Texas Instruments Derive 6
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Manual
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(English)Texas Instruments Derive 6 - Introduction, size: 2.4 MB |
Texas Instruments Derive 6
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User reviews and opinions
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Documents
Derive Tutorial One
Introduction
This set of activities is intended to illustrate the use of Derive 6, a powerful and easy to use Computer Algebra System (CAS). The initial activities assume that the reader is unfamiliar with CASs but is familiar with the use of a mouse and pop-down menus. As the activities progress, the amount of detail on how to use the Derive 6 interface decreases although more complicated commands are brought into play. We wish you well in becoming a competent user of mathematical tools Derive makes so easy to work with.
Factoring and Expanding
In this activity, you will factor polynomials in several ways. In addition, you will expand binomials in one and two variables. These activities may refresh your memory about factoring polynomials and multiplying polynomials together. You begin by authoring x 4. To author an expression, move the cursor to the Entry line at the bottom of the screen, type x^4-4 and press Enter on the keyboard. You will author many expressions and equations in these activities. Now find and click on Simplify on the Main Algebra Windows menu, and select Factor (in the future, this type of action will be described as use the Simplify>Factor command). In the Factor dialog box, notice that the Rational polynomial factoring button is selected. Now click on the
Factor button. The rational factors of your expression are displayed in the worksheet.
With this factored form of the expression highlighted, use the Simplify >Factor command, and then choose the Radical polynomial button by clicking on the appropriate button to the right of the screen. Now click the Factor button at the bottom of this dialog box. The radical factors of your expression are then displayed in the worksheet. With this new factored form of the expression highlighted, use the Simplify >Factor command, and then click on the Complex polynomial button. Finally, click on Factor at the bottom of this dialog box. The complex factors of your original expression are displayed. The History window of the worksheet should look like this.
Copyright 2004 Texas Instruments
You can expand a variety of expressions with Derive. The expression x^4-4 should be highlighted on the Entry line. Author the expression (x+y)^2 (type (x+y)^2, and press Enter). Use the Simplify >Expand command, and notice that the Rational polynomial button on the right is selected. Now click on Expand at the bottom of the dialog box. Your History window worksheet should look something like this.
. One way to do this is Notice that this expression (#6) has three terms. You can also expand to edit the expression on the Entry line using the Backspace, Del and other standard editing keys. Once you have edited the expression, press Enter. Using the commands given above, Author and Expand (x+y)^3. Your History window worksheet should look something like this.
( x + y )3
You might like to try expanding
( x + y ) n for larger values of n.
Summary
You should have learned how to author expressions. You can edit expressions on the Entry line using standard editing keys such as the Delete, Backspace, and arrow keys. You can find menu items on the Main Algebra Windows menu bar and on submenus. Also, when you point the mouse cursor (without clicking) on the Main Algebra Windows Icon menu toolbar at the top of the screen, a description of what each icon does is displayed near the cursor. Try this for one or more icons.
Derive Tutorial Ten
Differential Equations
Introduction to Standard Forms
You can write a first-order differential equation in the form p( x, y ) + q( x, y ) y = 0. For example,
( x 2 + 4) y + 3 xy = 6 x can be rewritten as 3 xy 6 x + ( x 2 + 4) y = 0 where p( x, y ) = 3 xy 6 x and
q( x, y ) = x 2 + 4. There is also a standard form for second-order differential equations. Derive uses these forms in the syntax for solving differential equations. Derive can give a general solution or a particular solution to a differential equation.
First-Order Differential Equations
Derive uses the DSOLVE1 command to solve first-order differential equations with p( x, y ) and q( x, y ) determined from the standard form. The syntax for DSOLVE1 is:
DSOLVE1(p(x,y),q(x,y),x,y)
where x is the independent variable and y is the dependent variable. For example, the simple first-order differential equation y = y can be rewritten in the standard form y + y = 0 where p( x, y ) = y and q( x, y ) = 1. You may recognize that a function that is a solution to this differential equation is the function y( x) = e x since the differential equation states that the derivative of the function is the function itself. Solve this differential equation by Authoring DSOLVE1(-y,1,x,y) and clicking on Simplify. The x0 and y 0 are the initial values of x and y (and can be consolidated into an arbitrary constant for the general solution). You can solve for y as follows: solve(DSOLVE1(-y,1,x,y),y) You may recall that the solution to this differential equation is y ( x) = Ae x which does not look like the solution Derive gives. With a little mental algebra, you can see that y 0 e x x0 = y 0 e x e x0 = Ae x where
A = y 0 e x0.
To obtain the particular solution to
y = y through the point (0, 1), Author the following:
solve(DSOLVE1(-y,1,x,y,0,1),y)
Slope Fields For Linear Differential Equations
To display the slope or direction field for a first-order differential equation y = x 2 , Author
DIRECTION_FIELD(x^2
, x, -3, 3, 12, y, -3, 3, 12),
Simplify and click on the 2D-Plot icon in the Main menu. To display the direction field correctly in the Graph window, do the following. 1. Click on Options and select Display 2. In the dialog box, click on Points 3. Click on the Yes button so that solid line segments will be displayed 4. Click OK 5. Reset the Plot Range 6. Click on the 2D-Plot icon in the Graph window.
Copyright 2004 Texas Instruments
The direction field will be displayed. The direction field for y = x 2 will be cubic in form as you know from your previous experience in calculus with antiderivatives. A more interesting direction field occurs for the differential equation y = y x. To display this equations direction field, you should work with the following:
DIRECTION_FIELD(y
- x, x, -3, 3, 12, y, -3, 3, 12).
You will notice in this direction field that there appears to be one solution to the differential equation that is a straight line. Can you estimate what the equation of this straight line is and check that it is a solution using substitution? Can you use the symbolic solution capability of Derive to solve this differential equation?
Second-Order Differential Equations
Derive can solve second-order differential equations that can be written in the form y + p( x ) y + q( x) y = r ( x ). The syntax for the Derive command is as follows:
DSOLVE2(p(x),q(x),r(x),x,c1,c2)
where c1 and c2 are arbitrary constants. For example, the second-order differential equation y = sin(x) can be rewritten as y + 0 y + 0 y = sin( x) where p( x) = 0 , q ( x ) = 0 , and r ( x) = sin( x). Thus the Derive command is
DSOLVE2(0,0,sin(x),x,c1,c2).
Author and Simplify this command to display the solution to the differential equation. Notice that with DSOLVE1 the result is an equation from which you can sometimes get an explicit solution whereas with DSOLVE2 the result is an expression for the solution. Initial-value differential equations can be solved with DSOLVE2_IVP and boundary-value differential equations can be solved with DSOLVE2_BV. The syntax for these commands can be found by clicking on Help in the Main menu and selecting Online.
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