Casio ALGEBRA FX 2.0 PLUS User Manual

Chapter

Computer Algebra System and Tutorial Modes (ALGEBRA FX 2.0 PLUS only)

7-1 Using the CAS (Computer Algebra System) Mode

7-2 Algebra Mode

7-3 Tutorial Mode 7-4 Algebra System Precautions

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7-1 Using the CAS (Computer Algebra System)

Mode

On the Main Menu, select the CAS icon to enter the CAS Mode.

The following table shows the keys that can be used in the CAS Mode.

COPY

k Inputting and Displaying Data

PASTE

H-COPY

REPLAY

Input in the Algebra Mode is performed in the upper part of the display, which is called the “input area.” You can input commands and expressions at the current cursor location.

Calculation results appear in the lower part of the display, which is called the “output area.” When a calculation produces an equation or inequality, the lower part of the display is divided between a “natural result display area” for the result, and a “formula number area” for the formula number as shown below.

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If all the result does not fit on the display, use the cursor keys to scroll it.

k Inputting List Data

List: {element, element, ..., element}

•Elements should be separated by commas, and the entire set of elements should be enclosed within {curly braces}.

•You can input numeric values and expressions, equations, and inequalities as list elements.

○○○○○

Example To input List {1, 2, 3}

!*( { )b,c,d

!/( } )w

k Inputting Matrix Data

Matrix (m × n): [[(1,1) entry, (1,2) entry, ..., (1,m) entry] [(2,1) entry, ......, (2,n) entry]...

[(m, n) entry, ..., (m, n) entry]]

• The above input is arranged to show the relative positions of entries in the matrix. Actual input is an unbroken line, from left to right.

•Entries should be separated by commas, and the entire set of elements should be enclosed within [square brackets]. And each line also should be enclosed within [square brackets].

•You can input numeric values and expressions as matrix entries.

○○○○○

Example To input the matrix shown below 1 2 3

4 5 6 7 8 9

!+( [ )!+( [ )b,c,d !-( ] )!+( [ )e,f,g

!-( ] )!+( [ )h,i,j

!-( ] )!-( ] )w

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k Inputting Vector Data

Vector: [component, component, ..., component]

•Components should be separated by commas, and the entire set of components should be enclosed within [square brackets].

•You can input numeric values and expressions as vector component entries.

○○○○○

Example To input Vector (1 2 3)

!+( [ )b,c,d !-( ] )w

k Performing an Algebra Mode Operation

There are two methods that you can use for input in the Algebra Mode.

• Function menu command input

•Manual formula and parameter input

k Menu Command Input

Press a function menu key to display the menu of functions for the type of operation you are trying to perform.

• TRNS ... {formula transformation menu}

• CALC ... {formula calculation menu}

• EQUA ... {equation, inequality menu}

• eqn ... {calls up an equation stored in Equation Memory in accordance with a specified input value}

• CLR ... {variable/formula delete menu}

Pressing the K key displays the menu shown below.

• LIST ... {list calculation menu}

• MAT ... {matrix calculation menu}

• VECT ... {vector calculation menu}

For details on commands and their formats, see the “Algebra Command Reference” on page 7-1-11.

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k Manual Formula and Parameter Input

You can use the function menus, K key, and J key in combination to input formulas and parameters as described below.

• 3(EQUA)b(INEQUA)

•{>}/{<}/{

t}/{

s} ... {inequality}

•Kkey

•{∞}/{Abs}/{x!}/{sign} ... {infinity}/{absolute value}/{factorial}/{signum function*1}

•{HYP} ... {hyperbolic}/{inverse hyperbolic} functions

• {sinh}/{cosh}/{tanh}/{sinh–1}/{cosh–1}/{tanh–1}

•Jkey

•{Yn}/{rn}/{Xtn}/{Ytn}/{Xn} ... input of graph memory {Yn}/{rn}/{Xtn}/{Ytn}/{Xn}

k Formula Memory

The CAS Mode has 28 formula variables. Variable names are the letters A through Z, plus r, and θ. CAS Mode formula variables are independent of standard value variables.

○○○○○

Example To assign a formula that differentiates sin(X) at X (cos(X)) to variable A

signum (A) A

2(CALC)b(diff)sv,

v)aav(A)w

1 (real number, A > 0) –1 (real number, A < 0)

(A= imaginary number)

|A|

Undefined (A = 0)

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○○○○○

Example To assign M to row 1 column 2 of variable A when the matrix

is assigned to it

123 XYZ

ah(M)aav(A) !+( [ )b,c!-( ] )w

○○○○○

Example To recall the value of variable A when the list {X, Y, Z} is assigned to it

av(A)w

○○○○○

Example To recall the first component (A [1]) of variable A when vector (X Y Z) is

assigned to it

av(A)!+( [ )b

!-( ] )w

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k Function Memory and Graph Memory

Function memory lets you store functions for later recall when you need them. With graph memory, you can store graphs in memory. Press the J key and then input the

name of the graph.

○○○○○

Example To differentiate f1 = cos(X), which is assigned to function memory f1,

at X

2(CALC)b(diff)K6(g)4(FMEM)

d(fn)b,v)w

○○○○○

Example To differentiate Y1 = cos(X), which is assigned to graph memory Y1,

at X

2(CALC)b(diff)

J1(Yn) b,v)w

k Eqn Memory

When a calculation result is an equation or inequality, its formula number is displayed in the formula number area, and the equation is stored in Eqn memory.*1 Stored equations can be recalled with the eqn command, rclEqn command or rclAllEqn command.

*1Up to 99 formulas can be stored in Eqn

memory. The error message “Memory ERROR” when you try to store an equation when there are already 99 equations in Eqn memory. When this happens, execute the ALLEQU (Delete All Equations) from the CLR menu.

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k Answer (Ans) Memory and Continuous Calculation

Answer (Ans) memory and continuous calculation can be used just as with standard calculations. In the Algebra Mode, you can even store formulas in Ans memory.

○○○○○

Example To expand (X+1)2 and add the result to 2X

1(TRNS)b(expand)

(v+b)x)w

Continuing:

+cvw

k Replay Contents

Replay memory can be used in the input area. After a calculation is complete, pressing d or e in the input area recalls the formula of the last calculation performed. After a calculation or after pressing A, you can press f or c to recall previous formulas.

k Moving the Cursor Between Display Areas

When ] ' ` $ indicates a calculation result that does not fit on the display, the cursor keys perform output area scrolling. To use the Replay Function from this condition, press 6(g)2(SW). ] ' ` $ change to a dotted line display to indicate that cursor key operations control the input area.

Pressing 2(SW) again moves the cursor back to the output area.

# Pressing 6(g)1(CLR)d(ALLEQU)

deletes Eqn memory, Ans memory, and Replay memory contents.

#You can input up to 255 bytes of data into the

input area.

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SET UP Items

uAngle ... Unit of angular measurement specification

• {Deg}/{Rad} ... {degrees}/{radians}

uAnswer Type ... Result range specification

• {Real}/{Cplx} ... {real number}/{complex number}

uDisplay ... Display format specification (for approx only)

• {Fix}/{Sci}/{Norm} ... {number of decimal places}/{number of significant digits}/

{normal display format}

k Graph Function

Pressing 5(GRPH) displays the graph formula screen, which you can use to input a graph formula. Press 4(G

•

VAR) if you want to input a graph memory.

You can also use the 1(SEL), 2(DEL), and 3(TYPE) functions while the graph formula screen is on the display.

Press 6(DRAW) to draw a graph.

k RECALL ANS Function

Pressing 6(g)3(R

•

ANS) recalls Ans Memory contents.

k Solution Memory

In the CAS Mode or ALGEBRA Mode, you can save the history of a calculation you perform (replay memory contents) into solution memory. This section describes how you can access and work with the contents of solution memory. Pressing 6(g)4(MEM) on the CAS Mode or ALGEBRA Mode main menu display the initial solution memory screen shown below.

• {SAVE} ... {saves the calculation history to solution memory}

• {DEL

•

A}... {deletes solution memory contents}

• {OPT} ... {optimizes solution memory}

• {DISP} ... {displays solution memory contents}

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u To save a calculation history to solution memory (Save)

On the initial solution memory screen, press 1(SAVE).

Press 1(YES) to save the calculation history to solution memory.

Pressing i returns to the solution memory initial screen.

•Pressing 6(NO) in place of 1(YES) returns to the solution memory initial screen without

saving anything.

u To clear solution memory contents (Clear Memory)

On the initial solution memory screen, press 2(DEL

•

A).

Press 1(YES) to clear solution memory contents.

Pressing i returns to the solution memory initial screen.

•Pressing 6(NO) in place of 1(YES) returns to the solution memory initial screen without

clearing anything.

• This clears both CAS Mode and ALGEBRA Mode memory contents. You cannot select the mode shows memory contents you want to delete.

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u To display solution memory contents (Display Memory)

On the initial solution memory screen, press 6(DISP).

This displays the oldest expression and result in solution memory. The bottom line shows the record number.

• 6(DISP) is disabled when there is no data in Solution memory.

•To display the next record

Press 6(NEXT).

•To display the previous record

Press 1(BACK).

•Pressing 1(BACK) while the oldest record is on the display returns to the solution memory initial screen.

•To display a particular record

Press 5(SEL) and then input the number of the record you want to display.

Pressing w displays the record whose number you input.

•To delete a single solution memory record

Display the record you want to delete, and then press 2(DEL). In response to the confirmation message that appears, press w(Yes) to delete the record

you displayed.

To clear the above screen without deleting anything, press i(No).

•To toggle record number display on and off

Press 4(NUM) to toggle display of the record number on and off.

u To o p timize solution memory (Optimization)

On the initial solution memory screen, press 3(OPT). Pressing i returns to the solution memory initial screen.

Optimizing solution memory rearranges data and can free up more storage space. Perform the above procedure when solution memory capacity starts running low.

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Algebra Command Reference

The following are the abbreviations used in this section.

• Exp ... Expression (value, formula, variable, etc.)

• Eq ... Equation

• Ineq ... Inequality

• List ... List

• Mat ... Matrix

• Vect ... Vector

Anything enclosed within square brackets can be omitted.

u expand

Function: Expands an expression.

Syntax: expand ( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To expand (X+2)

1(TRNS)b(expand)(v+c)xw X2 + 4X + 4

u rFactor (rFctor)

Function: Factors an expression up to its root.

Syntax: rFactor ( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To factor the X2– 3

1(TRNS)c(rFctor)vx-dw (X – 3) (X + 3)

u factor

Function: Factors an expression.

Syntax: factor ( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To factor X2– 4X + 4

1(TRNS)d(factor)vx-ev+ew (X – 2)

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u solve

Function: Solves an equation.

Syntax: solve( Eq [,variable] [ ) ]

solve( {Eq-1,..., Eq-n}, {variable-1,...,variable-n} [ ) ]

○○○○○

Example To solve AX + B = 0 for X

1(TRNS)e(solve)av(A)v+

al(B)!.(=)aw

○○○○○

Example To solve simultaneous linear equation 3X + 4Y = 5, 2X – 3Y = – 8

1(TRNS)e(solve)!*( { )

da+(X)+ea-(Y)!.(=)f,

ca+(X)-da-(Y)!.(=)-i

!/( } ),!*( { )a+(X), X = – 1

a-(Y)!/( } )w Y = 2

• X is the default when no variable is specified.

X =

– B

u tExpand (tExpnd)

Function: Employs the addition theorem to expand a trigonometric function.

Syntax: tExpand( {Exp/List/Mat/Vect} [ ) ]

○○○○○

Example To employ the addition theorem to expand sin(A+B)

1(TRNS)f(TRIG)b(tExpnd)

s(av(A)+al(B)w cos(B) • sin(A) + sin(B) • cos(A)

u tCollect (tCollc)

Function: Employs the addition theorem to transform the product of a trigonometric

function to a sum.

Syntax: tCollect( {Exp/List/Mat/Vect} [ ) ]

○○○○○

Example To employ the addition theorem to transform sin(A)cos(B) to

trigonometric sum

1(TRNS)f(TRIG)c(tCollc)

sav(A)cal(B)w

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u trigToExp (trigToE)

Function: Transforms a trigonometric or hyperbolic function to an exponential function.

Syntax: trigToExp( {Exp/List/Mat/Vect} [ ) ]

○○○○○

Example To convert cos(iX) to an exponential function

1(TRNS)f(TRIG)d(trigToE)c!a(i)vw

u expToTrig (expToT)

Function: Converts an exponential function to a trigonometric or hyperbolic function.

Syntax: expToTrig( {Exp/List/Mat/Vect} [ ) ]

○○○○○

Example To convert eix to a trigonometric function

1(TRNS)f(TRIG)e(expToT)

!I(ex)(!a(i)vw cos(X) + sin(X) • i

u simplify (smplfy)

Function: Simplifies an expression.

Syntax: simplify( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To simplify 2X + 3Y – X + 3 = Y + X – 3Y + 3 – X

1(TRNS)g(smplfy)ca+(X)+da-(Y)

-a+(X)+d!.(=)a-(Y) +a+(X)-da-(Y)+d-

a+(X)w X + 3Y + 3 = –2Y + 3

—

+ e

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u combine (combin)

Function: Adds and reduces rational expressions.

Syntax: combine( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To reduce the fraction (X + 1) / (X + 2) + X (X + 3)

1(TRNS)h(combin)(v+b)/

(v+c)+v(v+dw

u collect (collct)

Function: Rearranges an expression, focusing on a particular variable.

Syntax: collect( {Exp/Eq/Ineq/List/Mat/Vect} [,{Exp/variable}] [ ) ]

○○○○○

Example To rearrange X2 + AX + BX, focusing on the variable X

1(TRNS)i(collct)vx+av(A)v+ al(B)vw X2 + (A + B)X

• X is the default when nothing is specified for [,{Exp/variable}].

X3 + 5X2 + 7X + 1

X + 2

u substitute (sbstit)

Function: Assigns an expression to a variable.

Syntax: substitute( {Exp/Eq/Ineq/List/Mat/Vect}, variable=expression

[,..., variable=expression] [ ) ]

○○○○○

Example To assign 5 to X in 2X – 1

1(TRNS)j(sbstit)cv-b, v!.(=)fw 9

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u cExpand (cExpnd)

Function: Expands xth root of imaginary number.

Syntax: cExpand( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To expand 2 i

1(TRNS)v(cExpnd)!x( )c!a(i)w 1 + i

u approx

Function: Produces a numerical approximation for an expression.

Syntax: approx( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ]

○○○○○

Example To obtain a numerical value for 2

1(TRNS)l(approx)!x( )cw 1.414213562

○○○○○

Example 9

Normal:jMcaw 12157665459056928801

approx: 1(TRNS)l(approx)jMcaw 1. 215766546E+19 (Display: Norm1)

# About approx

With normal calculations (when approx is not used) in the CAS Mode, calculation results are displayed in full, without using exponents. When you use approx in the CAS Mode, however, results are displayed using the

exponential format range specified by the Display item of the SET UP screen.

This means approx displays results in the CAS Mode the same way they are displayed in the

•

RUN

MAT Mode.

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u diff

Function: Differentiates an expression.

Syntax: diff( {Exp/List} [, variable, order, derivative] [ ) ]

diff( {Exp/List}, variable [, order, derivative] [ ) ]

diff( {Exp/List}, variable, order [, derivative] [ ) ]

○○○○○

Example To differentiate X6 with respect to X

2(CALC)b(diff)vMgw 6X

• X is the default when no variable is specified.

• 1 is the default when no order is specified.

u ∫

Function: Integrates an expression. Syntax: ∫( {Exp/List} [, variable, integration constant] [ ) ]

∫( {Exp/List}, variable [, integration constant] [ ) ] ∫( {Exp/List}, variable, lower limit, upper limit [ ) ]

○○○○○

Example To integrate X2 with respect to X

2(CALC)c( ∫ )vxw

• X is the default when no variable is specified.

u lim

Function: Determines the limits of a function expression.

Syntax: lim( {Exp/List}, variable, point [, direction] [ ) ]

○○○○○

Example To determine the limits of sin(X)/X when X = 0

2(CALC)d(lim)sv/v,v,aw 1

• Direction can be positive (from right) or negative (from left).

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