Note: Before using the various commands from the Calculus1 package, execute the command with(Student:-Calculus1) (or with(Student[Calculus1])) so that the short form names of the package commands are available.

with(Student:-Calculus1):

infolevel[Student[Calculus1]] := 1:

Rule[sum](Diff(x^2+x*sin(x),x));

Creating problem #1

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=\frac{\ⅆ}{\ⅆx}\left(x^2\right)+\frac{\ⅆ}{\ⅆx}\left(x\sin \left(x\right)\right)$

Rule[power]((1));

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=2x+\frac{\ⅆ}{\ⅆx}\left(x\sin \left(x\right)\right)$

Rule[product]((2));

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=2x+\left(\frac{\ⅆx}{\ⅆx}\right)\sin \left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}\sin \left(x\right)\right)$

The first step in a single-step computation is the entry of an expression which is an inert form of a limit, differentiation, or integration problem, for example, $\underset{x\to 0+}{lim}x\ln \left(x\right)$, $\frac{\ⅆ}{\ⅆx}\left(x^2\right)$, or ${\int }_0^{1}x\sin \left(x\right)\ⅆx$.  In the previous example, the first call to Rule creates a new problem in the package problems table, determines that it is a differentiation problem, and then applies the sum rule for differentiation.

At any time, you can obtain the final answer by using value.  You can also ask to see the complete solution using the ShowSolution command.

You can also undo one or more steps of a problem (back to its original form) by using the Undo command.  This command allows you to experiment, which is useful in the study of integration techniques when it is not obvious which rule to apply.

If you do not know which rule to apply next in a single-step computation, ask for a hint by using the Hint command.  The output from Hint can be used as the rule parameter in a call to Rule.

It is recommended that you use the Maple infolevel facility by setting infolevel[Student] := 1 or infolevel[Student[Calculus1]] := 1. This displays informative messages from Calculus1 routines. In particular, explanatory text describing complicated hints is provided using this mechanism.

Create a new problem.

Int(x*cos(x), x);

$\int x\cos \left(x\right)\ⅆx$

Hint((4));

Creating problem #2

$\left[\mathrm{parts}\,x\,\sin \left(x\right)\,1\,\cos \left(x\right)\,x\right]$

Rule[(5)]((4));

$\int x\cos \left(x\right)\ⅆx=x\sin \left(x\right)-\left(\int \sin \left(x\right)\ⅆx\right)$

At this point you can continue working with this problem, return to the previous problem, get state information about either problem, or start a new problem.

Finish problem #2.

Rule[sin]((6));

$\int x\cos \left(x\right)\ⅆx=x\sin \left(x\right)+\cos \left(x\right)$

Return to problem #1.

GetProblem(1, internal);

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=2x+\left(\frac{\ⅆx}{\ⅆx}\right)\sin \left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}\sin \left(x\right)\right)$

Apply the identity rule.

Rule[identity]((8));

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=2x+\sin \left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}\sin \left(x\right)\right)$

Show the unsolved subproblem.

ShowIncomplete();

$\mathrm{Diff5}=\frac{\ⅆ}{\ⅆx}\sin \left(x\right)$

Apply the sin rule.

Rule[sin]((9));

$\frac{\ⅆ}{\ⅆx}\left(x^2+x\sin \left(x\right)\right)=2x+\sin \left(x\right)+x\cos \left(x\right)$

Use the Show routine to display the current state of problem #2.

Show(2);

$\int x\cos \left(x\right)\ⅆx=x\sin \left(x\right)+\cos \left(x\right)$

Create a new limit problem and apply the addition rule.

Rule[`+`](Limit(x^2+exp(x), x=1));

Creating problem #3

$\underset{x\to 1}{lim}\left(x^2+{\ⅇ}^x\right)=\left(\underset{x\to 1}{lim}{x}^2\right)+\left(\underset{x\to 1}{lim}{\ⅇ}^x\right)$

There are three commands in the Calculus1 package that you can use to obtain information about the current state of any or all problems that are defined in a Maple session: Show, ShowSteps, and ShowIncomplete.  Show is used to display the current state of a problem.  ShowSteps displays all the steps between the initial and current state of the problem. ShowIncomplete is explained in the following paragraphs.

Each problem in the Calculus1 internal problems table is represented in a form that package routines use to determine, for example, when rules are applied or what is displayed.  In particular, each incomplete subproblem of a problem is assigned a label. The ShowIncomplete command displays these subproblems with their labels.

You can refer to a subproblem directly by using its label as the argument to package routines, for example, Rule, Show, ShowIncomplete, and GetProblem.  This focuses the corresponding routine on a particular subproblem and its subproblems, if any. In integration, differentiation, or limit problems, if a rule applies to more than one subproblem of the current problem, use the ShowIncomplete command to select a subproblem. For example:

Int(sin(2*x) + cos(3*x),x);

$\int \left(\sin \left(2x\right)+\cos \left(3x\right)\right)\ⅆx$

Rule[`+`]((14));

Creating problem #4

$\int \left(\sin \left(2x\right)+\cos \left(3x\right)\right)\ⅆx=\int \sin \left(2x\right)\ⅆx+\int \cos \left(3x\right)\ⅆx$

ShowIncomplete();

$\mathrm{Int13}=\int \sin \left(2x\right)\ⅆx$

$\mathrm{Int14}=\int \cos \left(3x\right)\ⅆx$

Rule[change, u=3*x](%Int14);

Applying substitution x = 1/3*u, u = 3*x with dx = 1/3*du, du = 3*dx

$\int \cos \left(3x\right)\ⅆx=\int \frac{\cos \left(u\right)}{3}\ⅆu$

Note: The displays produced by these three commands are printed only; they are not the return values of the commands.  The return values are NULL. This means, for example, that the value of the history variable, %, is not changed by these commands.  (ShowIncomplete takes an option that requests that it return its output as the value of the routine; if this option is given, the history variable is updated.)

Note: Treat subproblem labels as temporary objects because the application of a rule to a problem can change the underlying problem representation, and hence the subproblem labels.  It is recommended that you call ShowIncomplete to verify the value of a label before passing it to a command.

The methods of single-variable calculus computation proceed from simple (for example, the constant rule for limits) to challenging (for example, change of variables for integration).  It would be tedious if you were required to explicitly apply the elementary rules when learning the more advanced methods.

The Calculus1 package provides the Understand command to alleviate this problem.  For example, if you execute the $\mathrm{Understand}\left(\mathrm{Diff}\,\mathrm{constant}\,\mathrm{c*}\,\mathrm{`+`}\right)$ command, whenever you invoke Rule on a differentiation problem it automatically checks whether one or more of the constant, constantmultiple, or sum rules can also be applied.  You can apply just the understood rules to a problem by calling Rule with an empty index (that is, as $\mathrm{Rule}\left[\right]\left(\mathrm{expr}\right)$).

Understand(Diff, constant, `c*`, `+`);

$\mathrm{Diff}=\left[\mathrm{constant}\,\mathrm{constantmultiple}\,\mathrm{sum}\right]$

Create a new problem. The understood rules are automatically applied.

Rule[](Diff(3*sqrt(x)-sin(x), x));

Creating problem #5

$\frac{\ⅆ}{\ⅆx}\left(3\sqrt{x}-\sin \left(x\right)\right)=3\frac{\ⅆ}{\ⅆx}\left(\sqrt{x}\right)-\frac{\ⅆ}{\ⅆx}\sin \left(x\right)$

Create a new problem.

Diff(sqrt(x^2+1), x);

$\frac{\ⅆ}{\ⅆx}\left(\sqrt{x^2+1}\right)$

Request a hint.

Hint((20));

Creating problem #6

$\left[\mathrm{chain}\right]$

Apply the hint.

Rule[(21)]((20));

$\frac{\ⅆ}{\ⅆx}\left(\sqrt{x^2+1}\right)=\left(\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_X0}}\left(\sqrt{\mathrm{\_X0}}\right)\right)}{\phantom{\mathrm{\_X0}=x^2+1}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_X0}}\left(\sqrt{\mathrm{\_X0}}\right)\right)}}{\mathrm{\_X0}=x^2+1}\right)\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)$

Note: The sum and constant rules have been applied to the derivative of the inner expression.

Rule[`^`]((22));

$\frac{\ⅆ}{\ⅆx}\left(\sqrt{x^2+1}\right)=\frac{\frac{\ⅆ}{\ⅆx}\left(x^2\right)}{2\sqrt{x^2+1}}$

For more introductory examples, see Calculus1 Single Stepping Example Worksheet.

The following is a list of the available Student:-Calculus1 single-stepping routines.