Math 173

Calculus I

Spring 2005

Dr. Constant J. Goutziers

Department of Mathematics, Computer Science and Statistics

goutzicj@oneonta.edu

Lesson 1

Maple Basics

Maple is an interactive mathematical software package which combines, Numerical Computation, Symbolic Computation and Graphics.

Initializations

>    restart;
with(plots): 

Warning, the name changecoords has been redefined

1.1 Numerical Computation

All arithmetic is exact, unless we explicitly ask for a decimal approximation using the evalf  command.

Examples

Example 1.1.1
Rational numbers, large integers, trigonometric values and decimal approximations.

>    n1:=(2/7)^4;

n1 := 16/2401

>    n2:=evalf(n1);

n2 := .6663890046e-2

>    n3:=73491097^6;

n3 := 157546362627224319599041649399034062211350126929

>    n4:=cos(Pi/6);

n4 := 1/2*3^(1/2)

>    n5:=evalf(n4);

n5 := .8660254040

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Example 1.1.2
Decimal approximations of arbitrary length.

Decimal approximations of arbitrary length can be obtained by specifying the desired number of significant digits as an option in the evalf  command.  The default is ten significant digits.

>    n6:=evalf(n4, 30);

n6 := .866025403784438646763723170755

Example 1.1.3
Non-trivial trigonometric values

Some non-trivial trigonometric values can be found by using the convert to radical routine.

>    n7:=cos(Pi/24);

n7 := cos(1/24*Pi)

>    n8:=convert(n7, radical);

n8 := 1/4*(8+2*6^(1/2)+2*2^(1/2))^(1/2)

Exercises

Exercise 1.1.1

Compute the exact value as well as a decimal approximation of sin(Pi/12) .

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1.2 Graphics

Maple contains an extensive array of plotting routines, most of which can be found in the plots  package which is loaded using the with(plots)  command which is conveniently placed in the Initializations group at the top of this worksheet.

Examples

Example 1.2.1
The plot  command.

>    plot(sin(x)/x, x=-10..10);

[Maple Plot]

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Example 1.2.2
The color  option with the plot  command.

We visualize the curve x^3-2*x^2  and color it blue.

>    plot(x^3-2*x^2, x=-2..3, color=blue);

[Maple Plot]

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Example 1.2.3
Limiting the Range of a plot.

The rational expression (3*x^2+5*x-4)/(x^2-9)   is not defined at x = 3  and x = -3 .  Moreover when x  is close to either one of these values the absolute value of the expression becomes extremely large.  In order to circumvent this complication, we limit the range of the plot with the option y = -10 .. 10 .

>    plot((3*x^2+5*x-4)/(x^2-9), x=-7..7, y=-10..10);

[Maple Plot]

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Example 1.2.4
Multiple plots in one picture.

We can create multiple plots within one frame by using Maple set notation { ... }  or list notation [ ... ] .

>    plot({x^2, 3-2*x}, x=-4..3);

[Maple Plot]

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Set notation will work with any plot routine within Maple, list notation is only implemented for some.  The advantage of list notation is that it allows for control of the color of individual curves.

>    plot([x^2, 3-2*x], x=-4..3, color=[blue, red]);

[Maple Plot]

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Example 1.2.5
Code and display the function

  [Maple OLE 2.0 Object]

Use the piecewise  command.

>    f:=piecewise(x<1, 1, x>=1, 3-x);

f := PIECEWISE([1, x < 1],[3-x, 1 <= x])

Observe that this function "jumps" at the point   x = 1 .  We say that it is discontinuous at 1 .  Maple can be informed about this discontinuity by adding the discont = true  option to the plot command.

>    plot(f, x=-4..4, discont=true);

[Maple Plot]

Example 1.2.6
Three dimensional images.

Maple can create three-dimensional graphs.  Here is the picture of a saddlepoint surface.

>    plot3d(x^2-y^2, x=-5..5, y=-5..5, style=patch, axes=boxed);

[Maple Plot]

Exercises

Exercise 1.2.1

Sketch the graph of f(x) = x^4-x^2  and use magenta as the color of the curve.

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Exercise 1.2.2

Sketch the graph of x*sin(1/x)  . Make sure that you choose the domain in such a way that the characteristics of the function around x = 0  are clearly displayed.

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1.3  Symbolics

The real power of Maple lies in its symbolic capabilities.  A computer algebra system allows the user to execute mathematical computations on the computer screen much like such they used to be performed with pencil and paper.  This section explores some of the capabilities of the program.

Examples

Example 1.3.1
This example illustrates the use of the expand, factor  and sort   commands.

>    e1:=(2*x-3*x^2)^2*(4*x-5)^3;

e1 := (2*x-3*x^2)^2*(4*x-5)^3

>    e2:=expand(e1);

e2 := 5836*x^5-5685*x^4+2700*x^3-500*x^2-2928*x^6+576*x^7

>    e3:=sort(e2, x);

e3 := 576*x^7-2928*x^6+5836*x^5-5685*x^4+2700*x^3-500*x^2

>    e4:=factor(e3);

e4 := x^2*(3*x-2)^2*(4*x-5)^3

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Example 1.3.2
Expressions and Functions.

All computer algebra systems make a clear distinction between expressions  and functions .  A function has to be associated with an argument (variable).

Maple syntax for an expression g is given by:  g:=expression_body; Maple syntax for a function f of variable x is:  f:=x->function_body;  A few examples say more than a thousand words.

We define the expression g equal to x^2+7  as well as the function f(x) = x^2+7 .  Then we evaluate f` for x=a,  x=2 and x=5t+1.

>    g:=x^2+7;

g := x^2+7

>    f:=x->x^2+7;

f := proc (x) options operator, arrow; x^2+7 end proc

>    f(a);

a^2+7

>    f(2);

11

>    f(5*t+1);

(5*t+1)^2+7

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Example 1.3.3
Solve the equation a*x^2+b*x+c = 0 .

>    eq1:=a*x^2+b*x+c=0;

eq1 := a*x^2+b*x+c = 0

>    sol1:=solve(eq1, x);

sol1 := -1/2*(b-(b^2-4*a*c)^(1/2))/a, -1/2*(b+(b^2-4*a*c)^(1/2))/a

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Example 1.3.4
Simultaneously solve the equations 3*x-4*y = 1  and 5*x+7*y = -3 .

>    eq2:={3*x-4*y=1, 5*x+7*y=-3};

eq2 := {3*x-4*y = 1, 5*x+7*y = -3}

>    sol2:=solve(eq2, {x, y});

sol2 := {y = -14/41, x = -5/41}

Exercises

Exercise 1.3.1

Compute the exact value and a decimal approximation of (1231/12)^11.

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Exercise 1.3.2

Plot the graph of   (x^2-2*x+5)/(x^2+1)  `.

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Exercise 1.3.3

Plot in one picture the graphs of x*sin(2*x)  and 5-3*x^3 .

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Exercise 1.3.4

Define the Maple function f(x) = x^3-5*x   and compute f(3*u+7) .

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Exercise 1.3.5

Solve the system of equations 5*x-3*y = 11  and -2*x+78*y = 13 .

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