Texas Instruments Titanium TI-89 Manuale utente

Computer Algebra Systems: An Introduction

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Topic: An Introduction to Computer Algebra Systems (CAS)

Notes to the Teacher:

This activity is designed to use the CAS on the TI-89/89T/92 calculator

family to enhance understanding and instruction. All screen shots are from the

TI-89.

This material may be used freely by teachers in their classrooms. The copyright

message must not be removed. Any other use or publication without the consent

of the author is a breach of copyright.

Computer Algebra Systems: An Introduction

What is a CAS?

A computer algebra system (CAS) is a program on a calculator or a computer

that can perform algebraic manipulations. The CAS may also be known as a

symbolic manipulator.

Starting a CAS

1. Press the ON key to turn the

calculator on. If you don't see the

Home screen, press the HOME

key. The Home screen will

appear.

2. Before using the CAS, it is

wise to clear the calculator of any

values left in the memory by a

previous use or user. Press 2nd,

then F1, to access the F6 or

Clean Up menu. Select

2.NewProb. Press ENTER.

3. Try some calculations to familiarize yourself with the keyboard and display.

a) 5(7 – 3)3Answer: 320

b) π(4.2)2Answer: 55.42

c) –25 – 3 Answer: –8

Tech Tip: If you press a key in error, you can usually cancel its effect by

pressing the ESC (Escape) key.

Keep in mind the difference between the negative key (-), which makes a number

negative, and the subtract key, which is used between two operands.

4. Investigate the Function keys at the top of the keypad.

a) Press F1 to see the Tools menu.

b) Press F2 to see the algebraic manipulations that the CAS can perform.

c) You can also use the left and right cursor keys to see the different function

menus. Look at the selections under F3, the calculate menu, and F4. You will

use some of these selections as you work through CAS tutorials.

d) If the cursor is in the command line, pressing the CLEAR key will delete the

contents of the command line. If you move the cursor into the work space,

pressing the CLEAR key will clear the line that the cursor is on.

The CAS will "remember" lines, even if they have disappeared off the top of the

workspace. You can use the cursor keys to scroll to these hidden lines.

Simplifying or expanding expressions

1. Consider an algebraic expression such as 7x+ 5y– 4x– 3y.

Type the expression, and press

ENTER.

Note that the CAS has collected

the like terms for you.

<< 89ss3te-4 >>

2. Try some other algebraic

expressions. Include terms with

exponents.

3. The CAS can expand

algebraic expressions such as

7(4x– 3)using the distributive

property.

Press F2, and select 3:expand(.

Type the expression, followed by

a close bracket ).

• Press ENTER.

Note that nested brackets are required. If you get an error message while using

this operation, check that you have the same number of open and close

brackets, and that they are placed properly.

4. Try some other expansions. Include a product of binomials, such as

(2x– 3)(3x+ 1), or even the product of a binomial and a trinomial such as

(5z– 2)(3z2+ z– 4).

Factoring expressions

1. Press F2, and select 2:factor(.

Type 28x+ 21, followed by a

close bracket ).

Press ENTER.

Compare the result to the

expansion in part 3 of the

previous section.

2. Try factoring some other expressions, including 6x2– 7x– 3.

Solving equations

The CAS can solve equations.

1. Press F2, and select 1:solve(.

Type 2x+ 1 = 3, then a comma,

then an xfollowed by a close

bracket ).

Press ENTER.

Note that the solution is

displayed. Note, also, that you must specify the variable that you would like the

CAS to solve for.

Try solving a few more equations.

The CAS can solve for a variable in a formula. Consider the motion equation

v= u+ at. Suppose that you want to solve this formula for time, t.

If you inspect the calculator keyboard, you will note that x, y, zand thave

dedicated keys. Other variables are entered by pressing the ALPHA key first. For

example, to access a, press ALPHA , then =.

2. Press F2, and select 1:solve(.

Type v= u+ a× t, then a comma,

then tfollowed by a close bracket

Tech Tip: be sure to type the

multiplication operation between

the aand the t. Otherwise, the

CAS will treat at as a single

variable with the name at.

Press ENTER.

Tech Tip: the new form of the formula is sometimes displayed in an unusual

manner.

Try solving other formulas, such as V= πr2hfor r.

Checking the solution to an equation

You can use the CAS to test a

solution to an equation. As an

example, test whether x= 2 is a

solution for the equation

3x– 1 = –7.

1. Type 3x– 1 = –7. Then, press

the |(such that) key, followed by

x= 2.

Press ENTER.

Note that the CAS returns a value of false. Hence, x= 2 is not a solution for the

equation. Try the check again with x= –2. Note that this time the CAS returns a

value of true. Hence, x= –2 is a solution.

Try checking a few more solutions for other equations.

Operations on equations

You can use the CAS to apply the same operation to both sides of an equation.

Consider the equation 7x– 3 = 3x+ 5. If you are solving this equation using

pencil and paper, you begin by adding 3 to both sides and subtracting 3xfrom

both sides. You can do this with the CAS.

1. Type the equation into the

CAS, inside a set of brackets.

Outside the brackets, type

+ 3 – 3x. Press ENTER.

Tech Note: the syntax is

somewhat different from that

normally used in a pencil and

paper approach.

The next step is to divide both sides by 4. You can make use of the ANS key to

do this.

2. Open a bracket.

Type 2nd, then (-) to access

ANS. This tells the CAS to use

the answer from the previous

operation.

Close the brackets.

Type ÷ 4.

Press ENTER. Note that the expected answer is displayed.

Copying and Pasting

If you have entered a complicated expression at some point, and need to use it

again, you can avoid retyping by using the COPY and PASTE operations.

Use the cursor keys to scroll up to the expression that you want. Press the ♦key,

and then the ↑key to access COPY.

Scroll back down to the command line. Press ♦and ESC to access PASTE.

The expression is pasted into the command line.

Defining functions

You can assign a definition to a function using the CAS.

As an example, define

f(x) = x+ x2+ x3.

Type the expression on the right of

the equals sign. Then, press the

STO (or store) key. Type f(x).

Remember that you need the

ALPHA key to access f.

Press ENTER.

Type f(1), and press ENTER.

Evaluate the function for other values of x.

You can also define functions of more than one variable.

As an example, consider the formula for the volume of a cylinder V= πr2h.

Define a function v(r, h) as πr2h.

Evaluate v(r, h) for various values of

rand h.

If you would like the answer

displayed as an approximate value,

rather than a multiple of π, you can

select the MODE menu, scroll down

to Exact/Approx, and select

APPROXIMATE.

Tech Tip: when you are finished

with the CAS, it is wise to return the

display mode to AUTO.

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