Casio CFX-9850G PLUS User manual

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
16

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16-1 Before Using the Recursion Table and
Graph Function
uu
uu
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}
• {SEL} ... {recursion formula generation/non-generation status}
• {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}
uu
uu
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).
•Inthisdisplay, “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.
•{an}/{an+1}/{an+2} ... {general term of sequence {an}}/{linear recursion between two
terms}/{linear recursion between three terms}
Selected storage area
Press f and c to move.
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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
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•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
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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.
•When Start = 0 and a1/b1is selected as the initial term, the calculator
changes to Start = 1 and generates the table.
•When Start > End, the calculator swaps the Start and End values and
generates the table.
•When Start = End, the calculator generates a table for Start values only.
•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

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16 - 2 Inputting a Recursion Formula and Generating a Table
uu
uu
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
c
1(SEL+C) 1(SEL) ... 1(SEL)
J
(Selects recursion formula to which
non-generation status is to be
assigned and then specifies non-
generation status.)
6(TABL)
(Generates table.)
•Each press of 1(SEL) toggles a table between generation and non-
generation.
uu
uu
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)
uu
uu
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

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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.
uu
uu
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.
kk
kk
kBefore Drawing a Graph for a Recursion Formula
You must first specify the following.
•Graph color (blue, orange, green) ............................... {BLUE}/{ORNG}/{GRN}
•Draw/non-draw status of for the recursion formula ..... {SEL}
•The type of data to be plotted ..................................... ΣDisplay
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uu
uu
uTo specify the color of the graph ({BLUE}/{ORNG}/{GRN})
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.
uu
uu
uTo specify the draw/non-draw status of a formula ({SEL})
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.
uu
uu
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
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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.
kk
kk
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.
w
↓
w
↓
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

227
2. Press wand then either for cto make the pointer appear at the pointer
start point (bnStr = 0.02).
•The Y value for the pointer start point is always 0.
3. Each press of wdraws web-like lines on the display.
w
↓
w
↓
This graph indicates that recursion formula bn+1 = 3bn+ 0.2 is divergent.
•Inputting bnor nfor the expression an+1, or Inputting anor nfor the expression
bn+1 for linear recursion between two terms causes an error.
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16 - 3 Editing Tables and Drawing Graphs
kk
kk
kDrawing a Recursion Formula Graph Using Dual Screen
Selecting “T+G” for the Dual Screen item of the set up screen makes it possible to
display both the graph and its numerical table of values.
Example To draw the graph of an+1 = 2an+1from Example 1, displaying
both the graph and its table
Display the set up screen and specify “T+G” for Dual Screen. Press
J.
6(TABL)
(Shows the table.)
6(G·PLT)
(Draws plot type graph.)
•Pressing !6 (G↔T) causes the graph on the left side of the Dual Screen
to fill the entire display. Note that you cannot use the sketch function while a
graph is displayed using !6 (G↔T).
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