The following instructions should enable you to run the computer programs that are needed on class projects. The set of programs we will use are the Mathematics Plotting Package (MPP), which was written at the United States Naval Academy, Derive, which is a commercial mathematics software package developed by Soft Warehouse Inc. of Honolulu, Hawaii, Maple, which is an on-going project of the Symbolic Computation Group at the University of Waterloo and commercially distributed by Waterloo Maple Software in Waterloo, Ontario, and Scientific Notebook, a commercial product of TCI Software Research distributed by Brooks/Cole. A separate tutorial for Scientific Notebook is available online at http://my.execpc.com/~aplehnen/tutorial.tex .

MPP is considered 'shareware', this means that anyone can make a copy of the programs. You can download your own version of the MPP programs from the link at http://my.execpc.com/~aplehnen/al or directly from the site : http://archives.math.utk.edu/software/msdos/calculus/mpp/.html . These programs are downloaded in zipped format and must be unzipped before use. 

Derive, Maple and Scientific Notebook are not shareware. To make and/or distribute copies of these programs is illegal! You may purchase student versions of Derive published by Addison-Wesley and student versions of Maple or Scientific Notebook published by Brooks/Cole.

There are many microcomputers scattered about the MATC Truax campus. However, the ones which are usually available are in the following locations:

General Studies Computer Lab (Room 264)

The MATC Information Resource Center (i.e., the library)

If there is a microcomputer to which you have personal access, you may wish to install MPP or your own personal (and purchased) copy of Derive on the computer's hard drive. If the MPP programs are on the computer's hard drive you can access them by typing MPP after the ' C prompt ', C:\> , and then pressing the enter key. If the programs are not on the hard drive put the appropriate program disk in drive A. If the A prompt, A:\> , is not on the screen, access the A drive. The programs can then be accessed, as before, by typing MPP after the A prompt.
 
 

The MPP programs are menu driven. This means that access to the programs as well as options within the programs are all specified by menus, which prompt you with instructions as to what to do next. The primary menu provides you with a list of the available programs and a number by which you can access them.

After you have found the program you want on the primary menu, type in its corresponding number. The computer will then load that program into its working memory. You now need to provide the program with the information needed to perform the mathematical operation desired. You may want a printed output (as opposed to results just being displayed on the screen). To get a printed outputfollow the procedure outlined later in this document. When you are finished, you can transfer control back (use the F10 key) to the primary menu. From there, you may quit.

The MPP programs were written in the PASCAL Programming Language. In using some of the programs you will need to enter explicit formulas for functions. The following conventions are used :

Multiplication is indicated by *

Division is indicated by /

Addition is indicated by +

Subtraction is indicated by

Exponentiation is indicated by placing the ^ (caret symbol) between the expression for the base and the expression for the exponent. The caret is the upper case of 6 on the keyboard. Actually, MPP does recognize algebraic notation. For example, ' two times ex ', can be entered either as 2x or 2*x . Similarly, ' three times ex cubed ', can be entered as 3x^3 or 3*x^3 or 3*x*x*x ; however, the combination, 3xxx, would not be properly interpreted. The only valid grouping symbols are parentheses ( ), but these may be nested as in (6A B^(E+3))/(5C + 3D).

The rules for the order of operations are the conventional ones used in mathematics and science. Certain standard functions are ' built into ' MPP, but the arguments must always appear in parentheses. For example, ' the sine of 3 times x ' should be written as sin(3x) .

abs(x) is the absolute value of x
sqrt(x) is the square root of x (for non-negative x)
exp(x) is the exponential function evaluated at x
ln(x) is the natural logarithm function for positive x
sin(x) is the sine of x (All trig functions assume the argument is in radians.)
cos(x) is the cosine of x
tan(x) is the tangent of x
arctan(x) is the arctangent of x
fn(x,L) is the function defined on function line L evaluated at x

A more detailed list is found in the MPP documentation or by typing Ctrl F in the plot specification menu .

The primary MPP plotting program is also called MPP. Typing 1 in the primary menu takes you to the plot specification menu.

This program can plot up to six different (color coded) curves on the same graph. A ' graph window ' must first be specified, by choosing values of Xmin (the leftmost boundary of the horizontal axis), Xmax (the rightmost boundary of the horizontal axis), Ymin (the bottom boundary of the vertical axis), and Ymax (the top boundary of the vertical axis). The values of Xtick and Ytick are used to space ' tick marks ' (grid markers) on the horizontal and vertical axes, respectively. There are preset or ' default ' values for these parameters. They are Xmin = Ymin = 10, Xmax = Ymax = 10 , and Xtick = Ytick = 1. These values can be changed to get a better view of the graph requested. For example, if the purpose of the graph is to show two periods of y = sinx , set Xmin = 0 , Xmax = 4pi (MPP will accept this input for 4),Ymin = 1.5 , = 1.5 (This shows the full y range of the plot, with a little bit of margin. Ymin = 1.2 , Ymax = 1.2 would also work.), Xtick = pi/2 (This allows the quarter period points to be easily identified), and Ytick = 0.5 .

To reset values of the plot parameters follow the procedure below :

1. Position the cursor with either the up-down arrow keys or the enter key.

2. Type over the existing entry and use either the space bar or the delete key to erase unwanted stuff.

To specify y as a function of x , use the default entry of y = f(x). (Continued pressing of the F4 key allows the different options of x as a function of y, polar coordinates, and parametric equations.) Enter the function desired, using the names of standard functions (e.g., sec(x) for secx ). Pi can be used for in any formula. All trig functions assume the argument is in radians. You now need to specify where in the plot x is to begin (xmin), where in the plot x is to end (xmax) and the difference between the x coordinates of successive points in the plot (xstep). Generally, you will want xstep to be small enough so that the curve appears continuous. A value of 0.01 works nicely for most graphs. In order to read off coordinates from a graph it is sometimes desirable to overlay a grid on the plot. This is the default option in the MPP programs, but it can be turned off prior to plot generation by accessing ' Printer Options ' (Ctrl O), and then typing F6 followed by a 1 , typing F10 then exits from ' Printer Options ' back to the plot specification menu. In order to turn the grid back on again type, from the plot specification menu, Ctrl O, F6, 2 and F10. To generate a graph, type the F9 key. To return to the plot specification menu to set up a new graph, type I or F10. Two useful and time saving options are the saving (F7) of current plot specifications and the retrieving (F8) of previous plot specifications. If you are running under DOS and are NOT running from a network typing F gives a printout of the graph. If you are running under Windows or from a network server do NOT press F, instead follow the directions given below. This procedure for printing graphs applies also to MPP3D and Derive..
 
 

Because of conflicts with the network and/or Windows 98, "old" DOS programs such as Derive, MPP, and MPP3D can not print directly to the network printers. Attempts to do so will just waste paper! Instead, once you have obtained the results you want on the computer screen do the following:

• Use the Print Scrn key on the keyboard to copy the screen as a bit map (bmp) image to the clipboard.
• Use the Alt Tab key combination to exit MPP, MPP3D, or Derive and return to Windows.
• Start a Windows program such as MS Paint (Use the command sequence Start, Programs, Accessories, Paint), Paint Shop Pro, Adobe Photo Shop, Microsoft Photo Editor, WordPerfect, etc.
• Paste the bmp image from the clipboard (use the Edit menu or simply type Ctrl v) in the Windows program.
• Reverse on the image to change the black background into white and thus save the print cartridges on the printer! In MS Paint choose Image Invert Colors from the main menu. In Paint Shop Pro choose Colors Negative Image. In Microsoft Photo Editor choose Effects Negative. In WordPerfect right click on the image, then from the pull down menu choose Image Tools Invert Colors. For other Windows applications consult the Help menu.
• Choose File Print from the top menu.

The MPP package includes as a separate program a three dimensional perspective plotter called MPP3D . To access this program enter MPP3d from the directory in which the MPP programs are stored. This takes you to the 3D plot specification menu where you can specify the 3D plot windows as well as the functions being plotted. The default mode is to plot z as a function of x and y, but by pressing F6 this can be changed to x as a function of z and y, y as a function of z and x, spherical and cylindrical coordinate plots as well as volumes of revolution. After setting up the viewing window and defining the type and formula for the function, type F9 to display the 3D graph. Typing R and using the arrow keys allows you to rotate your point of view.

Sample MPP3D Output
 
 

Using Derive

COMMAND: Author Build Calculus Declare Expand Factor Help Jump soLve Manage
Options Plot Quit Remove Simplify Transfer moVe Window approX

Choosing some of these options, such as Calculus, will generate a new or sub-menu. From a sub-menu you can always return to the previous higher level menu by pressing the Esc (Escape) key. Some sub-menus display one or more selection fields each with its own name, followed by a colon and a list of two or more selections. The position of the highlight in the first selection field indicates the option that is currently active. The current selections in the remaining selection fields are indicated by parentheses. For example, if from the COMMAND Menu you select the Options Display sequence of commands the following appears:

Mode: Text Graphics Reso: Medium (High) Text: (Large) Small Set: Std (Extended)
Adapter: MDA (CGA) EGA MCGA VGA Hercules AT&T T3100 PCjr Other

The five selection fields are Mode, Reso(lution), Text, Set and Adapter. Initially the active display mode, Text, is highlighted. The current selections in the remaining four fields are in parentheses. Use the Tab key to move the selection field one position to the right. (Shift-Tab moves one selection field to the left.) To make a selection within a selection field use the space bar (move to right) or backspace key (move to left) till the desired option is highlighted. Pressing the Enter key finalizes the choice of selections for all selection fields, pressing Esc cancels the selection process making no changes.

There are three types of Windows or screens in Derive.

1.The Algebra Window in which either symbolic or numeric operations are set-up and performed.

2.The 2D Plot Window for displaying one or more graphs in two dimensions.

3.The 3D Plot Window which can plot three dimensional perspective plots of z as a function of x and y .
 
 

The primary way to generate expressions in Derive is the using the Author option of the COMMAND Menu. Selecting Author (by pressing Enter when the highlight is over Author or typing A from the COMMAND Menu) causes a blinking cursor to appear in the Author line below the COMMAND Menu. Here you can type in an expression using the following conventions :

Multiplication is indicated by *

Division is indicated by /

Addition is indicated by +

Subtraction is indicated by

Exponentiation is indicated by placing the ^ (caret, upper case 6) between the expression for the base and the expression for the exponent. Grouping is indicated by parentheses, nested if necessary. The rules for the order of operations are the conventional ones used in mathematics and science.

Derive ' understands ' Algebraic implicit multiplication, i.e., you can enter 3x^2-5x+7 as a short cut for 3*x^2-5*x+7 .

To edit an expression in the Author line the following keystrokes apply:

Backspace Deletes the character to the left of the cursor.
Delete Deletes the character at the cursor.
Esc Cancels the expression.
Ins Toggles insert mode on or off so characters can be inserted at the cursor.
CtrlS (Control key pressed simultaneously with S) moves cursor to the left without erasing characters.
CtrlD (Control key pressed simultaneously with D) moves cursor to the right without erasing characters.
F3 Brings the expression highlighted in Algebra Window down into the Author line.
F4 Brings the expression highlighted in Algebra Window into the Author line and encloses it in parentheses.

After the enter key is pressed the Authored expression appears in the Algebra Window with an expression number in front. The expression will appear in standard mathematical notation and the expression numbers will increase sequentially as new expressions are generated. Expressions in the Algebra Window can be highlighted by using the up or down arrow keys. Parts of highlighted expressions can be highlighted by using the left or right arrow keys. The standard mathematical functions are 'built into' Derive , but the arguments must always appear in parentheses. The following is a partial list. The argument z is any valid real or complex expression.

abs(z) is the absolute value of z
sqrt(z) is the principal square root of z
exp(z) is the exponential function evaluated at z
ln(z) is the natural logarithm of z
log(z,w) is the logarithm base w of z
sin(z) is the sine of z (All trig functions assume the argument is in radians.)
cos(z) is the cosine of z
tan(z) is the tangent of z
cot(z) is the cotangent of z
sec(z) is the secant of z
csc(z) is the cosecant of z
asin(z) is the arcsine of z
acos(z) is the arccosine of z
atan(z) is the arctangent of z
acot(z) is the arccotangent of z
asec(z) is the arcsecant of z
acsc(z) is the arccosecant of z
sinh(z) is the hyperbolic sine of z
cosh(z) is the hyperbolic cosine of z
tanh(z) is the hyperbolic tangent of z
coth(z) is the hyperbolic cotangent of z
sech(z) is the hyperbolic secant of z
csch(z) is the hyperbolic cosecant of z
asinh(z) is the inverse hyperbolic sine of z
acosh(z) is the inverse hyperbolic cosine of z
atanh(z) is the inverse hyperbolic tangent of z
acoth(z) is the inverse hyperbolic cotangent of z
asech(z) is the inverse hyperbolic secant of z
acsch(z) is the inverse hyperbolic cosecant of z

Special mathematical symbols are also built into Derive . To enter them press the keys indicated. For example, Alt P means press the Alt key and P simultaneously.

Greek alpha = Alt A   Greek mu ( µ ) = Alt M    e (natural base) = Alt E    Greek beta = Alt B        Greek sigma = Alt S

Greek pi = AltP          Greek Gamma = Alt G     i (imaginary) = Alt I         Greek tau = Alt T          Greek delta = Alt D

Greek phi = Alt F        Greek epsilon = Alt N    infinity = inf                       Greek Omega = Alt O   Greek theta= Alt H

radical (square root) symbol= Alt Q         F(x):= Defines a function F(x) with the formula following the = .

Once an expression is in the Derive Algebra Window it can be manipulated by other Derive commands. From the COMMAND Menu

Simplify Simplifies the expression whose number is indicated.

Expand Expands out the expression whose number is indicated.

soLve Solves a relation for the variable indicated. If no= occurs in the expression, it solves for the zeros of the expression. This operation will attempt to find symbolic solutions unless the Options Precision Approximate command sequence is issued. If this command sequence is issued, a numerical solution on a specified interval is attempted.

approX Approximates an exact answer as a decimal to the number of digits specified in the O P sub-menu.

Factor Factors the expression whose number is indicated.

There are built in calculus functions which can be chosen from the Calculus sub-menu, or entered directly by typing in the function name in the Author line.

LIM(u, x, a) limit of the expression u as the variable x approaches a from both sides.

LIM(u, x, a,-1) limit of the expression u as the variable x approaches a from the left.

LIM(u, x, a, 1) limit of the expression u as the variable x approaches a from the right.

DIF(u, x, n) n'th order derivative of u with respect to x .

TAYLOR(u, x, a, n) n'th order Taylor polynomial approximation of u about x=a .

INT(u, x) Indefinite integral of u with respect to x (no arbitrary constant added).

INT(u, x, a, b) Definite integral of u from x=a to x=b (a and b can be variables).

SUM(u, n, k, m) Summation of u (as function of n) from n=k to n=m .

These commands will just set up the formal expression. To get the answer select Simplify for an exact or analytic answer or select approX for a numerical answer where the number of digits has been set by the Options Precision command sequence.
 
 

Using the Options Display command select Graphics as the Mode and VGA as the Adapter. Using Options Color you can adjust the color of the graphs on the screen. Highlight the expression you want plotted. The free variable in this expression (regardless of its name) will be interpreted as x, the horizontal variable, and the output of the expression will be interpreted as y, the vertical variable. If the expression to be plotted contains any free parameters other than the independent variable, assign values to these parameters using the Manage Substitute command sequence; the resulting expression should then be highlighted for plotting. Select Plot from the COMMAND Menu. You will now see a coordinate grid with center at the origin and equal scales on x and y of 1 unit per tick mark. Beneath the graph will appear the Plot Menu shown below.

Beneath this menu are the current coordinates of the ' Cross ' which can be moved vertically with the up or down arrow keys and horizontally with the left or right arrow keys. The Current x and y scale choices are also displayed. To see the graph of the highlighted expression from the previous Algebra Window select Plot from the menu. You can adjust the way the graph looks by using the Zoom command which allows you to either shrink (Zoom In) or expand (Zoom Out) the scales in either the vertical or horizontal directions. This can also be done directly by using the Scale command, but note that the tab key is used to move from the x axis scale value to the y axis scale value. The Center command replots the graph with the center at the current location of the Cross. The Move command repositions the Cross at a given set of coordinates without having to use the arrow keys. These features allow you to locate roots or points of intersections graphically. To plot a second expression on the same graph, issue an Algebra command to return to the Algebra Window, highlight the new expression, then Plot as before. To erase a plot use the Delete command from the Plot Menu. Polar plots can be generated by the Options State command to choose a Polar option. The variable in the expression plotted is then interpreted as the polar angle in radians and the output of the expression is the radial coordinate.

Using the Options Display command select Graphics as the Mode and VGA as the Adapter. Using Options Color you can adjust the color of the graphs on the screen. Now select Window Split from the COMMAND Menu. Choose a Vertical split at column 40. Typing the F1 key moves you from one window to the other. If there are any expressions in Window #2 erase them with a Transfer Clear command, then issue a Window Designate command and choose 3DPlot. Return to the first Window (F1 key) and highlight the expression which should be a function of two variables. Return to Window #2 and issue a Plot command. The perspective of the graph can be altered by using the other commands in the 3DPlot Menu.

Grids The number of wire frame panels appearing in the x and y directions (the larger these numbers, the greater the resolution of the plot).

Length Sets the lengths of the sides of the ' transparent box ' in which the surface is drawn.

Eye Sets the coordinates of the viewer's eye.

Derive comes equipped with special functions which, to save space, are not automatically loaded into the computers memory. They can be accessed through the Transfer Load Utility command followed by the utility's name. Some of the Utility Files of interest are as follows :

SOLVE.MTH which contains the function NEWTONS(u, x, x0 , n) which solves the vector equation u = 0 by Newton's Method. See page 142 of the Derive manual for details.

DIF_APPS.MTH which contains functions which perform applications of differentiation. See page 154 of the Derive manual for details.

INT_APPS.MTH which contains functions which perform applications of integration. Some of its functions are as follows:

AREA(x , x1 , x2 , y , y1 , y2 ) sets up the integral for the area between the curves y = y1(x) and y = y2(x) from x = x1 to
x = x2 . You should enter both y1 and y2 as formulas in x .

AREA_OF_REVOLUTION(y , x , x1 , x2 ) sets up the integral for the surface area of an expression y(x) from x = x1 to
x = x2 revolved about the x axis. You should enter y as formula in x .

AREAY_OF_REVOLUTION(y , x , x1 , x2 ) sets up the integral for the surface area of an expression y(x ) from x = x1 to
x = x2 revolved about the y axis. You should enter y as formula in x .

VOLUMEY_OF_REVOLUTION(y , x , x1 , x2 ) sets up the integral for the volume of an expression y(x) from x = x1 to
x = x2 revolved about the y axis. You should enter y as formula in x . If the volume of revolution has a hole of revolution given by the expression y = y1(x), then set up the calculation as follows :

VOLUMEY_OF_REVOLUTION(y, x , x1, x2 ) VOLUMEY_OF_REVOLUTION(y1, x, x1, x2 )
 
 

From the Author Line type x^2-5x+6 and press enter. This becomes expression #1 shown in Screen 1.

From the COMMAND Menu select Factor and designate Expression #1. This generates expression #2 in Screen 1.

From the Author Line enter x^2+7x-11 and press enter. This becomes Expression #3 in Screen 1.

From the COMMAND Menu select Factor and designate Expression #3 factored over Complex numbers. This generates Expression #4 shown in Screen 1.

From the COMMAND Menu select soLve and indicate Expression #3. This generates the two solutions displayed as Expressions #5 and #6 in Screen 1.

From the Author Line type 1/(x^2+1) and press enter. This becomes Expression #7 in Screen 1.

From the COMMAND Menu select Calculus Integrate and designate Expression #7 with respect to x and leave the limits of integration blank. This generates Expression #8 of Screen 2.

From the COMMAND Menu select Simplify and indicate Expression #8. This generates Expression #9 of Screen 2.

From the Author Line type (x+5)^4 and press enter. This generates Expression #10 of Screen 2.

From the COMMAND Menu select Expand and indicate Expression #10. This generates Expression #11 of Screen 2.

From the Author line type F(x):=xcos(x) and press enter. This generates the function definition shown in Expression #12 of Screen 2.

From the COMMAND Menu select Calculus Integrate and designate Expression #12 with respect to x and leave the limits of integration blank. This generates Expression #13 of Screen 2.

From the COMMAND Menu select Simplify and indicate Expression #13. This generates Expression #14 of Screen 2.

Now using the up arrow key highlight Expression 12 as shown in Screen 3 . Then from the COMMAND Menu select Options Precision Approximate Mode and 15 digit accuracy.

Screen 3

From the COMMAND Menu select Plot. From the PLOT Menu select Plot a second time. This generates the graph shown in Screen 4 . By using the arrow keys one sees from the Cross coordinates that the root of F(X) is approximately 0.736 .

From the Plot Menu select Algebra, then from the COMMAND Menu select soLve and indicate Expression #12. By pressing enter accept the default domain of -10 to 10 for locating the approximate root. This generates the 15 decimal digit solution shown in Expression #15 of Screen 5.

Like Derive, Maple is a computer algebra system. The version we have at MATC in Room 264 is a Windows based application. Launch Maple from the Start button. Detailed information on Maple commands is available through the Help feature. To access Help click on Help at the top of the screen or else choose the ? from the Menu bar. Help on the syntax of any Maple command can be obtained by typing ?command . For example, to get help on the solve command, type ?solve .

Each line of a Maple spreadsheet begins with a > . These can be inserted into an existing document by selecting the > from the Menu bar. Maple commands must end in either a colon : or a semi-colon ; The use of a ; generates a listing of Maple's resulting output, the : suppresses this output. Maple distinguishes between upper and lower case characters. Plots in Maple are displayed in their own windows which may be copied (via Clipboard) into the main Maple document. All variables are assigned values or expressions by assignment statements of the form variable := expression ; .

Functions of the form f(x) = expression in x are defined with the statement > f:=x> expression in x . Functions of more than one variable are defined in an analogous manner. As in Derive mathematical operations are indicated by the following conventions:

Multiplication is indicated by *

Division is indicated by /

Addition is indicated by +

Subtraction is indicated by

Exponentiation is indicated by placing the ^ (caret, upper case 6) between the expression for the base and the expression for the exponent. Grouping is indicated by parentheses, nested if necessary. The rules for the order of operations are the conventional ones used in mathematics and science.

Maple does not understand Algebraic implicit multiplication, i.e., you must enter 3*x^2-5*x+7 or a syntax error results. The following is a list of the common elementary functions defined in Maple .

abs(z) is the absolute value of z
sqrt(z) is the principal square root of z
exp(z) is the exponential function evaluated at z
ln(z) is the natural logarithm of z
log[b](z) is the log base b of z
sin(z) is the sine of z (All trig functions assume the argument is in radians.)
cos(z) is the cosine of z
tan(z) is the tangent of z
cot(z) is the cotangent of z
sec(z) is the secant of z
csc(z) is the cosecant of z
arcsin(z) is the arcsine of z
arccos(z) is the arccosine of z
arctan(z) is the arctangent of z
arccot(z) is the arccotangent of z
arcsec(z) is the arcsecant of z
arccsc(z) is the arccosecant of z
sinh(z) is the hyperbolic sine of z
cosh(z) is the hyperbolic cosine of z
tanh(z) is the hyperbolic tangent of z
coth(z) is the hyperbolic cotangent of z
sech(z) is the hyperbolic secant of z
csch(z) is the hyperbolic cosecant of z
arcsinh(z) is the inverse hyperbolic sine of z
arccosh(z) is the inverse hyperbolic cosine of z
arctanh(z) is the inverse hyperbolic tangent of z
arccoth(z) is the inverse hyperbolic cotangent of z
arcsech(z) is the inverse hyperbolic secant of z
arccsch(z) is the inverse hyperbolic cosecant of z

Pi is (The capital P is required!)
E is e (The natural exponential base and again capital E is required)
I is i (The imaginary unit)

Double quote, " , is used to refer to the last computed expression. Thus the command sequence
> eqn:=x^2-1; factor("); , causes x squared minus 1 to be factored as (x-1)(x-1) .

On a line all characters which follow the sharp character, # , are considered to be a part of a comment.

evalf(expression) Evaluates expression as a decimal (floating point number). The default number of digits is 10 .This can be changed by the assignment   > Digits := number of desired digits ;

expand(expression) Expands expression. For example, > expand((x+3)^2) ; generates x^2+6x+9 as output.

simplify(expression) Algebraically simplifies expression.

solve(equation, x) Tries to give an exact (analytic) solution of equation listing all x's that work. Maple will reprint the solve(equation, x ) command if it is unable to solve the equation.

fsolve(equation, x = a..b) Tries to give an approximate (numerical) solution of equation on the interval x = a to x = b. Maple will compute the root to an accuracy determined by the current value of Digits.

solve({eq1,eq2}, {x,y}) Tries to give an exact (analytic) solution of a system of equations.

factor(expression) Factors expression.

subs(x = a, expression) Substitutes a for x in the given expression.

plot(expression, x = a..b) Plots the values of expression along the vertical axis for x (the horizontal variable) in the interval x = a to x = b.

plot([f(t), g(t), t = a..b]) Plots the parametric curve given by x = f(t), y = g(t) for t in the interval t = a to t = b.

plot3d(expression, x = a..b ,y = c..d) Plots the surface defined by z = expression as a function of x and y.

limit(expression, x = a) Evaluates the limit of expression as x approaches a .

diff(expression, x) Evaluates the derivative of expression with respect to x .

int(expression, x) Evaluates the indefinite integral of expression as a function of x . No arbitrary constant is added to the answer.

int(expression, x = a..b) Evaluates the definite integral of expression as a function of x from x = a to x = b.

sum(expression, j = m..n) Evaluates the definite sum of expression as a function of j from j = m to j = n.

D(f) The differential operator. Returns the derivative of the function f as a function. This can be used in defining a differential equation.

dsolve(eq, y(x)) Solves the differential equation labeled eq for the unknown function y(x). The solution will include arbitrary constants.

dsolve(eq, init, y(x)) Solves the differential equation labeled eq subject to the initial condition labeled init for the unknown function y(x).
 
 

The following figures display a Maple work sheet which performs the same calculations as the Derive examples shown earlier in these notes.
.

Maple Screen 1

Maple Screen 2

Maple Screen 3

If you are having problems running the MPP programs, Derive, or Maple please contact me about it right away! I don't mind (especially in the beginning) giving a detailed guide.
 

These notes were authored by Al Lehnen, a math instructor at Madison Area Technical College in Madison, Wisconsin. I welcome any comments and suggestions. Please feel free to E mail me at alehnen@matcmadison.edu or write to me at the following address.