computer algebra system (ti calculators)...appendixb—computer algebra system (ti calculators) 709...

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052161547Xapxb.xml CUAU030-EVANS August 27, 2008 7:38

A P P E N D I X B

Computer AlgebraSystem (TI calculators)

B1 IntroductionThis appendix is written for use with TI calculators. Much of this material is also relevant to

the use of other calculators with a computer algebra facility. The TI calculators include the

TI-89, TI-89 Titanium, the TI-92 and TI-92 plus, and the Voyage. The screen dumps are from a

TI-89 or TI-89 Titanium.

The home screen is as shown. It is the default screen

when the calculator is first turned on. It can always be

reached by pressing the button HOME or 2ND QUIT .

The use of the 2ND key is very similar to its use with

the TI-83. The screen can be made lighter or darker by

pressing 2ND and holding down to make it darker,

or to make it lighter.

At the top of the home screen a toolbar appears.

Options are labelled F1 to F6. Different screens have

different toolbars. To view one of the submenus, press

the corresponding key from the top row of keys.

F6 is obtained from pressing 2ND F1 .

The following steps are useful to perform if the calculator has been used previously and you

are unsure of what has been stored in various memory locations.

Choose F6 and then choose 1:Clear a-z by selecting and pressing ENTER . This clears the

memory locations a to z. Return to the F6 menu and select 2:New Prob. NewProb appears in

the entry line of the home screen. Press ENTER again. This clears the home screen and turns

off any user-defined functions or stat plots.

The home screen can be cleared at anytime by choosing 8:Clear Home from the F1 menu.

703Cambridge University Press • Uncorrected Sample Pages • 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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704 Essential Mathematical Methods 3 & 4 CAS

Executing commands in the home screenNew commands are entered in the entry line. As new commands are entered in the entry line

the old commands and answers scroll up. These can be accessed with the ‘up’ and ‘down’

arrow keys.

The mode settings may be changed through the MODE menu.

For the screen shown to the right the

settings shown are:

Graph..................FUNCTION

Angle..................RADIAN

Exact/Approx......AUTO

mode settings

entry line

The 3/30 in the bottom right-hand corner of the screen informs the user that three

calculations have been made. The label MAIN to the left of the screen refers to the folder

being used. This option is not referred to in this chapter.

To change the mode EXACT/APPROX, press MODE and then F2 to get to the second page

of the MODE menu. Use the down arrow to go to AUTO, use the right arrow to see the

submenu and select 2:EXACT. Press ENTER twice.

Press ENTER again and note that 23.5 has been recalculated in EXACT mode. When in

EXACT mode you can press � ≈ (GREEN mode) to obtain approximations.

Errors, deleting and editing expressionsand answersAn error message appears if an expression is entered with

incorrect format.

Cambridge University Press • Uncorrected Sample Pages • 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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Appendix B — Computer Algebra System (TI calculators) 705

Press ESC to return to the entry line. Errors are deleted by using the back arrow key ← .

The whole entry can be deleted using CLEAR . The clear button can be used to clear other

expressions and the corresponding answers anywhere in the home screen. Expressions in the

home screen can be reached and selected in the home screen by using or , and then

ENTER . This returns the expression to the entry line and modifications can be made.

The following sequence of screens illustrates how to paste from the upper section of the

home screen for the purpose of editing or using a simplified expression. Note that the

calculator is still in EXACT mode.

Enter expression and press ENTER . Use the up arrow to select answer. Press ENTER to have answer in entry line.

The expression can be modified.

B2 Using the Algebra menu (F2)In this section the algebra menu is explored. You may work through this to become acquainted

with the menu.

The entire menu is as shown in these two screens.

It is important that the user is aware of any variables which have been assigned values. This

can be done by pressing 2ND VAR-LINK . For this machine the number 5 has been assigned to

the variable x. This is done in the entry line by 5 → x. The variable x has this value until some

action is taken. Press and ← to undo the assignment.


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Operations in the Algebra menu1:solve(This is used to solve equations, simultaneous equations and also linear inequations.

There are buttons for x, y, z and t; other letters may be accessed using the ALPHA button.

You may need to use the 2ND button to access other operators, e.g., press 2ND MATH and

find ‘and’ in the 8:Test sub-menu.

For example:

solve(a∗x + b = 0, x) results in x = −b

a

solve(x2 + x − 1 = 0, x) results in x = −(√

5 + 1)

2or

x =√

5 − 1

2Note: It is necessary to use ∗ in the expression a∗x + b as ax

without the operator is read as a new variable.

solve(a∗b∗t − w + t = w∗t, w) results in w = (ab + 1)t

t + 1solve(x3 − x2 − x + 1 = 0, x) results in x = 1 or x = −1

solve(2x + √2 < 3, x) results in x <

−(√

2 − 3)

2

solve(2x + 3y = 6 and x − y = 1, {x, y}) results in x = 9

5

and y = 4

5Note: There is no requirement to use ∗ between the 2 and x.

2:factor(This command is used for factorisation.

Factorisation over the rational numbers is obtained by

implementing the command without the separate designation

of the variable.

For example:

to factorise x3 − 2x over the rational numbers, the

command is factor(x3 − 2x).

to factorise over the real numbers, the command

is factor(x3 − 2x, x).

Some further examples are provided here. The results are

shown on the given screens.

factor(a2 − b2)factor(a3 − b3)

factor

(2

x − 1+ 1

(x − 1)2+ 1

)


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factor(2x4 − x2)factor(2x4 − x2, x)

The factor command can also be used to give the prime

decomposition of integers.

3:expand(This command is used in the expansion of expressions.

For example:

expand((a + b)3)

expand((a + √2b)2)

In the following, storing an expression to a memory location will also be utilised. If the

name of a variable is given as the second argument, then the expansion will be given in

decreasing powers of that variable.

Store (x + y)3 to memory location d and then expand

without, and then with, a designated variable.

The expand command can also be used to form partial fractions.

For example:

expand

(1

x2 − 1

)

expand

(x3 + 2x + 1

x2 − 1

)

Sometimes extra conditions are required for the expansion to be carried out as expected.


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For example:

expand((ax )b) gives the correct result only if a is stated

to be positive.

expand(ln(a*b)) does not result in the expected

ln(a) + ln(b). It requires the condition a > 0 or

b > 0 to be stated. It also works if both a and b are stated

to be positive.

4:zeros(This command solves an equation where one side of the equation is zero. The variable must be

named. The answer is given as a list.

zeros(x2 − 1, x)

zeros(x2 − y2, x)

zeros(x2 − y2, y)

zeros(x2 − y, y)

zeros(x2 − 4x + 8, x) No solutions

zeros(x2 − 4x + 1, x) Two solutions

zeros(x2 − 4x + 4, x) One solution

5:approx(Using this command is equivalent to choosing APPROXIMATE from the Exact/Approx

submenu of the MODE menu. When in EXACT mode you can also press � ≈ (GREEN

mode) to obtain approximations.

6:comDenom(This gives an expression with common denominator and in simplified ratio form. The screens

demonstrate the use of this command.

7:propFrac(This command changes a fractional expression to a proper fraction form.


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8:nSolve(This gives numerical solutions to equations. It will return only one solution. The syntax is

nSolve(equation,variable = guess). The guess is not necessary, but can make the process

faster. The screen to the left below shows the solutions obtained by giving suitable guesses.

The screen to the right below shows the solutions obtained by using a restriction. The positive

solution is obtained with the restriction x > 0 and the negative solution with x < 0. The

syntax for the latter is shown in the entry line.

9:Trig and A:Complex9:Trig will be used in a later section of this chapter, whereas A:Complex is not used.

B:ExtractThis contains a submenu. The first two functions of this menu are briefly discussed here.

1:getNum( This applies the common denominator function and then returns the

numerator of the result.

2:getDenom( This applies the common denominator function and then returns the

denominator of the result.

B3 GraphingGraphing using a TI CAS calculator is very similar to graphing with a TI graphing calculator.

The screens associated with graphing are listed in green directly below the screen.

Y= screenFor the Y= screen press � and then F1 .

The default Y= screen appears as shown.

The cursor appears at the y1= position. Press ENTER to take

the cursor to the Edit line.

This can also be achieved by pressing F3 for Edit.


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In the Edit line enter y1(x) = x∧2 and press ENTER again.

The result now appears in ‘pretty print’ format. Note the tick

to the left. This indicates that the function is selected and will

be graphed when � and then F3 is pressed.

Menus F2, F5 and F6 contain features similar to the TI graphing calculators. These are

reproduced here.

The F6 menu enables the user to change the style of the

graph. To change the style the cursor needs to be placed on

the required function as shown. Remember, to obtain the

F6 menu press 2nd F1 .

The user can experiment to find the use of each of the

menu items. To close the menu, and return to the Y= screen,

press ESC .

The F5 menu enables the Y= screen to be set.

The F2 menu has the same zoom features as the TI-83 with the addition of items

B:Memory and C:SetFactors.

B:Memory allows a window setting to be stored and recalled. From the F2 ZOOM menu

choose B:Memory.

The choices in this memory are:

1: ZoomPrev This returns to the viewing window displayed before the previous zoom.

2: ZoomSto This saves the current viewing window.

3: ZoomRcl This recalls the viewing window last stored with ZoomSto.

The screen on the right below is the ZOOM FACTORS screen. These are altered by first

using the keys and to highlight the change to be made and then by typing in the

required value.

The WINDOW screenThe WINDOW screen is very similar to that of the TI-83. The screen to the left below shows

the standard settings. By pressing F1 and 9 the GRAPH FORMATS screen can be obtained.


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This is also obtainable through the Y= screen and the GRAPH screen through the same

procedure.

The GRAPH screenIn the Y= screen enter y = 2x − 4. The WINDOW screen

has the settings as shown and TRACE (F3) has been activated.

The MATH (F5) menu contains options. These include

9:Distance, A:Tangent, and C:Shade. Some of these will

be discussed in a later section.

B4 Defining functionsExpressions can be stored in memory locations. For example x2 − 3x → p stores the

expression x2 − 3x in memory location p. This can be operated on using the functions

discussed previously. Make sure that memory location x has been cleared first.

Functions can be defined in terms of a variable through using store or the define command,

which is found by pressing F4 (to obtain F4 Other menu) and selecting 1:Define. In the

following, two functions are defined and then operations from the

Algebra and Calculus menus are applied. It is often efficient to

define functions at the beginning of a problem.

Functions can be defined in terms of a variable and arbitrary

constants. This is shown in the screen opposite and the method

of assigning values is also shown.


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Functions involving more than one variable can also be

defined.

B5 Circular functionsMODEThe choice of radian or degree measure is made from the

MODE menu.

Solving equationsSolve from the Algebra (F2) menu is used. Initially the solution of the equation sin x = 0.5

will be considered. The exact solution is given as x = 2k� + 5�

6or x = 2k� + �

6. The

parameter k is @n2 on the screens. The solution when the MODE chosen is AUTO is also

shown. The notation @ni indicates that it is the ith parameter to be used.

In the following, the syntax for finding the solutions for

sin x = 0.5 for 0 ≤ x ≤ 2� is shown. In the entry line:

solve(sin x = 0.5, x) | x > 0 and x < 2�

to give the solutions�

6and

5�

6.

It is also possible to give values to the parameter. For

example in the following @n2 is given values −1, 0 and 1.

The complete entry in the entry line is x = 2∗@n2� + 5�

6or

x = 2∗@n2� + �

6| @n2 = {−1, 0, 1}


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B6 Using the Calculus menuOperations in the Calculus menu (F3)Seven of the operations of the calculus menu are considered here. These are:

1:d( differentiate, 2:∫

(integrate, 3:limit(, 6:fMin(, 7:fMax(, A:nDeriv( and B:nInt(

1:d(differentiateThis operation is used to differentiate expressions. The required form is d(expression,

variable).

The followings screens illustrate its use. For the second derivative the form is d(expression,

variable, 2).

2:∫

(integrateThis operation is used to differentiate expressions. For indefinite integrals the required form is∫

(expression, var). If the family of antiderivatives is required enter∫

(expression, var, c).

Make sure that c has not been assigned a value. This is done by observing the VAR-LINK

screen. For definite integrals the required form is∫

(expression, var, upper, lower). The

following screens illustrate its use.

3:limit(The limits considered in this section will be those that are considered in the senior years of

school mathematics. Right and left limits as well as two-sided limits will be considered. The

syntax is as follows.

For limx→a

limit(expression, x, a)

For limn→∞ limit(expression, n, ∞)

For limx→0+

limit(expression, x, 0, 1)

For limn→0−

limit(expression, n, 0, −1)


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The following screens demonstrate this syntax.

6:fMin( and 7:fMax(These operations return the value for which the maximum or minimum value (or the least

upper bound or greatest lower bound) of a function occurs. fMax returns the value for which a

local maximum occurs only if this is the actual maximum for the interval being considered.

A similar statement holds for fMin.

The syntax for these functions is fMin(expression, var) and fMax(expression, var).

A:nDeriv(This operation will produce the numerical derivative of a function by determiningf (x + h) − f (x − h)

h. The derivative is then determined using the appropriate limit.

B:nInt(This returns an approximate value for a definite integral. This is illustrated in the following

screens. In the screen to the left below the technique for integration discussed above is

employed for comparison.


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B7 ProbabilityPress 2ND MATH to obtain the MATH menu.

Select 7: Probability to see the available functions.

The first four are 1: !, 2:nPr(, 3:nCr( and 4:rand(

The way of using 3:nCr( is to enter values

of n and r as follows.

In the home screen for 5C2, enter nCr(5, 2)

To generate a random integer between 1 and 6

inclusive enter rand(6)

For the binomial distribution, go to the Stats/List Editor

(Application) and go to F5 to see the Distribution menu.

Select B: Binomial Pdf. The screen appears as shown.

Enter the value of p and n required. In the screen

n = 6 and p = 0.7. Press ENTER again to see the

probability distribution.

A better way to do this is to define a binomial function which can be kept.

From F4 choose 1: Define and complete as bi(n, p, x) = nCr (n, x)∗ p∧x∗(1 − p)∧(n − x);

then this function can be used for any binomial distribution problem.

Also for the cumulative distribution define the function

cbi(n, p, x) = �(bi(n, p, x), x, 0, x). � is found in the CALC menu.

This will give the cumulative distribution. This is shown in the following section.

For the normal distribution, go to the Stats/List editor (Application) and go to F5 to see the

Distribution menu.

Select 4: Normal Cdf.

Enter the value of � and � required, for example � = 6 and � = 0.7. For Pr(X < 1) enter

upper value = 1 and lower value = −∞. Press ENTER again to see the probability.

For inverse choose 1: Inverse Normal after choosing 2: Inverse from the F5 menu. Enter

the probability, � and � to obtain the result.

Plotting a distributionThe example for the binomial distribution is used. In the Stats/List Editor the binomial

distribution values are found in the Pdf list. Enter 0 to 6 in List1. Press F2 plots and select Plot

Setup. Select Plot1 and then press F1. Complete as shown. Press ENTER twice. Go to the

Graph window and select 9. Zoom Data from the Zoom menu. The plot is as shown.


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