calculus0_10

7/31/2019 calculus0_10

1/15

Instructions for the Symbolic CalculatorReturn to Contents

The yellow field at the bottom of the page is a command line. After you type a command, pressEnter to execute it. To re -execute a command, put the caret at the end of the command and press enter.

The pink field just above it is a script window. Use this to define commands, programs or executewhole scripts. Type the command or program definition or script in the window, and then press the

button, and the definition is made. Use this for long commands or more complex statements.

First, let's do some simple numeric calculations. You should check the Reading mode for numbersbefore you do this. You can check or set the reading mode by right -clicking on any Command Line,

TextBox or on any MathEdit.

Right -click the Yellow Command Line to get its menu.

Choose: Settings, Read Numbers As...

Presently, it says Decimal. Uncheck that and close the box.

Now, since nothing is checked, the reading mode is Long , that is, our arithmetic will be with Javaintegers for now.

Some Arithmetic: There are 4 basic reading modes for numbers:

1) Long

2) Decimal

3) Big

4) Exact

And you can form various combinations of these, such as BigLong, BigDecimal , and so on. Decimal reading mode is most like what you find in ordinary calculators . Long reading mode is primarily forwork with whole numbers, but it also works just fine with graphs. Exact reading mode is for work withexact rational numbers (fractions). We will start with Long mode in this demonstration, but we'll needother settings shortly.

The reading mode affects the way that MathScript interprets all numbers, even those entered indialog boxes! The settings can also be made by command in a script or command line. We supplied thereadlong command as the first command below. Go ahead and execute it, it won't hurt anything. Place

Page 1 of 15

2/22/2004file://D:\Sites\Mathwright\librarya\galileo\galileo11.htm

7/31/2019 calculus0_10

2/15

the caret at the end of the line and press Enter.

Readlong; (Enter)

If you check the mode as before, you will see that it is still Long (nothing is checked). And if you check the precision just below it, you will see that it is 7. You can leave it there since precision does not affectinteger calculations. But notice the following:

To calculate a number, such as sqrt(10), precede with the word Calc , (case does not matter) so forexample, type:

calc sqrt(10); (enter)

you see: 3.1622776

If the precision were set to 4, you would see: 3.1622

So some decimal calculations work fine in Long mode. The system understands that functions like sqrt,sin, tan, etc. should return decimal numbers. And if you multiply or add an integer to a decimal number,the result will be decimal.

But be careful. If you try to get the cube root of 10 by typing:

calc 10^(1/3); (enter)

you will see: 1 in Long mode. Why? Because 1/3 is translated to 0 first, then the calculation 10^0 = 1 isdone! You should do it in decimal mode, but if you really want to do it in Long mode, you would haveto type something arcane like:

calc 10^(double(1)/double(3)); (enter)

And you see: 2.1544346

You may set the decimal precision (say, to 7) with the command:

precision 7; (enter)

There are many commands in Mathscript. These are a few to try:

Now type (and press Enter)

calc 2^300;

203703597633448608626844568840937816105146839366593625063614044935438129976333670618

calc 2^300/2^299;

2

Page 2 of 15


7/31/2019 calculus0_10

3/15

MathScript did both exponentiations and then did the division.

Now try:

calc (2/3)^2;

The answer 0 is reported. Again, this is because in long reading mode, 2/3 is projected to the integer 0.But try

calcexact (2/3)^5;

The correct answer: is reported now.

The calcexact command always does the operations in Exact reading mode, and it further simplifies(normalizes) its answer, so that for example, fractions are always reduced. Try

calcexact 1/2 + 1/3 + 1/4;

Now, what if we wanted a big number, say 2^1000

calc 2^1000;

107150860718626732094842504906000181056140481170553360744375038837035105112493612249

This is not a Java Integer! It is a Java BigInteger ! MathScript automatically translates an

exponentiation: long^long to the BigLong type. How about decimal calculations? These are controlledby the Precision setting in the TextBox menu, or by the precision command.

We will approximate Euler's number. We want to raise 1.005 to the 200th power.

For this, set the Reading mode to Decimal and Big Numbers. Set the Precision to 0. Ironically, this gives"unlimited" precision in a BigDecimal calculation. When you raise a BigDecimal number to a LongPower, you get a BigDecimal.

So we have to "force" the exponent to be a Long (and not a BigDecimal) number. The long() functiondoes that for us.

Try:

calc 1.005^long(200);

2.71151712292937479854899397016290412652651892864411079520031671964003264340274456721

Some Algebra:

Page 3 of 15


7/31/2019 calculus0_10

4/15

We will do a few algebraic calculations. For symbolic algebra, it is good to be in Exact Reading mode.We could do this from the menu, but let's use a command. Type: (then press Enter, of course.)

readexact;

If you check the menu, you'll see that it has been done. Now, let's do a simple calculation:

calc (x+1/2)^2;

You see that no expansion was done. The 1/2 was read as a fraction, but the multiplication was not done.But try:

calcexact (x+1/2)^2;

calcexact (x+y/2)^10;

Or the trinomial theorem:

calcexact (x+y+z)^10;

(Mathscript writes the result in prettier form than the following)

x^10+10*x^9*y+45*x^8*y^2+120*x^7*y^3+210*x^6*y^4+252*x^5*y^5+210*x^4*y^6+120*x^3*y^7

Let's try some other algebraic operations:

factor 144;

2^4*3^2

OK, how about a tougher one:

factor 2^(2^5)+1;

641*6700417

Now we can also factor polynomials in a single variable. Try:

factor y^10+y^5+1;

Page 4 of 15


7/31/2019 calculus0_10

5/15

(y^2+y+1)*(y^8+y^3 -1*y^7+y^5+1 -1*y^4 -1*y)

Let's check that. You may copy the result and paste it into the command line:

calcexact (y^2+y+1)*(y^8+y^3 -1*y^7+y^5+1-1*y^4-1*y);

y^10+y^5+1

Some Graphing

We can easily graph the built -in functions, like sine, cosine, etc.

graph sin;

graph cos;

graph sin+cos;

draw the graph of sin+2*cos using color red, width 2;

graph sin#cos color color(0,0,128), width 1;

(The # means composition, and we have the choice of 16,000,000 RGB colors with the color(R,G,B) function.

Each of the three arguments varies from 0 to 255.)

It is also easy to clear the screen.

clear;

If there was more than one window, we would say:

clear in "Graph2D";

to clear the Graph2D named "Graph2D". We might also have used the Actions, Clear Screen menu.

Let us define a few new functions and look at their graphs.

f(x) := (x+1)^2;

To see that the system knows the name "f" of this new function, right -click on the background and selectfrom the menu: Objects, Lexicon, functions... Scroll down the bottom to the User -defined functions,where you see F, and click on it.

The system shows:

Definition: F(X) is (X+1)^2

Page 5 of 15


7/31/2019 calculus0_10

6/15

All mathematical objects are stored in the Lexicon and may be checked in this fashion.

graph f width 2 color red;

We may ask for a numeric approximation for the solution of an equation using f.

For this, we use the roots() function. Here, the first argument is the equation (the variable must alwaysbe x), the second is the refinement for the initial partition of the interval, and the third is the interval tosearch for roots. It is entered as a MathScript vector, and so uses square brackets. Suppose we wanted anapproximate solution to

calc roots(f(x)+x=3, 50, [ -5,5]);

A vector of two approximate solutions is reported:

[-3.561552812808803, 0.5615528128088066]

Now let's experiment with graphing compositions of functions.

clear in "graph";

(Here's another way to define a function.)

make g(x) 1/(1- x);

h(x) := 1-1/x;

graph g color green;

graph g#g;

(It didn't do a good job with the singularity, but we'll come back to that in a moment..)

graph h color red;

graph h#h;

graph h#h#h;

This may be a surprise. It is essentially the identity function.

make u h(h(h(x)));

calc u;

Now, you'll notice that the built -in graph command did not handle the singularity well. You may fix thisby executing :

Page 6 of 15


7/31/2019 calculus0_10

7/15

curveconnected false;

Now all graphs and curves will be drawn without connecting points. Try

graph g#g using refinement 1000;

To restore the default state, enter:

curveconnected true;

This may be a good time to discuss user-defined commands. Let us define an interactive command thatdraws stars. For that we use the Command Protocol.

Execute the following command definition in the pink Script Window. Other commands may haveoutput their results to the window, so remove all trailing Ts, NILs and so on from the script below. Youexecute the script (thus, define the new

command) by pressing the at the top of the page.

The command is called Stars . If you right -click on the background and select from the menu:

Objects, Lexicon, commands... Scroll down the bottom to the User -defined commands, where you seeSTARS, and click on it.

The definition is:

command stars (ex num) [color ex col fillcolor col] {

let v be getpoints(true,num);

plot polygon v;

}

This command takes one mandatory argument: the number of points. It also may take one optionalargument: the fillcolor. This may be supplied or not, and in any order or placement. But if it is supplied,the fillcolor must be preceded immediately by the

word " color ". Try it out on the yellow command line:

clear;

stars 5 using color lred;

The word "using" is optional also. After you press Enter, you are expected to click 5 times on thegraph2D to make a star! You will see a pointing finger when you move the cursor over the graph2D andwhen you are finished, you will see something like:

Page 7 of 15


7/31/2019 calculus0_10

8/15

Some Calculus:

Next, let's create a curve, a vector-valued function of a single variable. We'll define an ellipse as afunction of a single variable t . The use of square brackets in the definition tells Mathscript that thevalues that this function takes are vectors.

First, change the reading mode to Decimal, Precision 7.

Type:

Readfloat;

Precision 7;

c(t) := [5*cos(t), 3*sin(t)];

To draw the image of this as t varies from 0 to , use the Curve command.

curve c;

or, more informally,

Draw the curve c;

will cause the curve to be drawn. Next, define a map

that takes

Page 8 of 15


7/31/2019 calculus0_10

9/15

Type:

tr(x,y) := [ x^2 - y^2 + x , 2*x*y -y ];

Next shrink the curve ccc(t) := c(t)/3;

curve cc;

The image of cc is the image of c scalar-multiplied by . Now, draw the curve which is the composition

of tr with cc.

curve tr#cc;

Next we look at the partial differentiation operator. For this, return to Exact reading mode:

readexact;

select "mathedit";

calcexact (x+y/2)^5;

you see:

Now, d(expr, var1, var2, ...) is the partial differentiation operator. Follow the expression with thedesired differentiation variables.

If you supply no variables, you have the 0th order operator, which is the identity. This has the handyproperty of normalizing expressions to make it possible to compare them. Thus, in a script, the returnvalue of d(x^2 - y^2 -(x+y)*(x -y)) is the same as the value printed by

calcexact x^2 - y^2 -(x+y)*(x-y);

That is, 0.

Since calcexact does not actually return a value (but prints one as a side -effect) the 0th orderdifferentiation operator is very useful in scripts.

calcexact d((x+y/2)^5,x);

Page 9 of 15


7/31/2019 calculus0_10

10/15

calcexact d((x+y/2)^5,y);

calcexact d((x+y/2)^5,x,y);

The numeric integration function is integrate( expression in x, refinement, interval) .

calc integrate(x^3,10,[1,2]);

3.757500000000004

calc integrate(x^3,20,[1,2]);

3.7518750000000036

Some Matrix calculations

This WorkBook has an 8x8 matrix called m defined for it. It is a rational matrix, and so we get intoexact reading mode to operate on it. With matrices, we use calc and not calcexact . Just follow alongbelow.

readexact;

calc m;

make n m^2;

calc n;

Page 10 of 15


7/31/2019 calculus0_10

11/15

calc n*inv(m);

calc inv(m);

calc m*inv(m);

Page 11 of 15


7/31/2019 calculus0_10

12/15

calc transpose(m);

calc det(m);

125411328000

calc det(inv(m));

readlong;

Now select, say the upper 4x4 submatrix of m. Select Actions, Copy .

Place the caret at the bottom line of "mathedit" and select Actions, Paste. You may assign that matrix aname in the following way:

make u getmatrix(false);

Getmatrix(n) returns the nth matrix in the window. Getmatrix(false) simply returns the last one. Thus uis the 4x4 matrix you just selected. Type:

calc u^2;

Page 12 of 15


7/31/2019 calculus0_10

13/15

calc det(u);

12

calc det(u^2);

144

And so on... You may also create matrices from the command line or from scripts.

readexact;

make mymatrix zmatrix(2,3);

calc mymatrix;

Now, modify the matrix by changing a few entries to nonzero values. Just click on an entry to highlightit, and then type in a new number. Type a fraction as say 2/3 . To change your mind, navigate with thearrow keys.

It takes a little getting used to, but is very intuitive.

Put only numeric entries in the matrix (fractions, whole numbers, or decimals). When you are finished,type.

make mynewmatrix getmatrix(false);

This assigns the name mynewmatrix to the modified matrix.

calc mynewmatrix ;

Now create a row matrix.

make myrow zmatrix(1,2);

make myrow(1,1) 2;

Page 13 of 15


7/31/2019 calculus0_10

14/15

make myrow(1,2) 3;

calc myrow;

Let's try some formatted output.

output "/v * /v = /v", myrow, mynewmatrix, myrow* mynewmatrix;

Just a little sprite animation

First, create two sprites for the "Canvas" Graph2D.

The command is:

sprite ;

sprite "red" "b6.bmp" [2,0];

sprite "blu" "b3.bmp" [ -2,0];

Now, let's move them around. For that we create a curve, a circle of radius 2.

make c(t) 2*[cos(t), sin(t)];

curve c;

Set decimal mode.

readfloat;

Now type or paste the following program in the pink script window (after you have erased the one that isthere).

do

i=0

until i>6.3 {

movesprite "red" to c(i);

movesprite "blu" to c(pi -i) ;

Page 14 of 15


7/31/2019 calculus0_10

15/15

i := i+pi/20;

}

Press the and watch them go!

There are over 270 built -in commands and functions in Mathscript. You can see them listed in theLexicon from the menu: Objects, Lexicon. In your browser, you get this menu by right- clicking on thebackground.

Page 15 of 15

Documents

calculus0_10