Texas Instruments Titanium TI-89 User manual
Computer Algebra Systems: An Introduction
© R. Meisel, April 7, 2007
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
© R. Meisel, April 7, 2007
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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