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Explorations in Precalculus

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Introduction

Composition of Functions

Conic Sections

Secant and Tangent Lines

Intuitive Limits

Straight Lines

Linear Inequalities

Polynomials

Rational Functions

Elementary Functions

Weighted Average

Distance between Points

Midpoint of a Line Segment

Slope of a Line

Algebraic Completion of the Square

See Also

Introduction

The Student[Precalculus] package enables the exploration of topics from a precalculus class. The tutors available in the package provide students with the ability to visualize the composition of functions, secant lines approaching a tangent line, numeric intuition about limits, equation and graph of a straight line, graph of a rational function with asymptotes, graphs of systems of linear inequalities, graphs and zeros of polynomials, graphs of elementary functions. In addition, the package enables students to compute the center of mass, distance, midpoint, slope, line, and complete the square.

 

See Student[Precalculus] for more information on all the commands and tutors available in this package.

 

The Student[Precalculus] package comes with four tutors: the Conic Sections Tutor, the Linear Inequalities Tutor, the Rational Functions Tutor, and the Line Tutor. These tutors can be accessed by:

 

1. 

Launching the tutor of choice from the Tools > Tutors menu and typing the equation.

2. 

Loading the Student[Precalculus1] 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[Precalculus1] > Tutors.

 

withStudentPrecalculus:

 

    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.

 

StudentSetColorsred, black,green,blue:

 

Composition of Functions

To study the composition of two functions such as

 

x2,x+1

 

use the Function Composition tutor. The easiest way to launch this tutor with the functions fx and gx already embedded, is via the Context Panel. If the Student[Precalculus] package has already been loaded, then the Context Panel will provide access to all relevant tutors.

 

The Context Panel for a sequence of two expressions such as x2,x+1 will contain the options Student Precalculus > Tutors > Function Composition. Figure 1 shows the Function Composition tutor for this pair of expressions.

Figure 1: Function Composition Tutor for fx=x2 and gx=x+1

As per the indicated color-coding, the graph shows fx=x2 in red, gx=x+1 in black, gfx=x2+1 in green, and fgx=x+12 in blue.

The graph provided by the tutor shows both compositions simultaneously. The associated CompositionPlot command draws just one of the two possible compositions, with gfx being the default. To obtain the graph of fgx,use the syntax

 

 

CompositionPlotx2, x+1, compositiontype='fg'

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 x and y) or in polar coordinates using r=gt or gt,where gt=constant or gt=ab+c dt, with dt either sint or cost. In Cartesian coordinates, the quadratic can have an xy-term, which rotates the conic.

 

Figure 2 shows the Conic Sections tutor applied to the Cartesian equation. To open the tutor, from the Context Panel for the equation select Student Precalculus  > Tutors > Conic Sections.

 

3x2+2xy+5y218x+40y+93=0 

 

Figure 2: Conic Sections Tutor applied to 3x2+2xy+5y218x+40y+93=0

 

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-.665

-----------------------------

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' = -.255

Figure 3: Contents of analysis window of the Conic Sections Tutor

 

The graph of the conic is drawn in the xy-plane. The analysis window (see Figure 3) provides details for the conic as drawn in the xy-plane, and as it would appear in the x'y'-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", located in Tools > Tasks > Browse > Algebra > 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 Conics Tutor

Enter a quadratic equation:

Figure 4: Task template for launching the Conic Sections Tutor

Secant and Tangent Lines

To superimpose secant and tangent lines on the graph of the function

3 x2+x3

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 x=1.

 

Figure 5: The Function Slope Tutor applied to 3 x2+x3

In general, ten secant lines are drawn, each passing through the point of contact where x=a. The other point coincident with the graph of the function has x-coordinate a ±52k,k=0,,4. 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 x-coordinate common to the curve and secant line, and the slope of the corresponding secant line.

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 x=a ±5/2k,k=0,,4,can be activated.

 

Click the Play button  and watch the animation. To step through the frames of the animation, click .

FunctionSlopePlot3x2+x3,1,'view'=4.50..6.50,16.4..26.

Figure 6: Secant and tangent lines on a graph of fx=3 x2+x3

Intuitive Limits

To obtain some sense of what the word "limit" means mathematically, apply the Limits tutor to a function such as sinx.

 

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 x-coordinate of the point at which the limit is being investigated is x=a, then the neighboring points at which the function is sampled are a ±52k,k=0,,4.

For instance, you can apply the Limits tutor to

sinx

Figure 7: The Limits Tutor applied to sinx

The animation in the Limits tutor can be generated by the LimitPlot command, as shown in Figure 8. Click on the plot to access the animation toolbar.

LimitPlotsinx,x=0

Figure 8: Animation illustrating "limit"

Straight Lines

To graph and otherwise analyze the line y=2 x3, launch the Line tutor via the Context Panel. The result is shown in Figure 9.

Figure 9: The Line Tutor applied to the equation y=2 x3

 

The line is graphed, and its equation is rendered in point-slope, two-point, slope-intercept, and general forms. The point-slope form uses the y-intercept for the point, while the two-point form uses the y-intercept and the point 1,y1.

The Line command, shown at the bottom of the tutor, provides a graph of the line. In addition, the Line command can be used to obtain the equation, slope, y-intercept, and x-intercept if given any of the data listed in Table 1. Table 2 illustrates these uses.

 

1. 

 point and slope (in any order)

2. 

 two points

3. 

 slope and y-intercept (in that order)

Line4,5,2;Line1,1,4,5;Line2,3

y=2x3,2,3,32

(1)

Table 1: Computational inputs to the Line command

Table 2: 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.

Linear Inequalities

The Linear Inequalities tutor shows the feasible region for a set of linear inequalities. It can be accessed via the Context Panel 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. Clearing the check box beside an inequality removes it 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 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.

 

 

 

You can launch this tutor from the Context Panel applied to inequalities entered in math mode, or use the Task Template Tools > Tasks > Browse > Algebra > Graph Linear Inequalities, as per Figure 11.

 

Graph Linear Inequalities

Enter up to six linear inequalities separated by commas:

 

Figure 11: The Task Template Graph Linear Inequalities

 

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 x36 x2+7 x13. When the real zeros are not simple integers, the tutor reverts to floats to express them. They (the x-intercepts on the graph) are displayed in the box labeled "Roots". To access the Tutor from Context Panel,  from the Context Panel for the equation select Student Precalculus  > Tutors > Polynomials.

Figure 12: The Polynomial Tutor applied to x36 x2+7 x13

 

Rational Functions

The Rational Function 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 fx=x3+x2x+1x12. To access the Tutor from the menu, select Tools > Tutors > Precalculus > Rational Functions.

Figure 13: Rational Functions Tutor applied to the function fx=x3+x2x+1x12

Direct entry of the rational function requires separating the numerator and denominator. The alternative methods of launching the tutor as per Table 2 do not require this separation.

 

Closing the tutor returns just the graph. To get the equations of the asymptotes, use the Task Template Tools > Tasks > Browse > Algebra > Rational Function - Graph and Asymptotes, shown in Figure 14.

Rational Function Tutor

Enter a rational function PxQx:

 

       

 

Asymptotes

 

Horizontal

 

 

Oblique

 

 

Vertical

 

 

Plot

           

 

Figure 14: The Task Template Rational Function - Graph and Asymptotes

Using the Task Template, you can enter the Rational Function tutor. When you close the tutor, the graph is returned. When you click Asymptotes, the information about asymptotes is likewise preserved.

Note: As shown at the bottom of the tutor, the graph is drawn with the RationalFunctionPlot command. For more information on this command and its options, see RationalFunctionPlot.

Elementary Functions

Explore the graphs of elementary functions. The Standard Functions tutor lets you graph any of these elementary functions: the six trigonometric functions and their inverses, the six hyperbolic trigonometric functions and their inverses, the natural and common logarithmic functions, and the exponential functions. To access this tutor from the menu, select Tools > Tutors > Precalculus > Standard Functions.

 

Given an elementary function fx, draw its graph and experiment with transformations of the form afbx+c+d. Figure 15 shows this tutor applied to the function sinx.

 

Figure 15 The Standard Functions Tutor applied to sinx, (in red), with the black curve curve corresponding to a horizontal shift of sinx

The tutor defaults to a=b=1,h=k=0. Changing one or more of these values and pressing the Display button adds the transformed curve (in black) to the graph.

 

The graph is drawn with the basic Maple plot command, as shown in the Maple Command window at the bottom of the 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 uses this to computes the weighted average using lists (for coordinates) or vectors. To give a weight (or, a mass), enclose the object and its weight in list brackets. If no weights are explicitly given, the weights are assumed to be 1. Table 3 contains examples.

The center of mass for uniform masses at a,b and c,d is a+c2,b+d2, computed to the right.

CenterOfMassa,b,c,d

12c+12a,12d+12b

(2)

The center of mass for masses 110,1,12, respectively located at 0,1,2, 1,2,3, 1,3,0, is 516,94,2, as computed at the right. Note how weights are included and not, and how coordinate and vector notation can be mixed.

CenterOfMass0,1,2,110,1,2,3,1,3,0,12

516,94,2

(3)

Table 3 : 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 4.

 

The distance between a,b and c,d

Distancea,b,c,d

a22ac+c2+b22bd+d2

(4)

The distance between 1,2,3 and 3,5,7

Distance1,2,3,3,5,7

41

(5)

Table 4: 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 xk,yk,zk,k=1,2, are given by

 

xm,ym,zm=x1+x22,y1+y22,z1+z22 

As with the CenterOfMass and Distance commands, Cartesian points can be described as lists or vectors. Table 5 illustrates the use of the Midpoint command.

Midpoint of the segment connecting P1=a,b and P2=c,d

Midpointa,b,c,d

12c+12a,12d+12b

(6)

Midpoint of the segment connecting P1=2,3,4 and P2=5,7,8

Midpoint2,3,4,5,7,8

32,5,6

(7)

Table 5: 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 P1=x1,y1 and P2=x2,y2 is given by

 

m=y2y1x2x1 

As with the CenterOfMass, Distance and Midpoint commands, points in the Cartesian plane can be described as lists or vectors. Table 6 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 P1=x1,y1 and P2=x2,y2 is given by

Slopex2,y2,x1,y1

y2y1x2x1

(8)

The slope of the line passing through the points P1=3,5 and P2=7,2 is given by any of the formalisms at the right

Slope3,5,7,2; Slope3,5,7,2; Slope3,5,7,2

710

(9)

Table 6: Computation of the slope by the Slope command

 

Algebraic Completion of the Square

The CompleteSquare command will write the quadratic expression a x2+b x+c as axb2 a2+cb24 a. This command can be applied to expressions and equations, to one or more variables, and to functions such as sinx or y'x. Table 7 lists a number of examples of the functioning of this command.

Q1 3 x2+2 x: 

CompleteSquareQ1

3x+13213

(10)

Q2 x2 1y3+2 x y2y: 

CompleteSquareQ2

1y3x+y21y32yy41y3

(11)

Q31sin2t+2 sint+1:

CompleteSquareQ3;CompleteSquareQ3,sint

1sint+12

(12)

Q4x2x=y2+2 y+3:

CompleteSquareQ4

x12214=y+12+2

(13)

Q5u2+2 uv23 v:

CompleteSquareQ5 ⅆv ⅆu

∫∫u+12v322941v32294ⅆvⅆu

(14)

Table 7: The CompleteSquare command illustrated

Note: To execute this table, select the entire table and press the execute button, .

 

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. Normally, the Maple int command would immediately evaluate the integral before passing it to the CompleteSquare command, 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 (Tools > Complete Command), or enter the indefinite integral template from the Calculus palette, and use the Context Panel to convert it to its inert form (2-D Math > Convert To > Inert Form).

See Also

 inequal, SetColors, Student[Precalculus], Tutors in Student[Precalculus], Using Tasks

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