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layout="Title" style="Title"><Font executable="false">New Tools for Educators and Students in Maple 9.5</Font></Text-field></Input></Group><Group><Input><Text-field firstindent="0.0" layout="Author" leftmargin="0.0" linebreak="space" rightmargin="0.0" style="Author"><Font bold="false" encoding="ISO8859-1" executable="false" foreground="[0,0,0]" italic="false" subscript="false" superscript="false" underline="false">\251 Maplesoft, a division of Waterloo Maple Inc., 2004</Font></Text-field></Input></Group><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field></Title><Group><Input><Text-field layout="Normal" style="Text">Maple 9.5 offers more tools to help educators effectively deliver course material, and to facilitate and increase students' understanding of mathematical and engineering concepts.  The following Maple enhancements are highlighted::</Text-field><Text-field layout="Normal" style="Text"><Font encoding="UTF-8">	\342\200\242 New Mutivariate Calculus Student package</Font></Text-field><Text-field layout="Normal" style="Text"><Font encoding="UTF-8">	\342\200\242 Updated Precaculus Student Package</Font></Text-field><Text-field layout="Normal" style="Text"><Font encoding="UTF-8">	\342\200\242 Access to the interactive tutors from the </Font><Font bold="true">Tools</Font> menu</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field bookmark="ex1" layout="Heading 1" style="Heading 1">Multivariate Calculus</Text-field></Title><Text-field layout="Normal" style="Text">The new <Font bold="true">MultivariateCalculus</Font> subpackage of the <Font bold="true">Student </Font>package will help teachers present and students understand the basic material of multivariate calculus. There are three principal components to the package: interactive tutors, visualization routines and computation routines. The tutors act as graphical point-and-click interfaces to the other <Font bold="true">MultivariateCalculus</Font> routines</Text-field><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">restart: interface(warnlevel=0):
interface(displayprecision=4):
with( Student[MultivariateCalculus] ):</Text-field></Input></Group><Section collapsed="true"><Title><Text-field layout="Heading 2" style="Heading 2"><Font executable="false">Integrals in Multivariate Calculus</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">MultiInt</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine of the <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">MultivariateCalculus</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> provides an interface for the construction or evaluation of multiple integrals.</Font></Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">MultiInt( x^2*y, x=0..4, y=x-1..x+1, output = integral );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">MultiInt( x^2*y, x=0..4, y=x-1..x+1 );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">MultiInt( x^2+y^2+z, x=2..4, y=x-1..x+6, z=-2..4+y^2, output = integral );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">MultiInt( x^2+y^2+z, x=2..4, y=x-1..x+6, z=-2..4+y^2 );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 2" style="Heading 2"><Font executable="false">Integral Approximations</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">Multivariate integral approximations using Riemann sums can be observed using the <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">ApproximateInt</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> and <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">ApproximateIntTutor</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routines.  The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">ApproximateIntTutor</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine provides a Maplet application interface to the <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">ApproximateInt</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine.</Font></Font></Font></Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateIntTutor( );
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leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateIntTutor( x^3-y^2+x*y, y=-4..4 );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateIntTutor( r, r=0..2, theta=0..Pi/2, coordinates=polar );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">ApproximateInt</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine allows you to plot, animate, or obtain the value of an integral approximation, without using the Maplet application interface.</Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateInt( r, r=0..5, t=0..Pi/2, 
	coordinates=polar, output=plot, showfunction=false, prismoptions=[color=green], method=upper );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">The following command demonstrates an animation of approximations to the integral of the implicit function f( x, y) = x^2 + y^2. over the region [-4..4, -6..6] using a midpoint Reimann sum. See the animation by clicking on the graphic, then choosing play from the buttons that appear on the top panel. You can also change the speed of the animation in frames per second, or view the animation frame by frame.</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateInt( x^2+y^2,x=-4..4,y=-6..6, output = animation, partition = [6,6], frames=2..7 );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">ApproximateInt( x^2+y^2, x=-4..4, y=-6..6, output = value, partition = [6,6] );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 2" style="Heading 2"><Font executable="false">Taylor Approximations</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">MultivariateCalculus</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> package routine <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">TaylorApproximation</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> calculates, plots, or animates Taylor approximations to multivariate functions.  The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">TaylorApproximationTutor</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine provides an interface to the <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">TaylorApproximation</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine.</Font></Font></Font></Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">TaylorApproximationTutor( ):
</Text-field></Input></Group><Group><Input><Text-field alignment="centred"><Image height="568" 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leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">TaylorApproximationTutor( sin( x+y), [x,y]=[1,2], 8 );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">TaylorApproximation</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> routine allows you to calculate, plot, or animate the Taylor approximation, without using the Maplet application interface.</Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">TaylorApproximation( sin( x+y), [x,y]=[1,0], 5 );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">The following command displays an animation of the Taylor Approximation of sin( x + y). You can rotate the plot to have a better view of the animation. </Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">TaylorApproximation( sin( x+y), [x,y]=[1,0], 5, output=animation );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 2" style="Heading 2"><Font executable="false">Lagrange Multipliers</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">LagrangeMultipliers</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> command calculates or plots the local minima, local maxima, and saddle points of a function subject to constraints, using the method of Lagrange multipliers.</Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">LagrangeMultipliers( x*y, [x^2/8+y^2/2-1], [x,y] );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">LagrangeMultipliers( x*y, [x^2/8+y^2/2-1], [x,y], output=plot );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">The following command finds the local minima, maxima, or saddle points of the function x*y. The output option is used to display a plot showing the objective function, the constraints, the solution points in pink dots, and the level curves of the objective function through those solution points. </Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">LagrangeMultipliers( x*y, [x^2/8+y^2/2-1], [x,y], output=plot, showlevelcurves = false );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 2" style="Heading 2"><Font executable="false">Applications</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">MultivariateCalculus</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> package provides routines to calculate quantities such as function average, surface area, and center of mass, using multivariate calculus.</Font></Font></Text-field></Input></Group><Section collapsed="true"><Title><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" rightmargin="0.0" spaceabove="8.0" spacebelow="2.0"><Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" size="14" underline="false">Function Average</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">FunctionAverage</Font></Font></Text-field><Text-field alignment="left" leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> command returns the integral used to determine the average of a function over a region or calculates the average of the function over the region.</Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">FunctionAverage( x^2+y^2, x=0..1, y=0..sqrt( 1-x^2), output=integral );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">FunctionAverage( x^2+y^2, x=0..1, y=0..sqrt( 1-x^2) );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">FunctionAverage( r^2, r=0..1, t=0..Pi/2, coordinates=polar[r,t], output=integral );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" rightmargin="0.0" spaceabove="8.0" spacebelow="2.0"><Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" size="14" underline="false">Surface Area</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">SurfaceArea</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> command plots, constructs the integral for, or calculates the surface area of, a curve within a region. </Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">SurfaceArea( x^2+y^2, x=-2..2, y=-1..1, output=integral );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">SurfaceArea( x^2+y, x=-2..2, y=-1..1 );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" rightmargin="0.0" spaceabove="8.0" spacebelow="2.0"><Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" size="14" underline="false">Center of Mass</Font></Text-field></Title><Group><Input><Text-field leftmargin="0.0" rightmargin="0.0"><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">The <Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" underline="false">CenterOfMass</Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false"> command plots, constructs the integral for, or calculates the center of mass of a curve within a region. The plot option <Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false">specifies that a plot displays, which shows the region, the density function, and the center of mass as a green dot. </Font></Font></Font></Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">CenterOfMass( x^2+y^2, x=1..4, y=-1..6, output=plot );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">CenterOfMass( x^2+y^2, x=1..4, y=-1..6, output=integral );</Text-field></Input></Group><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">CenterOfMass( x^2+y^2, x=1..4, y=-1..6 );</Text-field></Input></Group></Section></Section><Text-field layout="Normal" style="Text"/></Section><Section collapsed="true"><Title><Text-field bookmark="ex2" layout="Heading 1" style="Heading 1">Precalculus</Text-field></Title><Text-field layout="Normal" style="Text">Another improved sub-package in the <Font bold="true">Student</Font> package is the <Font bold="true">Precalculus</Font> sub-package, which now includes more functions for visual explorations of topics in greater detail. The package has added visualization commands that allow exploration of concepts in greater depth. The basic<Font bold="true"> Line</Font> command computes lines from various different forms of input and returns information about slope and intercepts, as well as the line itself. Also, the visualization routines provide tools to create plots related to precalculus concepts. They allow you to see the geometric interpretation of these concepts. </Text-field><Text-field layout="Normal" style="Text"/><Group><Input><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">restart;</Text-field><Text-field leftmargin="0.0" prompt="&gt; " rightmargin="0.0" style="Maple Input">with( Student[Precalculus] ):</Text-field></Input></Group><Section collapsed="true"><Title><Text-field bookmark="examples" layout="Heading 2" style="Heading 2">Line</Text-field></Title><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Line( -1, 2 );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Line( [1,1], -1/2)[1];</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Line( [1,1/2],&lt;4,1&gt; );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Line( -1, 2, output=plot, lineoptions=[color=green] );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field bookmark="examples" layout="Heading 2" style="Heading 2">CompositionPlot</Text-field></Title><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">CompositionPlot( x^3+2, 1/x^2 );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">CompositionPlot( x^2, x+5, view=[-10..6, -2..10], function2options=[color=blue] );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field bookmark="examples" layout="Heading 2" style="Heading 2">FunctionSlopePlot</Text-field></Title><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">FunctionSlopePlot( x^3+3*x^2+x-3, x=2 );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">FunctionSlopePlot( x^2, x=2, view=[-10..10, -10..10], animation=true, pointoptions=[color=blue] );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field bookmark="examples" layout="Heading 2" style="Heading 2">RationalFunctionPlot</Text-field></Title><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">RationalFunctionPlot( ( x^3+x^2-x+1)/( x^3) );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">RationalFunctionPlot( 1/x^2, verticalasymptoteoptions=[color=red], functionoptions=[color=blue] );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input"/></Input></Group></Section></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Expanded Tools Menu</Text-field></Title><Group><Input><Text-field layout="Normal" style="Text">Another improvement in Maple 9.5 is the expanded <Font bold="true">Tools</Font> menu, which provides easy access to 40 interactive tutors that guide students through mathematical explorations in precalculus, calculus, multivariate calculus, and linear algebra. 
</Text-field></Input></Group></Section><Group><Input><Text-field layout="Normal" style="Text"><Font italic="true">
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.</Font>
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