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Explorations in Calculus

Introduction

Over 15 interactive tutors are included in the Student[Calculus1] subpackage to help students gain insight into first year Calculus concepts. These tutors make it easier for students, who may not be familiar with Maple syntax, to learn the concepts presented in a Calculus 1 course. Virtually every visualization and single-stepping command that is available in the Calculus package is available in an interactive tutor.



The tutors provided in the Student[Calculus1] package can be accessed by:

 1 Launching the tutor of choice from the Tools > Tutors menu and typing the expression.
 2 Loading the Student[Calculus1] package using the with command (or by selecting the package from the Tools > Load Package menu). After the package is loaded, you can access the tutorials from the Context Panel for the expression of interest under Student[Calculus1] > Tutors.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$



Single-stepping through Calculus Problems

Single-stepping with Tutors

The DiffTutor is an interactive tutor to walk through the stages of differentiation problems.  While walking through a problem, you can apply a rule, ask for a hint to the problem, display a full solution, or jump to the final answer all with the simple click of buttons.  To get further clarification to the particular rules, select the rule from the Rule Definition menu.  You can control what rules the Tutor will use: if students already understand a specific rule, select that rule from the Understood Rules menu.

$\mathrm{DiffTutor}\left(\mathrm{ln}\left(x\right)\mathrm{sin}\left(x\right)\right)$



 Figure 1:  DiffTutor steps you through a problem

Interactive tutors are also available for integration and limit problems using the IntTutor and LimitTutor respectively.  The following shows the results single-stepping through an indefinite integration problem.  To walk through the this integration problem with definite integration, such as  int(csch(x),x=.5..1.5), simply enter the integration limits .5 and 1.5 at the top of the tutor and press Start to start over.  See ?DiffTutor, ?IntTutor and ?LimitTutor for more information about the tutors.

$\mathrm{IntTutor}\left(\mathrm{csch}\left(x\right)\right)$



 Figure 2:  IntTutor steps you through a problem

In the step-by-step tutors, you can ask for a hint.  In the following example, the LimitTutor is used for the limit $\underset{x\to -3}{lim}\frac{{x}^{2}+x-6}{x+3}$.

In the Limit Tutor, click Get Hint.  The message is "Hint: is there any way of simplifying the numerator?"  You can then choose Factor or Next Step to do the factoring of the expression.

Record the Steps and Rules Applied for Each Step

The ShowSolution command will show all the steps to solve a problem, as well as record the rule or method used for each step.  The ShowSolution command can be used on a single-variable differentiation, limit, or integration problem.

$\mathrm{ShowSolution}\left(\frac{{ⅆ}}{{ⅆ}x}\left(x\cdot \mathrm{sin}\left(x\right)\right)\right)$

 $\begin{array}{cccc}\multicolumn{4}{c}{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({x}{}{\mathrm{sin}}{}\left({x}\right)\right)}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}{\mathrm{sin}}{}\left({x}\right){+}{x}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{sin}{}\left({x}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{product}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\mathrm{sin}}{}\left({x}\right){+}{x}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{sin}{}\left({x}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{identity}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\mathrm{sin}}{}\left({x}\right){+}{x}{}{\mathrm{cos}}{}\left({x}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{sin}}\right]\end{array}$ (1)



$\mathrm{ShowSolution}\left(\underset{x\to 9}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{{x}^{2}-81}{\sqrt{x}-3}\right)$

 $\begin{array}{cccc}\multicolumn{4}{c}{\underset{{x}{\to }{9}}{{lim}}{}\frac{{{x}}^{{2}}{-}{81}}{\sqrt{{x}}{-}{3}}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \underset{{x}{\to }{9}}{{lim}}{}{4}{}{{x}}^{{3}}{{2}}}\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{lhopital}}{,}{{x}}^{{2}}{-}{81}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {4}{}\left(\underset{{x}{\to }{9}}{{lim}}{}{{x}}^{{3}}{{2}}}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{constantmultiple}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {4}{}{\left(\underset{{x}{\to }{9}}{{lim}}{}{x}\right)}^{{3}}{{2}}}\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{power}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {36}{}\sqrt{{9}}\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{identity}}\right]\end{array}$ (2)



 $\begin{array}{cccc}\multicolumn{4}{c}{{\int }\frac{{{ⅇ}}^{{x}}}{{{ⅇ}}^{{x}}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\int }\frac{{1}}{{u}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{change}}{,}{u}{=}{{ⅇ}}^{{x}}{,}{u}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\int }\frac{{1}}{{\mathrm{u1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{u1}}\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{change}}{,}{\mathrm{u1}}{=}{u}{-}{1}{,}{\mathrm{u1}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\mathrm{ln}}{}\left({\mathrm{u1}}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{power}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\mathrm{ln}}{}\left({u}{-}{1}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{revert}}\right]\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {\mathrm{ln}}{}\left({{ⅇ}}^{{x}}{-}{1}\right)\hfill & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left[{\mathrm{revert}}\right]\end{array}$ (3)



The derivative, limit, and integral entered in these examples appear in gray.  This indicates they are the inert form of the functions, corresponding to the Maple Diff, Limit, and Int commands.  Normally, Maple command would immediately evaluate the derivative (or limit or integral) before passing it to the ShowSolution command, so it is essential that the inert form be used.  To set this inert form in math notation, either type Diff and use command completion (e.g.,Tools > Complete Command) or enter the differentiation template from the Expression palette, and use the Context Panel to convert it to its inert form (e.g. 2-D Math>Convert To>Inert Form).

You can control what rules are understood, enabling you to control the level of detail, from elementary rules to more advanced methods.

For more information, see ShowSolution and Single Step Computations.



Learning with Graphs in Interactive Tutors

Numerous tutors also exist for the various visualization commands available in the Calculus1 package.  Not only is the concept conveyed graphically, but the corresponding integrals, numerical approximations, and required syntax to display this image into the Maple worksheet are also available.

Curve Analysis

The CurveAnalysisTutor command launches a tutor interface that analyzes the graph of $f$ on the interval $\left[a,b\right]$.

$\mathrm{CurveAnalysisTutor}\left(\right)$

 Figure 3: Curve Analysis Tutor

Tangent and Secant

The TangentSecantTutor command launches a tutor interface that shows how the secant line between point $a$ and point $b$ on the graph of a function approaches a tangent line, as $b$ approaches $a$.

$\mathrm{TangentSecantTutor}\left({x}^{3}-2,x=1\right)$



 Figure 4:  The Tangent Secant Slope Tutor applied to ${\mathbit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}$

Mean Value Theorem

The MeanValueTheoremTutor command launches a tutor interface that illustrates the Mean Value Theorem applied to a function on an interval.  Change the interval and see the results.

$\mathrm{MeanValueTheoremTutor}\left(\frac{1}{x},x=0.5..2.5\right)$



 Figure 5:   The Mean Value Theorem Tutor applied to $\frac{\mathbf{1}}{\mathbit{x}}$ on the interval [0.5, 2.5].

Approximate Integration

The ApproximateIntTutor command launches a tutor interface that computes, plots, and animates approximated definite integrals of a function on an interval .

$\mathrm{ApproximateIntTutor}\left(2x\mathrm{sin}\left(x\right),x=0..2\mathrm{\pi }\right)$



 Figure 6:  Visualize different methods of approximating an integral

Volume of Revolution

The VolumeOfRevolutionTutor displays the plot, interval, and area of the volume of revolution of a function where the axis of revolution can be either the horizontal or vertical axis.  The corresponding Maple command is displayed in the bottom of the maplet application.  Copy this command from the maplet application and paste the result into a Maple worksheet.

You can also view the graph of the Riemann sum approximation for specified number of partitions.

$\mathrm{VolumeOfRevolutionTutor}\left(\mathrm{sin}\left(x\right)x,x=-10..10\right)$



 Figure 7: Demonstrate the volume of revolution

Taylor Approximation

The TaylorApproximationTutor displays the Taylor Approximation of a function and allows you to animate the approximation.

$\mathrm{TaylorApproximationTutor}\left(\mathrm{tanh}\left(x\right)\right)$



 Figure 8:  Taylor Approximation Tutor for $\mathbf{tanh}\left(\mathbit{x}\right)$

 See Also