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 singlevariable
calculus
. There are three principal components to the subpackage: interactive, visualization, and singlestep computation. These components are described in the following sections. General commands that are not related to visualization, interactive, or singlestep 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 contextmenu. 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 singlevariable calculus, and in particular, realvariable 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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${\∫}\frac{1}{x}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}{\ⅆ}x\=\mathrm{ln}\left(x\right)$, not $\mathrm{ln}\left(\x\\right)$. The latter answer, often found in calculus textbooks, is actually a shorthand representation of two different antiderivatives of $\frac{1}{x}$: $\mathrm{ln}\left(x\right)$ if $0<x$ and $\mathrm{ln}\left(x\right)$ if $x<0$. While convenient in the context of singlevariable 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 $\mathrm{ln}\left(x\right)$ when appropriate for a definite integral.

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The general power function, ${x}^{r}$ where $r$ is a given real number, is realvalued only for $0<x$ unless $r$ is an integer. This follows from the fact that the power function is multivalued when $r$ is not an integer. Maple implements what is known as the principal branch of this function. For negative $x$ and noninteger $r$, this principal branch is complexvalued. This can conflict with your expectations for some common functions, for example, the cube root function ${x}^{1\/3}$. 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 realvariable 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, ${\∫}{x}^{2}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}{\ⅆ}x\=\frac{1}{3}{x}^{3}\+C$. 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.


The visualization commands are:



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.


The interactive commands are:



SingleStep Computation



To help you study the techniques of computation in singlevariable 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 singlestep 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.


The commands available for use in singlestep computation are:



Additional Commands



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


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



Getting Help with a Command in the Package



