Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors. Content of the Precalculus Subpackage

Transcription

1 Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This article provides a systematic exposition of the functionalities available in the Student Precalculus package. The commands and tutors comprising the package are exhaustively listed and detailed. Content of the Precalculus Subpackage Table 1 lists the mathematical functionalities addressed in the Precalculus subpackage of the Student package. The first column describes the functionality; the second lists the available commands; the third indicates if the commands draw graphs ("V" for visualization), or output mathematical calculations ("C" for computational); the fourth column lists the interactive tutors that implement the relevant functionality in a Maplet format; and the last column provides the commnd with which the tutor can be launched from the keyboard. Table 2 in the next section (Launching Tutors) provides alternatives for these commands. Column four in Table 1 uses for the name of the tutor, the term seen in the Tools/Tutors/Precalculus menu. Unfortunately, these terms do not match the titles displayed within the tutors themselves, and these titles often do not match the name embedded in the command that launches the tutor from the keyboard. Hence, Table 1 has its fifth column, no pun intended. Mathematical Task Command TypeInteractive Tutor Keyboard Command Composition of functions CompositionPlot V Function Composition CompositionTutor Graphing conics Conic Sections ConicsTutor Secant line becoming tangent line FunctionSlopePlotV Slopes FunctionSlopeTutor

2 Numeric intuition about limits Equation and graph of a straight line Graphing linear inequalities LimitPlot V Limits LimitTutor Line V, C Lines LineTutor Linear Inequalities LinearInequalitiesTu tor Graph and zeros of polynomials Polynomials PolynomialTutor Graph of rational function and its asymptotes RationalFunctionPV lot Rational Functions RationalFunctionTut or Graphs of elementary functions Standard Functions StandardFunctionsT utor Weighted average CenterOfMass C Distance between points Midpoint of line segment Distance C Midpoint C Slope of a line Slope C Complete the square CompleteSquare C Table 1 Content of the Student Precalculus subpackage. Command types: V = visualization; C = computational The Line command outputs either the equation of a line, or a graph of the line. Launching Tutors Table 2 lists three ways any of the Precalculus tutors can be launched. 1. Launch tutor and type equation 2. Copy equation (even in 2D

3 math) Launch tutor and paste equation 3. Load Student[Precalculus] package Launch context menu on tutor input Select Tutors/Precalculus/... Table 2 Options for launching a tutor A tutor can be launched interactively from the Tools/Tutors menu, or can be launched by the command listed in the last column of Table 1. Executing the command with empty an argument launches the tutor with its default data. Alternatively, provide the input as argument to the command. This is another way to avoid having to enter data in text mode. This is the import of Table 2, namely, ways to supply to the tutors data typed in math mode. If the tutor is launched first from the Tools/Tutors menu, then any data that is entered into the tutor must by typed in text mode. Thus, multiplications must be made explicit, must be entered as Pi, etc. If input to a tutor already exists in math mode, it can be copied and pasted into the tutor. Because the tutor is primal, the copy must be made before the tutor is launched. If the Student[Precalculus] package has been loaded, then the context menu for an appropriate mathematical expression will contain the Tutor option from which all relevant tutors can be accessed and launched. This is usually the most convenient way to launch a tutor. For selected tutors, there are Task Templates that provide access to the tutor and some of its related calculations. These tutors are the Conic Sections Tutor, the Linear Inequalities Tutor, the Rational Functions Tutor, and the Line Tutor. Initializations The SetColors command is available for all the Student subpackages. It sets a color sequence for commands and tutors that use default colors to distinguish various mathematical objects.

4 Composition of Functions To study the composition of two functions such as use the Function Composition tutor. The easiest way to launch this tutor with the functions and already embedded, is via the context menu. As per Table 2, if the Student Precalculus package has already been loaded, then the context menu will provide access to all relevant tutors. Load the Student Precalculus package via the Tools/Load Package menu. The context menu for a sequence of two expressions such as will contain the options Tutors/Function Composition. Figure 1 shows the Function Composition tutor for this pair of expressions. Figure 1 Function Composition tutor for and As per the indicated color-coding, the graph shows in red, in black, in green, and in blue.

5 The graph provided by the tutor shows both compositions simultaneously. The associated CompositionPlot command draws just one of the two possible compositions, with being the default. To obtain the graph of use the syntax Conic Sections The Conic Sections tutor will analyze and graph the conic section determined by its associated quadratic equation. The equation can be given in Cartesian coordinates (using and ) or in polar coordinates using or where or, with either or. In Cartesian coordinates, the quadratic can have an -term, which rotates the conic. Figure 2 shows the Conic Sections tutor applied to the Cartesian equation Figure 2 Conic Sections tutor applied to

6 The totality of the information in the analysis window appears in Figure 3. class: ellipse eccentricity:.723 semimajor axis (a): 4.28 semiminor axis (b): 2.96 latus rectum: 4.09 angle: 3/8*Pi In the xy-plane: vertices: [(8.60,-6.57), (.689,- 3.29)] foci: [(7.50,-6.11), (1.79,- 3.74)] center (h,k): (4.64,-4.93) directrix: y = 2.41*x In the x'y'-plane: vertices: [(-2.78,-10.5), (- 2.78,-1.90)] foci: [(-2.78,-9.27), (-2.78,- 3.08)] center (h',k'): (-2.78,-6.18) directrix: y' = Figure 3 Contents of analysis window of the Conic Sections tutor The graph of the conic is drawn in the -plane. The analysis window (see Figure 3) provides details for the conic as drawn in the -plane, and as it would appear in the - plane where the conic assumes the standard form shown in Figure 2. The eccentricity, major and minor axes, and length of the latus rectum are the same for any orientation of the conic. The coordinates of the center, vertices and foci change with orientation, as does the equation of the directrix. The Task Template "Conic - Analysis and Graph" appears in Figure 4. Pressing the launch button after the equation is dragged, pasted, or typed into the template's math container launches the tutor with the equation embedded. Closing the tutor afterwards writes the graph to the plot window of the template. Analyze a Quadratic Equation Using the

7 Conics Tutor Enter a quadratic equation: Figure 4 Task Template for launching the Conic Sections tutor

8 Secant and Tangent Lines To superimpose secant and tangent lines on the graph of the function use the Function Slope tutor. Launching this tutor from the Context Menu yields Figure 5 in which the default point of contact for the tangent line is at. Figure 5 The Function Slope tutor applied to In general, ten secant lines are drawn, each passing through the point of contact where. The other point coincident with the graph of the function has -coordinate The tangent line is drawn in green, and its equation is given on the right in the tutor. The table of values in the tutor lists the -coordinate common to the curve and secant line, and the slope of the corresponding secant line. Unfortunately, there is no way to control the location or spacing of the secant lines. Figure 6 is created with the FunctionSlopePlot command, as per the display at the bottom of the tutor shown in Figure 5. This command defaults to an animation. Click on the graph to access the animation toolbar, with which the animation of the secant line with intersections at can be activated.

9 > Figure 6 Secant and tangent lines on a graph of > Intuitive Limits To obtain some sense of what the word "limit" means mathematically, apply the Limits tutor to a function such as Figure 7 shows the use of this tutor. The graph in the tutor defaults to an animation in which a point moves along the curve according to the values listed in the tables to its right. If the -coordinate of the point at which the limit is being investigated is, then the neighboring points at which the function is sampled are

10 Figure 7 The Limits tutor applied to The animation in the Limits tutor can be generated by the LimitPlot command, as shown in Figure 8. >

11 Figure 8 Animation illustrating "limit" > Straight Lines To graph and otherwise analyze the line menu. The result is shown in Figure 9., launch the Line tutor via the context

12 Figure 9 The Line tutor applied to the equation The line is graphed, and its equation is rendered in point-slope, two-point, slopeintercept, and general forms. The point-slope form uses the -intercept for the point, while the two-point form uses the -intercept and the point As per the display at the bottom of the tutor, the Line command provides a graph of the line. In addition, the Line command will provide the equation, slope, -intercept, and - intercept if given any of the data listed in Table 3. Table 4 illustrates these uses. point and slope (in any order) two points slope and - intercept (in that order) Table 3 Computational inputs to the Line command Table 4 Examples of the use of the Line command The Line tutor can also be launched by the LineTutor command with any of the arguments in Table 4, or with the equation of a line.

13 Linear Inequalities The LinearInequalitiesTutor shows the feasible region for a set of linear inequalities. It can be accessed via the context menu applied to a sequence, list, or set of up to six linear inequalities. It can also be launched by the LinearInequalitiesTutor command with argument either a set or list of no more than six linear inequalities. Figure 10 shows the default content of this tutor, the feasible region for six linear inequalities. Unchecking any of the inequalities removes them from the set whose feasible region is graphed when the Display button is pressed. Figure 10 Default content of the Linear Inequalities tutor, with feasible region shown in red The graph is actually drawn with the inequal command from the plots package. This command is not restricted in the number of linear inequalities it can resolve, and has numerous options for coloring the feasible and infeasible regions and their boundaries. The Linear Inequalities tutor is simply a more convenient front-end to this command. However, data entry into the tutor must be in text mode. To provide the Linear Inequalities tutor data in math mode, launch it from the context menu applied to inequalities entered in math mode, or use the Task Template Algebra/Graph Linear Inequalities, as per Figure 11. Graph Linear Inequalities

14 Enter up to six linear inequalities separated by commas: Figure 11 The Task Template Algebra/Graph Linear Inequalities

15 The inequalities are entered using math mode, and the tutor launched by pressing the obvious button. When the tutor is closed, the graph it generates is embedded in the box on the right, thus preserving a view of the inequalities and the feasible region. Polynomials The real zeros and graph of a polynomial are provided by the Polynomial tutor, which can be launched by any of the methods in Table 2. Figure 12 shows the Polynomial tutor applied to. When the real zeros are not simple integers, the tutor reverts to floats to express them. They (the -intercepts on the graph) are displayed in the box labeled "Roots". Figure 12 The Polynomial tutor applied to Rational Functions

16 The Rational Functions tutor draws a graph of a rational function - complete with all its asymptotes - and provides the equations of the asymptotes. Figure 13 shows this tutor applied to the function. Figure 13 Rational Functions tutor applied to the function Direct entry of the rational function requires separating the numerator and denominator. Alternate methods of launching the tutor as per Table 2 do not require this separation. Closing the tutor returns just the graph, and the equations of the asymptotes are lost. To compensate, use the Task Template Algebra/Rational Function - Graph and Asymptotes, shown in Figure 14.

17 Rational Function Tutor Enter a rational function Asymptotes Horizontal Plot Oblique Vertical

18 Figure 14 The Task Template Algebra/Rational Function - Graph and Asymptotes In Figure 14, the Rational Function Tutor button was pressed, as was the Asymptotes button. The tutor was launched, and then closed. The graph is written to the Task Template, and the information about asymptotes is likewise preserved.

19 As shown at the bottom of the tutor, the graph is drawn with the RationalFunctionPlot command. Users familiar with the syntax of the plot command and the commands in the plots package will find the syntax of this, and the other visualizaiton commands in the Student package awkward. Before using any of these commands, the reader is advised to look up the syntax on the relevant help page. The paradigms in use for these Student visualization commands is radically different from Maple's older plot commands. For example, in these older commands, the plot range is given by an equation of the form, and the plot window in which the graph appears is trimmed by the view option. In the Student package, the plot range is not given separately, but rather, it is extracted from the view option! Moreover, for the older commands, there is no title by default. To obtain a title, syntax for it must be included. The Student package commands default to a title, and to eliminate the title, the syntax title = "" (the empty string) must be explicitly given! Elementary Functions Given an elementary function, the StandardFunctionsTutor will draw its graph and allow the user to experiment with transformations of the form. Figure 15 shows this tutor applied to the function Figure 15 The Standard Functions tutor applied to curve curve corresponding to, (in red), with the black The tutor defaults to. Changing one or more of these values and pressing the Display button adds the varied curve (in black) to the graph.

20 The graph is drawn with the basic Maple plot command, as shown in the Maple Command window at the bottom of the tutor. The drop-down listing in the window includes the six trig functions and their inverses, the six hyperbolic functions and their inverses, the natural and common logarithmic functions, and the exponential functions. Polynomial functions are not included, and no other functions can be entered into this tutor. Weighted Average The center of mass of a discrete system of particles is the weighted average of their Cartesian coordinates or position vectors. The CenterOfMass command in the Student Precalculus package computes this weighted average using lists (for coordinates) and/or vectors. To give a weight (i.e., mass), enclose the object and its weight in list brackets. If no weights are explicitly given, the weights are assumed to be 1. Table 5 contains examples. The center of mass for uniform masses at and is, computed to the right. The center of mass for masses, respectively located at,,, is, as computed at the right. Note how weights are included and not, and how coordinate and vector notation can be mixed. Table 5 Examples of the CenterOfMass command used to compute the center of mass for discrete systems. Distance between Points The (Euclidean) distance between Cartesian points is obtained with the Distance command, which accepts locations given as points and/or vectors in any number of dimensions. Several examples are listed in Table 6. The distance between and

21 The distance between and Table 6 Examples of the distance between two points computed by the Distance command Midpoint of a Line Segment The coordinates of the midpoint of the line segment connecting the points, are given by As with the CenterOfMass and Distance commands, Cartesian points can be described as lists or vectors. Table 7 illustrates the use of the Midpoint command. Midpoint of the segment connecting and Midpoint of the segment connecting and Table 7 Examples of the midpoint of a line segment computed by the Midpoint command Slope of a Line The slope of the line passing through the points and is given by

22 As with the CenterOfMass, Distance and Midpoint commands, points in the Cartesian plane can be described as lists or vectors. Table 8 gives examples of the use of the Slope command for computing the slope between two planar points. The slope of the line passing through the points is given by and The slope of the line passing through the points and is given by by any of the formalisms at the right Table 8 Computation of the slope by the Slope command Algebraic Completion of the Square The CompleteSquare command will write the quadratic expression. This command can be applied to expressions and equations, to one or more variables, and to functions such as or. Table 9 lists a number of examples of the functioning of this command. as

23 Table 9 The CompleteSquare command illustrated The iterated integrals in the last example in Table 9 appear in gray. This indicates they are the inert form of the integral, corresponding to the Maple Int command. The Maple int command would immediately evaluate the integral, so it is essential that the inert form be used. To set this inert form in math notation, either type Int and use command completion (e.g.,tools/complete Command) or enter the indefinite integral template from the Expression palette, and use the context menu to convert it to its inert form. Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.

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