details and examples of each tool are available in the three articles listed in Table 1. Stepwise Solutions Precalculus and Calculus Linear Algebra Vector Calculus Table 1 Links to Tips & Techniques Reporter articles on stepwise solutions in Maple For each applicable subject area, we list the calculation and the appropriate stepwise tools in the following sections.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 2 lists the two areas in basic algebra for which Maple provides stepwise solutions. Calculation Tool Category Tool Solving Equations Assistant use DocumentTools in ContextMenu:-Interactive:-Equation(); end use; Partial Fraction Decomposition Task Template Stepwise Partial Fraction Decomposition Table 2 Stepwise tools in algebra LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 3 lists the areas in differential calculus for which Maple provides stepwise solutions. Calculation Tool Category Tool Limits Tutor Command Context Menu use DocumentTools in Student[Calculus1][LimitTutor](); end use; ShowSteps Solve\342\211\273Show Solution Steps Derivatives Tutor use DocumentTools in Student[Calculus1][DiffTutor](); end use; Derivatives by Definition Task Template Command Context Menu Derivatives by Definition ShowSteps Solve\342\211\273Show Solution Steps Difference (Newton) Quotient Task Template Tutor Difference (Newton) Quotient use DocumentTools in Student[Calculus1][TangentTutor](); end use; Implicit Differentiation Task Template Implicit Differentiation Tangent Line Task Template Tangent Line Normal Line Task Template Normal Line Mean Value Theorem Task Template Tutor Mean Value Theorem use DocumentTools in Student[Calculus1][MeanValueTheoremTutor](); end use; Rolle's Theorem Task Template Command Rolle's Theorem RollesTheorem Curve Sketching: Intercepts Max/Min Intervals of Increase/Decrease Inflection Points Concavity Tutor Command Task Template use DocumentTools in Student[Calculus1][CurveAnalysisTutor](); end use; FunctionChart (or FunctionPlot) Find Special Points on a Function Table 3 Stepwise tools in differential calculus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 4 lists the areas in integral calculus for which Maple provides stepwise solutions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Calculation Tool Category Tool Riemann Sums Tutor Task Templates use DocumentTools in Student[Calculus1][ApproximateIntTutor](); end use; Riemann Sums Left Right Lower Upper Midpoint Random Numeric Integration Tutor use DocumentTools in Student[Calculus1][ApproximateIntTutor](); end use; Surface Area of a Surface of Revolution Tutor use DocumentTools in Student[Calculus1][SurfaceOfRevolutionTutor](); end use; Volume of a Solid of Revolution Tutor use DocumentTools in Student[Calculus1][VolumeOfRevolutionTutor](); end use; Table 4 Stepwise tools in integral calculus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 5 lists the areas in multivariate calculus for which Maple provides stepwise solutions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Calculation Tool Category Tool Iterated Multiple Integration Command MultiInt Visualizing Regions of Integration Task Templates Cartesian Coordinates: 2D Cartesian Coordinates: 3D Polar Coordinates Cylindrical Coordinates Spherical Coordinates Critical Points and the Second Derivative Test Task Template Critical Points and the Second Derivative Test Center of Mass Task Templates Command Center of Mass Cartesian 2-D Polar Cartesian 3-D Cylindrical Spherical CenterOfMass Table 5 Stepwise tools in multivariate calculus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH Table 6 lists the areas in linear algebra for which Maple provides stepwise solutions. JSFH Calculation Tool Category Tool Row-Reduction to Upper Triangular Form Tutor use DocumentTools in Student[LinearAlgebra][GaussianEliminationTutor](); end use; Row-reduction to Reduced Row-Echelon Form Tutor use DocumentTools in Student[LinearAlgebra][GaussJordanEliminationTutor](); end use; Solve a System of Linear Equations Tutor use DocumentTools in Student[LinearAlgebra][LinearSolveTutor](); end use; Matrix Inversion Tutor use DocumentTools in Student[LinearAlgebra][InverseTutor](); end use; Determine Eigenvalues Tutor use DocumentTools in Student[LinearAlgebra][EigenvaluesTutor](); end use; Determine Eigenvectors Tutor use DocumentTools in Student[LinearAlgebra][EigenvectorsTutor](); end use; Geometric Interpretation of Matrix-Vector Multiplication Task Template Matrix Action 2-D Graphing a Plane and Its Normal Task Template Command Plane Plot PlanePlot Projection of a Vector onto and Orthogonal to another Vector Task Template Projection onto a 1-D Subspace Projection of a Vector onto and Orthogonal to a Plane Task Template Projection onto a 2-D Subspace Visualize the Geometry of Vector Addition Task Template Command Vector Sum Plot VectorSumPlot Table 6 Stepwise tools in linear algebra JSFH

JSFH Table 7 lists the areas in vector calculus for which Maple provides stepwise solutions. JSFH Calculation Tool Category Tool Integrate a Plane Vector Field Task Template Tutor Integrate Planar Vector Field use DocumentTools in Student[VectorCalculus][VectorFieldTutor](); end use; Flux through a Plane Curve Task Templates Command Flux Through: Circle Ellipse Line Segment or Polygonal Line General Plane Curve Flux Flux though a Surface Task Templates Command Flux Through: Box Sphere Parametric Surface Surface Defined over a Disk Surface Defined over an Ellipse Surface Defined over a Triangle Surface Defined over a General Planar Region Flux Surface Integrals Task Templates Command Integrate over: Box Sphere Parametric Surface Surface Defined over a Rectangle Surface Defined over a Disk Surface Defined over an Ellipse Surface Defined over a Triangle Surface Defined over a General Planar Region SurfaceInt Iterated Double Integrals Task Templates Command Maplet Integrate over: Rectangle Disk Ellipse Triangle General Planar Region int Multiple Integral Maplet Iterated Triple Integrals Task Templates Command Maplet Integrate over: Cube Sphere Tetrahedron General 3-D Region int Multiple Integral Maplet Visualize a Space Curve and Its Frenet-Serret Basis Vectors Tutor use DocumentTools in Student[VectorCalculus][SpaceCurveTutor](); end use; Table 7 Stepwise tools in vector calculus JSFH In the VectorCalculus packages, the int command is modified to generate multiple integrals. The special geometric regions that the command then recognizes are then rectangle, disk, ellipse, triangle, cube sphere, tetrahedron, and general planar and three-dimensional regions. The Multiple Integral Maplet is launched from a Task Template of the same name. JSFH

JSFH JSFH JSFH JSFH Solu\303\247\303\265es Stepwise no Maple Uma vis\303\243o geral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Robert J. Lopez Em\303\251rito Professor of Mathematics, RHIT Maple Fellow, Maplesoft Tr\303\252s artigos recentes na s\303\251rie Tips & Techniques abordaram a quest\303\243o das solu\303\247\303\265es passo a passo no Maple. O que o Maple fornece por meio de solu\303\247\303\265es graduais para c\303\241lculos padr\303\243o nos curr\303\255culos de matem\303\241tica? Existem comandos, assistentes, tutores e modelos de tarefas que fornecem c\303\241lculos graduais em pr\303\251-c\303\241lculo, c\303\241lculo, \303\241lgebra linear e c\303\241lculo vetorial. Al\303\251m disso, uma vez que o Maple pode implementar quase qualquer opera\303\247\303\243o matem\303\241tica, qualquer c\303\241lculo passo a passo pode ser reproduzido no Maple pela montagem das etapas intermedi\303\241rias apropriadas, da mesma forma que seriam montadas ao trabalhar com l\303\241pis e papel. Esta planilha resume as ferramentas graduais dispon\303\255veis. Detalhes completos e exemplos de cada ferramenta est\303\243o dispon\303\255veis nos tr\303\252s artigos listados na Tabela 1. Solu\303\247\303\265es Stepwise Precalculus and Calculus Linear Algebra Vector Calculus Tabela1 Links para artigos do rep\303\263rter de dicas e t\303\251cnicas sobre solu\303\247\303\265es graduais no Maple Para cada \303\241rea de assunto aplic\303\241vel, listamos o c\303\241lculo e as ferramentas passo a passo apropriadas nas se\303\247\303\265es a seguir.

JSFH A Tabela 2 lista as duas \303\241reas da \303\241lgebra b\303\241sica para as quais o Maple fornece solu\303\247\303\265es graduais. C\303\241lculo Categoria de ferramenta Ferramenta Resolvendo Equa\303\247\303\265es Assistente use DocumentTools in ContextMenu:-Interactive:-Equation(); end use; Decomposi\303\247\303\243o parcial da fra\303\247\303\243o Modelo de Tarefa Stepwise Partial Fraction Decomposition Tabela 2 Ferramentas passo a passo em \303\241lgebra JSFH

JSFH A Tabela 3 lista as \303\241reas de c\303\241lculo diferencial para as quais o Maple fornece solu\303\247\303\265es graduais. C\303\241lculo Categoria de ferramenta Ferramenta Limite Tutor Comando Menu contextual use DocumentTools in Student[Calculus1][LimitTutor](); end use; Mostrar etapas Resolver\342\211\273Mostrar etapas da solu\303\247\303\243o Derivada Tutor use DocumentTools in Student[Calculus1][DiffTutor](); end use; Derivados por defini\303\247\303\243o Modelo de Tarefa Comando Context Menu Derivatives by Definition ShowSteps Resolver\342\211\273Mostrar etapas da solu\303\247\303\243o Quociente de diferen\303\247a (Newton) Modelo de Tarefa Tutor Difference (Newton) Quotient use DocumentTools in Student[Calculus1][TangentTutor](); end use; Diferencia\303\247\303\243o impl\303\255cita Modelo de Tarefa Implicit Differentiation Reta tangente Modelo de Tarefa Tangent Line Reta normal Modelo de Tarefa Normal Line Teorema do valor m\303\251dio Modelo de Tarefa Tutor Mean Value Theorem use DocumentTools in Student[Calculus1][MeanValueTheoremTutor](); end use; Teorema de Rolle Modelo de Tarefa Comando Rolle's Theorem RollesTheorem Esbo\303\247o de curva: Interceptar M\303\241ximo/M\303\255nimo Intervalos de aumento / diminui\303\247\303\243o Pontos de Inflex\303\243o Concavidade Tutor Comando Modelo de Tarefa use DocumentTools in Student[Calculus1][CurveAnalysisTutor](); end use; FunctionChart (ou FunctionPlot) Find Special Points on a Function Tabela 3 Ferramentas passo a passo em c\303\241lculo diferencial JSFH

JSFH Table 4 lists the areas in integral calculus for which Maple provides stepwise solutions. JSFH C\303\241lculo Categoria de ferramenta Ferramenta Soma de Riemann Tutor Modelos de Tarefa use DocumentTools in Student[Calculus1][ApproximateIntTutor](); end use; Soma de Riemann Left Right Lower Upper Midpoint Random Integra\303\247\303\243o Num\303\251rica Tutor use DocumentTools in Student[Calculus1][ApproximateIntTutor](); end use; \303\201rea de superf\303\255cie de uma superf\303\255cie de revolu\303\247\303\243o Tutor use DocumentTools in Student[Calculus1][SurfaceOfRevolutionTutor](); end use; Volume de um s\303\263lido de revolu\303\247\303\243o Tutor use DocumentTools in Student[Calculus1][VolumeOfRevolutionTutor](); end use; Tabela 4 Ferramentas passo a passo em c\303\241lculo integral JSFH

JSFH A Tabela 5 lista as \303\241reas em c\303\241lculos multivariados para as quais o Maple fornece solu\303\247\303\265es passo a passo. JSFH C\303\241lculo Categoria de ferramenta Ferramenta Integra\303\247\303\243o M\303\272ltipla Iterada Comando MultiInt Visualizando Regi\303\265es de Integra\303\247\303\243o Modelos de Tarefa Cartesian Coordinates: 2D Cartesian Coordinates: 3D Polar Coordinates Cylindrical Coordinates Spherical Coordinates Critical Points and the Second Derivative Test Modelos de Tarefa Critical Points and the Second Derivative Test Centro de massa Modelos de Tarefa Comando Centro de massa Cartesian 2-D Polar Cartesian 3-D Cylindrical Spherical CenterOfMass Tabela 5 Ferramentas passo a passo em c\303\241lculo multivariado JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A Tabela 6 lista as \303\241reas da \303\241lgebra linear para as quais o Maple fornece solu\303\247\303\265es graduais. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= C\303\241lculo Categoria da ferramente Ferramenta Redu\303\247\303\243o de linha para a forma triangular superior Tutor use DocumentTools in Student[LinearAlgebra][GaussianEliminationTutor](); end use; Redu\303\247\303\243o de linha para forma escalonada de linha reduzida Tutor use DocumentTools in Student[LinearAlgebra][GaussJordanEliminationTutor](); end use; Resolva um sistema de equa\303\247\303\265es lineares Tutor use DocumentTools in Student[LinearAlgebra][LinearSolveTutor](); end use; Invers\303\243o de matriz Tutor use DocumentTools in Student[LinearAlgebra][InverseTutor](); end use; Determinar os autovalores Tutor use DocumentTools in Student[LinearAlgebra][EigenvaluesTutor](); end use; Determine os autovetores Tutor use DocumentTools in Student[LinearAlgebra][EigenvectorsTutor](); end use; Interpreta\303\247\303\243o geom\303\251trica da multiplica\303\247\303\243o do vetor-matriz Modelo de Tarefa Matrix Action 2-D Representando graficamente um plano e seu normal Modelo de Tarefa Comando Plane Plot PlanePlot Proje\303\247\303\243o de um vetor e ortogonal a outro vetor Modelo de Tarefa Projection onto a 1-D Subspace Proje\303\247\303\243o de um vetor e ortogonal a um plano Modelo de Tarefa Projection onto a 2-D Subspace Visualize a geometria da adi\303\247\303\243o de vetores Modelo de Tarefa Comando Vector Sum Plot VectorSumPlot Table 6 Ferramentas passo a passo em \303\241lgebra linear LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Table 7 lists the areas in vector calculus for which Maple provides stepwise solutions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= C\303\241lculo Categoria da ferramenta Ferramenta Integrar um campo vetorial plano Modelo de Tarefa Tutor Integrate Planar Vector Field use DocumentTools in Student[VectorCalculus][VectorFieldTutor](); end use; Fluxo atrav\303\251s de uma curva plana Modelo de Tarefa Comando Fluxo atrav\303\251s: Circle Ellipse Line Segment or Polygonal Line General Plane Curve Flux Fluxo atrav\303\251s de uma superf\303\255cie Modelo Tarefa Comando Fluxo atrav\303\251s: Box Sphere Parametric Surface Surface Defined over a Disk Surface Defined over an Ellipse Surface Defined over a Triangle Surface Defined over a General Planar Region Flux Integrais de superf\303\255cie Modelo Tarefa Comando Integrar em: Box Sphere Parametric Surface Surface Defined over a Rectangle Surface Defined over a Disk Surface Defined over an Ellipse Surface Defined over a Triangle Surface Defined over a General Planar Region SurfaceInt Integrais Duplas Interadas Modelo Tarefa Comando Maplet Integrar em: Rectangle Disk Ellipse Triangle General Planar Region int Multiple Integral Maplet Integrais Triplas Interadas Modelo Tarefa Comando Maplet Integrate over: Cube Sphere Tetrahedron General 3-D Region int Multiple Integral Maplet Visualize uma curva de espa\303\247o e seus vetores base Frenet-Serret Tutor use DocumentTools in Student[VectorCalculus][SpaceCurveTutor](); end use; Tabela 7 Ferramentas passo a passo em c\303\241lculo vetorial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= No VectorCalculus pacotes, o comando int \303\251 modificado para gerar m\303\272ltiplos integrais. As regi\303\265es geom\303\251tricas especiais que o comando reconhece s\303\243o ent\303\243o ret\303\242ngulo, disco, elipse, tri\303\242ngulo, esfera c\303\272bica, tetraedro e regi\303\265es planas e tridimensionais gerais. O Multiple Integral Maplet \303\251 iniciado a partir de um Modelo de Tarefa com o mesmo nome. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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