Using Derive

Derive is a computer algebra system which can perform symbolic, numeric and graphical operations. Derive is a menu driven program in which you select operations or options by positioning the highlight (shown in reverse video) over the appropriate name and then pressing the Enter key. By pressing the space bar or Tab key you advance the highlight forward to the next menu option. In addition, each menu option in Derive has one uppercase letter in its name; typing this letter will immediately select the associated menu item. The highest level COMMAND Menu consists of the menu title, COMMAND followed by nineteen options as shown below.

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

Derive Command Menu

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 Options Display Menu

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 .
 
 

Using Derive's Algebra Window

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 Derive's 2D Plot Window

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.

COMMAND: Algebra Center Delete Help Move Options Plot Quit Scale Ticks Window
                       Zoom

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 Derive's 3D Plot Window

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's Utility Files

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 )
 
 

Examples of Using Derive

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.

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.

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 .

Screen 4

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.

Screen 5