Overview of the Student[Calculus1] Subpackage - Maple Help

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Overview of the Student[Calculus1] Subpackage

Calling Sequence

Student[Calculus1][command](arguments)

command(arguments)

Description

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The Student[Calculus1] subpackage is designed to help teachers present and students understand the basic material of a standard first course in single-variable calculus .  There are three principal components to the subpackage: interactive, visualization, and single-step computation.  These components are described in the following sections. General commands that are not related to visualization, interactive, or single-step computation are described in the Additional Commands section.

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This subpackage has associated tutors. To access Student[Calculus1] tutors, from the main menu select the Tools>Tutors>Calculus.

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Note: Throughout the help pages for the Calculus1 subpackage, the terms function and expression are generally used interchangeably, and refer to the mathematical objects that can be manipulated using the operations of calculus.  The routines in the subpackage are referred to using the terms routine or command.

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Each command in the Student[Calculus1] subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.

  

As the underlying implementation of the Student[Calculus1] subpackage is a module, it is also possible to use the form Student[Calculus1]:-command or Student:-Calculus1:-command to access a command. For more information,  see Module Members.

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The Maple Command Completion facility is helpful for entering the names of Student package commands.

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Many of the commands and tutors in the Student[Calculus1] package can be accessed through the context-menu. These commands are consolidated under the Student[Calculus1] name.

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Note on boolean options: Many of the commands in the Student[Calculus1] subpackage take boolean options, that is, options of the form option_name = value, where value is true or false.  These options are used, for example, to control which components are included in a plot.  In all cases, the form option_name =true can be abbreviated as option_name.

Calculus1 in the Maple System

  

While the focus of the Calculus1 subpackage is single-variable calculus, and in particular, real-variable calculus, it exists as a part of the Maple system, whose normal domain of computation is the field of complex numbers.  The routines in the Calculus1 subpackage compute only over the real domain and do not accept input expressions that contain complex numbers. However, it is also a goal to introduce as few conflicts with the main Maple system as possible.

  

This has resulted in the following consequences.

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&int;1x&DifferentialD;x&equals;lnx, not ln&verbar;x&verbar;.  The latter answer, often found in calculus textbooks, is actually a shorthand representation of two different antiderivatives of 1x: lnx if 0<x and lnx if x<0.  While convenient in the context of single-variable calculus, this form is of questionable value in general. For example, it creates difficulties when studying the integration theory of complex numbers.  The Rule routine returns lnx when appropriate for a definite integral.

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The general power function, xr where r is a given real number, is real-valued only for 0<x unless r is an integer.  This follows from the fact that the power function is multi-valued when r is not an integer. Maple implements what is known as the principal branch of this function.  For negative x and non-integer r, this principal branch is complex-valued.  This can conflict with your expectations for some common functions, for example, the cube root function x1&sol;3.  If you plot the cube root function over the interval 5..5 using the Maple plot command, you see only the part of the graph on the positive interval 0..5 because the rest of the graph consists of complex numbers.  To avoid this situation, Maple has another power function, surd, that returns a real value whenever possible.

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Certain transformations that are valid for real-variable calculus are not valid for general complex numbers. Therefore, Maple may not perform them. If Maple does not perform an operation (for example, by calling simplify) that is valid for your expression, use the assuming operator.

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The antiderivative, or indefinite integral, of a function is an entire class of functions, whose members differ by an additive constant.  For example, &int;x2&DifferentialD;x&equals;13x3&plus;C. Maple, in general, does not explicitly include the constant C. For consistency, the Calculus1 integration rules also omit the constant.

Visualization

  

The visualization routines are designed to assist in the understanding of basic calculus concepts, theorems, and computations.  These routines normally produce a Maple plot, and most can optionally return one or more symbolic representations of the studied quantity.

  

You have considerable control over the presentation of plots produced by the visualization routines.  The display of each object included in the plot can be adjusted by using a corresponding option in the calling sequence.  See the help pages for the individual commands for details.

  

For a more in-depth introduction to the visualization routines in this subpackage, see Student[Calculus1][VisualizationOverview].

  

The visualization commands are:

AntiderivativePlot

ApproximateInt

ArcLength

DerivativePlot

FunctionAverage

FunctionChart

InversePlot

MeanValueTheorem

NewtonQuotient

NewtonsMethod

PointInterpolation

RiemannSum

RollesTheorem

SurfaceOfRevolution

Tangent

TaylorApproximation

VolumeOfRevolution

 

 

 

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For introductory examples, see Calculus1 Visualization Example Worksheet.

Interactive

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The interactive routines use the Maple Maplet technology to assist you to work through the standard problems of calculus in a visually directed manner.  These commands display one or more dialog boxes allowing you to plot a function and change the various plot options.

  

The DiffTutor, IntTutor, and LimitTutor display dialog boxes presenting the current state of the problem you are working on and a set of controls which let you move to the next step in the solution of that problem.

  

For more information on this functionality, see Student[Calculus1][InteractiveOverview].

  

The interactive commands are:

AntiderivativeTutor

ApproximateIntTutor

ArcLengthTutor

CurveAnalysisTutor

DerivativeTutor

DiffTutor

FunctionAverageTutor

IntTutor

InverseTutor

LimitTutor

MeanValueTheoremTutor

NewtonsMethodTutor

SurfaceOfRevolutionTutor

TangentSecantTutor

TangentTutor

TaylorApproximationTutor

VolumeOfRevolutionTutor

 

 

 

Single-Step Computation

  

To help you study the techniques of computation in single-variable calculus, the Calculus1 subpackage provides facilities to proceed in small steps through calculations.  For example, you can request that a differentiation problem be solved one step at a time, where you specify the differentiation rule applied at each step.

  

The operations that can be stepped through in this manner are: limit, differentiation, and integration.

  

At any time during a single-step computation, you can ask for a hint about the next step to take, which you can then apply to the problem.

  

The subpackage maintains a list of all the problems that you have stepped through in this manner in the current session. You can ask for a review of any or all of these problems.

  

For a more in-depth introduction to single-step computation using this subpackage, see Student[Calculus1][SingleStepOverview].

  

The commands available for use in single-step computation are:

Clear

GetMessage

GetNumProblems

GetProblem

Hint

Rule

Show

ShowIncomplete

ShowSolution

ShowSteps

Understand

Undo

WhatProblem

 

 

 

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For introductory examples, see Calculus1 Single Stepping Example Worksheet.

Additional Commands

  

Some additional commands are available for obtaining information about functions or expressions.

Asymptotes

CriticalPoints

ExtremePoints

InflectionPoints

Integrand

Roots

Summand

 

  

Some additional pages that explain conventions and the implementation of common Calculus methods and theorems  are available.

Boole's Rule

Differentiation Rules

Integration Rules

Limit Rules

Newton-Cotes Formula

Simpson's Rule

Simpson's 3/8 Rule

Trapezoidal Rule

Getting Help with a Command in the Package

  

To display the help page for a particular Student[Calculus1] command, see Getting Help with a Command in a Package.

See Also

Calculus1 Single Stepping Example Worksheet, Calculus1 Visualization Example Worksheet, Command Completion, Student, Student[Calculus1][SingleStepOverview], Student[Calculus1][VisualizationOverview]


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