Casio CFX-9970G Manuale utente

Recursion Table and

Graph

You can input two formulas for any of the three following types of

recursion, which you can then use to generate a table and draw

graphs.

• General term of sequence {an}, made up of anand n

• Formulas for linear recursion between two terms, made up of

an+1, an, and n

• Formulas for linear recursion between three terms, made up of

an+2, an+1, an, and n

16-1 Before Using the Recursion Table and Graph Function

16-2 Inputting a Recursion Formula and Generating a Table

16-3 Editing Tables and Drawing Graphs

Chapter

218

16-1 Before Using the Recursion Table and

Graph Function

uTo enter the RECUR Mode

On the Main Menu, select the RECUR icon and enter the RECUR Mode. This

causes the Recursion Menu to appear.

• All recursion formulas that are stored in memory appear in the Recursion

Menu.

• {SEL+C} ... {menus for control of table generation and graph color}

• {DEL} ... {recursion formula delete}

• {TYPE} ... {recursion formula type specification}

• {n, an ···} ... {menu for input of variable nand general terms anand bn}

• {RANG} ... {screen for setting of table range}

• {TABL} ... {recursion formula table generation}

uTo specify the recursion formula type

Before inputting a recursion formula, you must first specify its type.

1. In the Recursion Menu, press 3(TYPE).

• In this display, “an=An+ B” is the general term (an= A×n+ B) of {an}.

2. Press the function key for the recursion formula type you want to set.

•{a

n}/{an+1}/{an+2} ... {general term of sequence {an}}/{linear recursion between two

terms}/{linear recursion between three terms}

Selected storage area

Press

and

to move.

219

16-2 Inputting a Recursion Formula and

Generating a Table

Example 1 To input an+1 = 2an+ 1 and generate a table of values as the

value of nchange from 1 to 6

Make a1= 1.

1. Specify the recursion formula type as linear recursion between two terms and

then input the formula.

c4(n, an···) 2(an) +b

2. Press w5 (RANG) to display the table range setting screen, which contains

the following items.

•{a0}/{a1} ... setting of value for {a0(b0)}/{a1(b1)}

The table range settings specify the conditions that control the value of variable n

in the recursion formula, and the initial term of the numeric value table. You should

also specify a starting point for the pointer when drawing a convergence/diver-

gence graph (WEB graph) for a formula for linear recursion between two terms.

Start................ Starting value of variable n

End ................. Ending value of variable n

a0, b0............... Value of 0th term a0/b0(a1, b1.... Value of 1st term a1/b1)

anStr, bnStr ...... Pointer starting point for convergence/divergence graph

(WEB graph)

• The value of variable nincrements by 1.

3. Specify the range of the table.

2(a1)

bwgwbw

4. Display the table of the recursion formula.At this time, a menu of table

functions appears at the bottom of the screen.

J6(TABL)

Currently selected cell (up to six digits)

Value in currently highlighted cell

P.225

220

• Displayed cell values show positive integers up to six digits, and negative

integers up to five digits (one digit used for negative sign). Exponential display

can use up to three significant digits.

• You can see the entire value assigned to a cell by using the cursor keys to

move the highlighting to the cell whose value you want to view.

• You can also display the sums of the terms (Σanor Σbn) by turning on ΣDisplay.

Example 2 To input an+2 = an+1 + an(Fibonacci series) and generate a table

of values as the value of nchange from 1 to 6

Make a1= 1 and a2= 1.

1. Specify the recursion formula type as linear recursion between three terms and

then input the formula.

3(TYPE)3(an+2) 4(n, an···)

3(an+1)+2(an)

2. Press wand then press 5(RANG) to display the table range setting

screen, which contains the following items.

• {a0}/{a1} ... setting of value for {a0(b0) and a1(b1)}/{a1(b1) and a2(b2)}

The table range settings specify the conditions that control the value of variable n

in the recursion formula, and the initial term of the numeric value table.

Start................ Starting value of variable n

End ................. Ending value of variable n

a0, a1, a2.......... Values of 0th term a0/b0, 1st term a1/b1, and 2nd term a2/b2.

• The value of variable nincrements by 1.

3. Specify the range of the table.

2(a1)

bwgwbwbw

16 - 2 Inputting a Recursion Formula and Generating a Table

P.7

221

4. Display the table of the recursion formula.At this time, a menu of table

functions appears at the bottom of the screen.

J6(TABL)

• There can be only one recursion table stored in memory at one time.

• Except for linear expression n, any of the following can be input for general

term {an} to generate a table: exponential expressions (such as an= 2n– 1),

fractional expressions (such as an= (n+ 1)/n), irrational expressions (such as

an= n– ), trigonometric expressions (such as an = sin 2nπ).

n– 1

• Note the following points when specifying a table range.

• If a negative value is specified as a start or end value, the calculator

drops the negative sign. If a decimal value or fraction is specified, the unit

uses only the integer part of the value.

• If the value of a0/b0(or a1/b1) is greater than the start value, the calculator

makes the starting value of variable xthe same as the value of a0/b0(or

a1/b1) before generating the table.

• If the start value is greater than the end value, the calculator swaps the

two values before generating the table.

• If the start value is the same as the end value, the calculator generates a

table using the start value of variable xonly.

• If the start value is very large, it may take a long time to generate a table

for linear recursion between two terms and linear recursion between

three terms.

• Changing the angle unit setting while a table generated from a trigonometric

expression is on the display does not cause the displayed values to change.

To cause the values in the table to be updated using the new setting, display

the table, press 1(FORM), change the angle unit setting, and then press

6(TABL).

Inputting a Recursion Formula and Generating a Table 16 - 2

Currently selected cell (up to six digits)

Value in currently highlighted cell

222

16 - 2 Inputting a Recursion Formula and Generating a Table

uTo specify the generation/non-generation status of a formula

Example To specify generation of a table for recursion formula an+1 =

2an+ 1 while there are two formulas stored

c1(SEL+C)

(Selects recursion formula to which

non-generation status is to be

assigned.)

1(SEL)

(Specifies non-generation status.)

J6(TABL)

(Generates table.)

• To change the status of a recursion formula from non-generation to generation,

select the formula and press 1(SEL+C), 1(SEL).

uTo change the contents of a recursion formula

Changing the contents of a recursion formula causes the values in the table to be

updated using the current table range settings.

Example To change an+1 = 2an+ 1 to an+1 = 2an– 3

e(Displays the cursor.)

ee-dw

(Changes the formula contents.)

6(TABL)

uTo delete a recursion formula

1. Use fand cto highlight the formula you want to delete. Press 2(DEL).

2. Press 1(YES) to delete the formula or 6(NO) to abort the operation

without deleting anything.

Unhighlights this formula

223

16-3 Editing Tables and Drawing Graphs

You get a choice of four options for editing tables and drawing graphs.

• Deletion of a recursion formula table

• Drawing of a connect type graph

• Drawing of a plot type graph

• Drawing of a graph and analysis of convergence/divergence (WEB)

You can access these options from the function menu that appears at the bottom

of the screen whenever a table is displayed.

•{FORM} ... {returns to Recursion Menu}

•{DEL} ... {table delete}

•{WEB} ... {convergence/divergence (WEB) graph draw}

•{G·CON}/{G·PLT} ... {connected type}/{draw plot type} recursion graph draw

• The {WEB} item is available only when a table generated using a formula for

linear recursion between two terms (an+1 =, bn+1 =) is on the display.

uTo delete a recursion table

1. Display the recursion table you want to delete and then press 2(DEL).

2. Press 1(YES) to delete the table or 6(NO) to abort the operation without

deleting anything.

kBefore Drawing a Graph for a Recursion Formula

You must first specify the following.

• Graph color (blue, orange, green)

• Draw/non-draw status of for the recursion formula

• The type of data to be plotted

To specify the graph color and the draw/non-draw status, display the Recursion

Menu and then press 1(SEL+C) to display the following menu items.

•{SEL} ... {draw/non-draw status}

•{BLUE}/{ORNG}/{GRN} ... {blue}/{orange}/{green}

P.225

P.128

224

uTo specify the color of the graph

The default color for a graph is blue. Use the following procedure to change the

graph color to orange or green.

1. Display the Recursion Menu and then use fand cto highlight the formula

whose graph color you want to change.

2. Press 1(SEL+C).

3. Press the function key that corresponds to the color you want to specify.

uTo specify the draw/non-draw status of a formula

There are two options for the draw/non-draw status of a recursion formula graph.

• Draw the graph for the selected recursion formula only

• Overlay the graphs for both recursion formulas

To specify the draw/non-draw status, use same procedure as that for specifying

generation/non-generation status.

uTo specify the type of data to be plotted (ΣDisplay: On)

You can specify one of two types of data for plotting.

•anon the vertical axis, non the horizontal axis

•Σanon the vertical axis, non the horizontal axis

In the function menu that appears while a table is on the display, press 5

(G·CON) or 6(G·PLT) to display the plot data menu.

•{

an}/{Σan} ... {an}/{Σan}on vertical axis, non horizontal axis

Example 1 Draw a graph of an+1 = 2an+ 1 with anon the vertical axis and n

on the horizontal axis, and with the points connected.

Set the following parameters in the View Window.

Xmin = 0 Ymin = 0

Xmax = 6 Ymax = 65

Xscale = 1 Yscale = 5

6(TABL)5(G·CON)

(Selects connected type.)

1(an)

(Draws graph with anon the vertical

axis.)

16 - 3 Editing Tables and Drawing Graphs

P.222

225

Editing Tables and Drawing Graphs 16 - 3

Example 2 Draw a graph of an+1 = 2an+ 1 with Σanon the vertical axis and n

on the horizontal axis, and with the points unconnected.

Use the same View Window parameters as those provided in

Example 1.

6(TABL)6(G·PLT)

(Selects plot type.)

6(Σan)

(Draws graph with Σanon the vertical

axis.)

• To input a different recursion formula after a graph is drawn, press !Q.

This displays the Recursion Menu where you can input a new formula.

kDrawing a Convergence/Divergence Graph (WEB graph)

With this feature, you can draw a graph of an+1 = f

(an) where an+1 and anare the

terms of linear recursion between two terms, substituted respectively for yand xin

the function y= f

(x). The resulting graph can then be viewed to determine whether

or not the graph is convergent or divergent.

Example 1 To determine whether or not the recursion formula an+1 = –3an2

+ 3anis convergent or divergent.

Use the following table range.

Start = 0 End = 6

a0= 0.01 anStr = 0.01

b0= 0.11 bnStr = 0.11

Use the following View Window parameters.

Xmin = 0 Ymin = 0

Xmax = 1 Ymax = 1

Xscale = 1 Yscale = 1

This example assumes that the following two recursion formulas are already

stored in memory.

1. Press 6(TABL) 4(WEB) to draw the graph.

226

2. Press w, and the pointer appears at the pointer start point (anStr = 0.01).

• The Y value for the pointer start point is always 0.

3. Each press of wdraws web-like lines on the display.

↓

This graph indicates that recursion formula an+1 = –3an2+ 3anis convergent.

Example 2 To determine whether or not the recursion formula bn+1 =

3bn+ 0.2 is convergent or divergent.

Use the following table range.

Start = 0 End = 6

b0= 0.02 bnStr = 0.02

Use the View Window parameters from Example 1.

1. Press 6(TABL) 4(WEB) to draw the graph.

16 - 3 Editing Tables and Drawing Graphs

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