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An Introduction to MapleV, Release 5.1

Douglas B. Meade ( meade@math.sc.edu )

Plotting

In this section we will learn different methods of entering functions in Maple. Then, the user interface will be used to create and manipulate a graph of the function. Unless someone has a better choice, let's use the function .

> f := x/2 + sin(x) + exp(-x/6);

> smartplot(1/2*x+sin(x)+exp(-1/6*x));

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3D Plots

Here is a brief sample of 3D plots. For further details and examples, please consult the on-line help for plot3d and the plots package.

> plot3d( sin(x*y), x=-Pi..Pi, y=-Pi..Pi );

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> plot3d([x*sin(x)*cos(y),x*cos(x)*cos(y),x*sin(y)],x=0..2*Pi,y=0..Pi);

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Calculus

Derivatives and Antiderivatives

In this section we will emphasize the graphical aspects of differential calculus. The general outline is to use the GUI to: plot the function, construct the derivative and antiderivative, and then add these functions to the original graph. Let's begin by recalling the definition of the function.

> f;

> 1/4*x^2-cos(x)-6*exp(-1/6*x);

> 1/2+cos(x)-1/6*exp(-1/6*x);

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> smartplot(1/2*x+sin(x)+exp(-1/6*x));

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Definite Integrals & Riemann Sums

There are a number of nice tools that can be used to enhance the discussion of integral calculus. The Maple spreadsheet is a new addition in Release 5; the graphical and symbolic tools have existed for much longer, but are still very effective. Let's begin by examining the definite integral: . where f(x) is the function introduced earlier.

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Using a Spreadsheet to Create a Table of Approximate Values for a Definite Integral

> f;

> x0 := 0;

> x1 := Pi;

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Graphical View

> f, x0, x1;

Method 1: Explicit Construction of Sequence of Plots

> R0 := rightbox( f, x=x0..x1, 1 ):

> R1 := rightbox( f, x=x0..x1, 2 ):

> R2 := rightbox( f, x=x0..x1, 4 ):

> R3 := rightbox( f, x=x0..x1, 8 ):

> R4 := rightbox( f, x=x0..x1, 16 ):

> display( [R0,R1,R2,R3,R4], insequence=true );

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Method 2: Programmed Construction of Sequence of Plots

> Nframe := 10;

> LL := plot( f, x=x0..x1 ):

> for n from 0 to Nframe do

> LL := LL, leftbox( f, x=x0..x1, 2^n );

> od:

> display( LL, insequence=true );

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Symbolic and Numeric Evaluation

Now that we have some ideas about how the definite integral is defined in terms of approximate sums, let's turn our attention to an explicit evaluation of the integral. The context menus provide a simple means to construct the definite integral of from to . This integral can now be evaluated either symbolically or numerically.

> f;

> F := Int(1/2*x+sin(x)+exp(-1/6*x),x = x0 .. x1);

> F := 1/4*exp(-1/6*Pi)*Pi^2*exp(1/6*Pi)+exp(-1/6*Pi)*exp(1/6*Pi)-6*exp(-1/6*Pi)+7;

> simplify( % );

> evalf( % );

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Differential Equations

The DEtools package (and the odeplot command in the plots package) are very useful when preparing graphical demonstrations about differential equations.

> with( DEtools );

First-Order Differential Equation

Consider the first-order ordinary differential equation

> ODE := diff( x(t), t ) + 2*x(t) = 1+sin(t)+exp(-2*t);

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The general solution to this problem (or to the related IVP) can be found using the context menu - or by direct use of the dsolve command.

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> R5 := dsolve(ODE,{x(t)});

> dsolve( { ODE, x(0)=alpha }, x(t) );

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Personally, I believe the direction field is more informative:

> DEplot( convert(ODE,piecewise), x(t), t=0..5, [[0,1],[0,2],[0,-5],[0,5]], arrows=THIN );

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Second-Order Differential Equation

Consider the second-order linear ODE with constant coefficients and a piecewise-defined forcing term:

> ODE2 := diff( x(t), t,t ) + x(t) = t + (5-t)*Heaviside( t-5 );

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Prior to constructing a solution, it might be useful to create a plot of the forcing function.

> plot( rhs(ODE2), t=0..10 );

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Now, the solution can be obtained "directly" and then massaged into a more reasonable form:

> dsolve( ODE2, x(t) );

> combine( % );

> collect( %, Heaviside(t-5) );

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A more natural means of obtaining the solution is to use Laplace transforms:

> dsolve( ODE2, x(t), method=laplace );

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First-Order System of Differential Equations

> with( plots, odeplot );

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Here we illustrate another method of creating a graphical view of the solution to ODEs by considering a familiar first-order system of ODEs:

> sys := diff(y(x),x)= z(x),

> diff(z(x),x)=-y(x);

> ic := y(0)=0, z(0)=1;

> fcns := { y(x), z(x) };

> p:= dsolve( {sys,ic}, fcns, type=numeric );

> odeplot( p, [x,y(x)], -4..4 );

> odeplot( p, [ [x,y(x)], [x,z(x)] ], -4..4 );

> odeplot( p, [ [x,y(x)], [x,z(x)], [y(x),z(x)], [x,y(x)^2+z(x)^2] ], -4..4, scaling=constrained );

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The exact solution can, of course, be found:

> dsolve( { sys, ic }, fcns );

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