Casio FX-9860GII - SOFTWARE VERSION 2-00 User manual

Using a graphic display calculator
© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute
Casio fx-9860GII
Table of contents
 Functions
.
Graphing linear functions 2
Finding information about the graph
. Finding the gradient (slope) of a line 3
. Solving simultaneous equations
graphically 3
Simultaneous and quadratic equations
. Solving simultaneous linear equations in
two unknowns 4
. Solving simultaneous linear equations in
three unknowns 5
Quadratic functions
. Drawing a quadratic graph 9
. Solving quadratic equations 10
. Finding a local minimum or
maximum point 11
Complex numbers
. Operations with complex numbers 15
. Conjugate, modulus and argument 16
. Solving equations with complex roots 16
. Polar form 17
Exponential functions
. Drawing an exponential graph 18
. Finding a horizontal asymptote 19
Logarithmic functions
. Evaluating logarithms 20
. Finding an inverse function 21
. Drawing a logarithmic graph 22
Trigonometric functions
. Degrees and radians 22
. Drawing trigonometric graphs 23
More complicated functions
. Solving a combined quadratic and exponential
equation 24
Sequences and series
. Summation of a series 25
Modelling
. Using sinusoidal regression 27
. Drawing a piecewise function 28
Use this list to help you nd the topic you needInstructions for the Casio fx-9860GIl calculator
 Dierential calculus
.
Finding the gradient at a point 28
. Drawing a tangent to a curve 29
. Finding maximum and minimum
points 30
. Finding a numerical derivative 31
. Graphing a numerical derivative 32
. Using the second derivative 34
 Integral calculus
.
Finding the value of a definite integral 35
. Finding the area under a curve 36
 Vectors
Scalar product
. Calculating a scalar product 37
. Calculating the angle between two vectors 38
Vector product
. Calculating a vector product 39
 Statistics and probability
Entering data
. Entering lists of data 39
. Entering data from a frequency table 40
Drawing charts
. Drawing a frequency histogram from a list 40
. Drawing a frequency histogram from
a frequency table 41
. Drawing a box and whisker diagram
from a list 42
. Drawing a box and whisker diagram from a
frequency table 42
Calculating statistics
. Calculating statistics from a list 43
. Calculating statistics from a frequency table 44
. Calculating the interquartile range 45
. Using statistics 45
Calculating binomial probabilities
. Use of nCr 46
. Calculating binomial probabilities 47
Calculating Poisson probabilities
. Calculating poisson probabilities 49
Calculating normal Probabilities
. Calculating normal probabilities from X-values 52
. Calculating X-values from normal probabilities 53
Using a graphic display
calculator
CHAPTER OBJECTIVES:
This chapter shows you how to use your graphic display calculator
(GDC) to solve the different types of problems that you will meet in your
course. You should not work through the whole of the chapter – it is simply
here for reference purposes. When you are working on problems in the
mathematical chapters, you can refer to this chapter for extra help with
your GDC if you need it.
1

Using a graphic display calculator
© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute
Casio fx-9860GII
1 Functions
1.1 Graphing linear functions
Example 1
Draw the graph of the function y= 2x+ 1.
Press MENU . You will see the dialog box as shown on the right.
Choose 5: GRAPH and press
EXE .
The default graph type is Function, so the form Y= is displayed.
Type 2x+ 1 and press
EXE .
Press SHIF TF3 V-Window and choose F3 STD to use the default axes
which are −10 ≤x≤10 and −10 ≤y≤10.
Press EXE and
F6 DRAW.
The graph of y= 2x+ 1 is now displayed on the screen.
Use X,i,T to
enter x.
Before you start
You should be familiar with:
● Important keys on the keyboard: AC/ON SHIF TDEL MENU X,i,T EXE OPTN EXIT
● The home screen
● Using the Main Menu to enter the mode you want
● Changing window settings in GRAPH mode
● Using zoom tools in GRAPH mode
● Using trace in GRAPH mode
For a reminder of how to
perform the basic operations
have a look at your GDC
manual.
2

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Casio fx-9860GII
Finding information about the graph
The GDC can give you a lot of information about the graph of a function,
such as the coordinates of points of interest and the gradient (slope).
1.2 Finding the gradient (slope) of a line
The correct mathematical notation for gradient (slope) is
d
d
y
x. You will find
out more about this in the chapter on differential calculus. Here we just need
to know this is the notation that will give us the gradient (slope) of the line.
Example 2
Find the gradient of y= 2x+ 1.
First draw the graph of y= 2x+ 1 as in Example 1.
Press SHIF T(SET UP).
Set Derivative to On.
Press EXE and
F6 DRAW to return to the graph.
Press SHIF T
F1 Trace.
The calculator displays the coordinates of the point and the gradient.
Move the point along the line using the and
keys.
The gradient (slope) is shown by dy/dxand is 2 at every point along
the line.
1.3 Solving simultaneous equations graphically
To solve simultaneous equations graphically you draw the straight lines and then
find their point of intersection. The coordinates of the point of intersection give
you the solutions xand y.
Note: The calculator will only draw the graphs of functions that are expressed
explicitly. By that we mean functions that begin with ‘y=’ and have a function that
involves only xto the right of the equals sign. If the equations are written in a different
form, you will need to rearrange them before using your calculator to solve them.
Example 3
Solve the simultaneous equations 2x+ y= 10 and x− y= 2 graphically with your GDC.
First rearrange both equations in the form y=
2x− y= 10 x− y= 2
y= 10 − 2x−y= 2 − x
y= x− 2
Solving simultaneous
equations using a
non-graphical method is
covered in section 1.5.
{Continued on next page
3

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To draw graphs y= 10 − 2xand y= x− 2:
Press MENU and choose 5: GRAPH and press EXE .
The default graph type is Function, so the form Y= is displayed.
Type 10 – 2xand press EXE and x– 2 and press EXE .
Press SHIF TF3 V-Window and choose F3 STD to use the default axes
which are −10 ≤x≤10 and −10 ≤y≤10.
Press EXE and
F6 DRAW
The calculator displays both straight line graphs
Y1 = 10 – 2xand Y2 = x– 2.
Press F5 G-Solv and
F5 ISCT.
The calculator displays the intersection of the two straight lines at the
point(4, 2).
The solutions are x= 4, y= 2.
Simultaneous and quadratic equations
1.4 Solving simultaneous linear equations in two unknowns
When solving simultaneous equations in an examination, you do not need to show any
method of solution. You should simply write out the equations in the correct form and then
give the solutions. The calculator will do all the working for you.
Example 4
Solve the equations:
2x+ y= 10
x− y= 2
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
From the menu, choose Simultaneous and press
F1 .
If the calculator displays a
graph, press EXIT to return
to this screen.
If there are previous
equations in the
memory, press EXIT
until you return to this
menu.
{Continued on next page
4

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Casio fx-9860GII
Choose equations in two unknowns and press
F1 .
Note: This is how you will use the linear equation solver in your
examinations. In your project, you might want to solve a more complicated
system with more equations and more variables.
You will see the template on the right.
Type the coefficients from two equations into the
template, pressing
EXE after each number.
Press
F1 and the calculator will solve the equations, giving the solutions as
X and Y.
The solutions are x= 4, y= 2.
1.5 Solving simultaneous linear equations in three unknowns
When solving simultaneous equations in three unknowns there might be a
unique solution, infinitely many solutions or no solutions at all. Geometrically,
if the equations represent planes in three-dimensions, then their solutions
would be intersection at a point, intersection on a line (or plane) or
non-intersecting planes.
The equations must
be in the correct order.
5

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Example 5
Solve the equations 2x− 3y+ 4z= 1
x− y− z= −1
−x+ 2y− z= 2
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
From the menu, choose Simultaneous and press
F1 .
Choose equations in three unknowns and press
F2 .
You will see the template on the right.
Type the coefficients from the three equations into the template,
pressing
EXE after each number.
Press
F1 and the calculator will solve the equations,
giving the solutions as X, Y and Z.
The solutions are x= 3, y= 3 and z= 1
In this example, the solutions represent a point.
If there are previous
equations in the memory,
press EXIT until you return
to this menu.
The equations must be in
the correct order.
6

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Example 6
Solve the equations 2x+ 4y+ 2z= 8
x+ 2y+ z= 4
3x− y+ z= −9
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
From the menu, choose Simultaneous and press
F1 .
Choose equations in three unknowns and press
F2 .
You will see the template on the right.
Type the coefficients from the three equations into the template,
pressing
EXE after each number.
Press
F1 and the calculator will solve the equations,
giving the solutions as X, Y and Z.
The GDC solves the equations by an inverse matrix method.
In this example there is an error because the matrix is singular,
which means that there will be no unique solution.
To find the solution you can use the reduced row echelon form
method instead.
Press MENU 1
Press OPTN | F2 MAT | F6 | F5 Rref
Press EXIT EXIT | F4 MATH | F1 MAT | F3 m × n
Choose m as 3 and n as 4 and press
EXE .
If there are previous
equations in the memory,
press EXIT until you return
to this menu.
The equations must be in
the correct order.
{Continued on next page
7

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Type the coefficients from the three equations into the
matrix, pressing after each number.
Press
EXE .
The reduced row echelon form gives the solutions to the equations
of the line if the final row of the matrix is all zeroes.
The solutions are equivalent to
xz+=−
3
72and yz+=
2
73
In this example, the solutions represent a straight line.
The equations of the line can be written
72
3
73
2
()()xy
z
+
−
−
−
==
xy z+−
−
==
2
3
3
27
Example 7
Solve the equations x+ 2y− 3z= 13
2x− y+ x= 4
x+ 2y− 3z= 7
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
From the menu, choose Simultaneous and press
F1 .
Choose equations in three unknowns and press
F2 .
You will see the template on the right.
Type the coefficients from the three equations into the template,
pressing
EXE after each number.
Press
F1 and the calculator will solve the equations,
giving the solutions as X, Y and Z.
If there are previous
equations in the memory,
press EXIT until you return
to this menu.
The equations must be in
the correct order.
{Continued on next page
8

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© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute
Casio fx-9860GII
The GDC solves the equations by an inverse matrix method.
In this example there is an error because the matrix is singular,
which means that there will be no unique solution.
To find the solution you can use the reduced row echelon form
method instead.
Press MENU 1.
Press OPTN | F2 MAT | F6 | F5 Rref
Press EXIT EXIT | F4 MATH | F1 MAT | F3 m × n
Choose m as 3 and n as 4 and press
EXE .
Type the coefficients from the three equations into the matrix,
pressing after each number.
Press
EXE .
The reduced row echelon form gives the solutions
to the equations of the line if the final row of the
matrix is all zeroes.
There are no solutions and the final row is not all zeroes.
In this example the equations are inconsistent.
Quadratic functions
1.6 Drawing a quadratic graph
Example 8
Draw the graph of y= x
2
− 2x+ 3 and display it using suitable axes.
Press MENU and choose 5: GRAPH and press EXE .
The default graph type is Function, so the form Y= is displayed.
Type y = x2− 2x+ 3 and press EXE .
Press SHIF TF3 V-Window and choose F3 STD to use the default axes
which are −10 ≤x≤10 and −10 ≤y≤10.
Press EXE and
F6 DRAW.
The calculator displays the curve with the default axes.
If the calculator
displays a graph,
press EXIT to return
to this screen.
{Continued on next page
9

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Adjust the window to make the exponential curve fit the screen better.
1.7 Solving quadratic equations
When solving quadratic equations in an examination, you do not need to show any
method of solution. You should simply write out the equations in the correct form
and then give the solutions. The GDC will do all the working for you.
Example 9
Solve 3x
2
− 4x− 2 = 0
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
From the menu, choose Polynomial and press
F2 .
You will see the dialog box as shown on the right. Choose equations of
Degree2 and press
F1 .
Another dialog box opens for you to enter the equation.
The general form of the quadratic equation is aX
2
+ bX + c= 0
so we enter the coefficients in a, band c.
Here a= 3, b= −4 and c= −2. Be sure to use the (–) key
to enter the negative values. Press EXE after each value.
Press
F1 and the calculator will find the roots of the equation.
The solutions are x= −0.387 or x= 1.72 (3 sf ).
For help with changing
axes, see your GDC
manual.
If there are previous
equations in the
memory, press EXIT
until you return to this
menu.
10

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1.8 Finding a local minimum or maximum point
Example 10
Find the minimum point on the graph of y= x
2
− 2x+ 3.
Draw the graph of y= x
2
− 2x+ 3 (See Example 8).
Method  - using a table
You can look at a table of the values of the function.
Press MENU and choose 7: TABLE and press
EXE .
Press
F5 SET.
Choose a start and end point for the table and a step of 1.
Use the x-values from the graph you drew.
Press
EXIT .
Press SHIF TSET UP.
Scroll down to Dual Screen and press
F1 T + G.
Press EXE
F6 to view the table.
Press
F5 G·CON to display the graph alongside the table.
{Continued on next page
11

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Press OPTN
F2 GLINK.
When you scroll through the table the cross moves along the curve.
The minimum value shown in the table is 2 when x= 1.
Look more closely at the values of the function around x= 1.
Press SHIF TF3 V-Window.
Choose values −0.9 ≤x≤1.1 and 1.99 ≤y≤2.01 and press EXE .
Change the settings in the table: Press F1 F5 SET.
Set Start to 0.9
End to 1.1
 to 0.01
Press EXE
F6 to view the graph and the table.
Press F5 G·CON to display the graph alongside the table.
Press OPTN
F2 GLINK and scroll through the table.
The table shows that the function has larger values at points around (1, 2).
We can conclude that this is a local minimum on the curve.
Method  – Using the minimum function
Press F5 G-Solv
F3 MIN.
The calculator displays the minimum point on the curve at (1, 2).
12

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Example 11
Find the maximum point on the graph of y= −x
2
+ 3x− 4.
Press MENU and choose 5: GRAPH and press EXE .
The default graph type is Function, so the form Y = is displayed.
Type −x
2
+ 3x− 4 and press EXE .
Press SHIF TF3 V-Window and choose F3 STD to use the default axes
which are −10 ≤x≤10 and −10 ≤y≤10.
Press EXE and
F6 DRAW.
The calculator displays the curve with the default axes.
Adjust the window to make the quadratic curve fit the screen better.
Method  - using a table
You can look at a table of the values of the function.
Press MENU and choose 7: TABLE and press
EXE .
Press
F5 SET.
Choose a start and end point for the table and a step of 1.
Use the x-values from the graph you drew.
Press
EXIT .
Press SHIF TSET UP.
Scroll down to Dual Screen and press
F1 T+G.
If the calculator displays a
graph, press EXIT to return
to this screen.
For help with changing
axes, see your GDC
manual.
{Continued on next page
13

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Press EXE
F6 to view the table.
Press
F5 G·CON to display the graph alongside the table.
Press OPTN
F2 GLINK.
When you scroll through the table the cross moves along the curve.
The minimum value shown in the table is –2 when x= 1 and x= 2.
Look more closely at the values of the function around x= 1 and x= 2.
Press SHIF TF3 V-Window.
Choose values 1 ≤x≤2 and −2.5 ≤y≤−1.5 and press EXE .
Change the settings in the table: Press F1 F5 SET.
Set Start to 0.9
End to 1.1
Step to 0.01
Press EXE
F6 to view the graph and the table.
Press F5 G·CON to display the graph alongside the table.
Press OPTN
F2 GLINK and scroll through the table.
The table shows that the function has larger values at points around (1.5,
–1.75). We can conclude that this is a local maximum on the curve.
Method  – Using the maximum function
Press F5 G-Solv
F3 MIN.
{Continued on next page
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The calculator displays the maximum point on the curve at (1.5, −1.75).
Complex numbers
1.9 Operations with complex numbers
Example 12
Evaluate the following expressions
i 27 4 2
1
2
()( )++ −ii
ii (2 + 3i ) ⋅(3 − 4i )
iii 34+i
iv 1
3
−
+
i
i
v (1 − i )
3
Press MENU . You will see the dialog box as shown on the right.
Choose 1: RUN·MAT and press
EXE .
Complex calculations are entered the same way as you
would enter a real expression.
To enter the imaginary number symbol ipress SHIF Ti.
Enter the expressions and then press
EXE .
The results are as shown.
15

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1.10 Conjugate, modulus and argument
Example 13
Let zi=+13
Find
i z∗ ii |z| iii arg (z)
Press MENU . You will see the dialog box as shown on the right.
Choose 1: RUN·MAT and press
EXE .
i Press OPTN | F3 CPLX | F4 Conj
Enter the complex number in brackets. To enter the imaginary number
symbol ipress F1 i.
Press
EXE
ii Press OPTN | F3 CPLX | F2 Abs
Enter the complex number. To enter the imaginary number symbol i
press F1 i.
Press
EXE
iii Press OPTN | F3 CPLX | F3 Arg
Enter the complex number in brackets.
To enter the imaginary number symbol ipress F1 i.
Press
EXE
1.11 Solving equations with complex roots
Example 14
Solve the equation 2x
3
− 15x
2
+ 44x− 39 = 0
Press MENU . You will see the dialog box as shown on the right.
Choose A: EQUA and press
EXE .
Press SHIF TSET UP
Select a+ bias the Complex mode
Press
EXIT
{Continued on next page
16

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From the menu, choose Polynomial and press
F2 .
You will see the dialog box as shown on the right.
Choose equations of degree 3 and press
F2 .
Another dialog box opens for you to enter the equation.
The general form of the quadratic equation is aX
3
+ bX
2
+ cX + d= 0
so we enter the coefficients in a, b, cand d.
Be sure to use the (–) key to enter the negative values.
Press EXE after each value.
Press
F1 SOLV and the calculator will find the roots of the equation.
The solutions are,
x= 1.5, x= 3 + 2iand x= 3 – 2i.
1.12 Polar form
The GDC displays complex numbers in either Cartesian form (z= x+ yi ) or
in a modulus, argument form r∠
T
– see 1.11 for how to find the modulus and
argument of a complex number expressed in Cartesian form.
Example 15
i Change 2 + 2ito polar form.
ii Change
S
§·
¨¸
©¹
2
3
3cis to Cartesian form.
Press MENU . You will see the dialog box as shown on the right.
Choose 1: RUN·MAT and press EXE .
Complex calculations are entered the same way as you would
enter a real expression.
To enter the imaginary number symbol ipress
SHIF Ti.
i Enter (2 + 2i ) and then press OPTN | F3 CPLX | F6 | F3 r∠ θ
Press
EXE
The result is
SS
§·
¨¸
©¹
i
44
22cos sin or
S
4
22cis
If there are previous
equations in the memory,
press EXIT until you return
to this menu.
{Continued on next page
17

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ii Enter
3SHIF T∠ ab/c2SHIF T
π
3
and then press OPTN | F3 CPLX | F6 | F4 a+ bi
Press
EXE .
You can also change the mode that the calculator uses to display complex
results in settings.
Press SHIF TSET UP
Select Real, Rectangular (a+ bi ) or Modulus Argument (r∠
θ ).
For example, in Polar mode, typing 2 + 2iEXE would result in the number
being displayed in Modulus Argument form without entering F3
r∠ θ .
Exponential functions
1.13 Drawing an exponential graph
Example 16
Draw the graph of y= 3x+ 2.
Press MENU and choose 5: GRAPH and press EXE .
The default graph type is Function, so the form Y= is displayed.
Type 3x+ 2 and press EXE .
Press SHIF TF3 V-Window and choose F3 STD to
use the default axes which are −10 ≤x≤10
and −10 ≤y≤10.
Press EXE and
F6 DRAW.
The calculator displays the curve with the default axes.
Adjust the window to make the exponential curve fit the screen better.
The ∠ symbol is above the
X,i,T key
Note: Type 3
>
X,i,T to enter 3
x
.
The returns you to
the baseline from the
exponent.
18

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1.14 Finding a horizontal asymptote
Example 17
Find a horizontal asymptote to the graph of y= 3x+ 2.
Draw the graph of y= 3x+ 2 (See Example 16).
You can look at a table of the values of the function.
Press MENU and choose 7: TABLE and press
EXE .
Press
F5 SET
Choose a start and end point for the table and a step of 1.
Use the x-values from the graph you drew.
Press
EXIT .
Press SHIF TSET UP.
Scroll down to Dual Screen and press
F1 T + G.
Press EXE
F6 to view the table.
Press
F5 G·CON to display the graph alongside the table.
{Continued on next page
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Press OPTN
F2 GLINK.
As the value of xgets smaller, Y
1
gets closer and closer to 2.
Press EXIT and then press F5 SET.
Change the minimum value of xto –12.
EXE F6 F5 G·CON OPTN
F2 GLINK.
Eventually the value of Y
1
displayed in the table reaches 2.
You can see, at the bottom of the screen, that the actual value of Y
1
is
2.000001882...
We can say that Y
1
→2 as x→− ∞ .
The line x= 2 is a horizontal asymptote to the curve y= 3x+ 2.
Logarithmic functions
1.15 Evaluating logarithms
Example 18
Evaluate log
10
3.95, ln 10.2 and log
5
2.
Press MENU . You will see the dialog box as shown on the right.
Choose 1: RUN·MAT and press
EXE .
Press EXIT until you see the menu on the right, then press
F4 MATH.
Press
F2 logab to open the logarithm template.
For natural logarithms it is possible to use the same method, with the base
equal to e, but it is quicker to press ln . Similarly for logarithms to base
10 you can press
log .
Note that the GDC will evaluate logarithms with any base without having
to use the change of base formula.
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