How do I ……………on the ClassPad 300? For Units 1 - 4 Mathematics VCE Study Design, accreditation period 2006 - 2009. Content refers to Operating System 3.0 of the ClassPad 300. Written by Elena Zema Former Head of Mathematics - Mildura Senior College. Edited by Anthony Harradine Baker Centre, Prince Alfred College. Work in progress, version 2.0. For Operating System 3.0
171
Embed
How do I ……………on the ClassPad 300? Working with ungrouped univariate data .....68 6.1.2 Working with grouped univariate data .....69 6.1.3 6.1.4 Box plot.....71 Box plot
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
How do I ……………on
the ClassPad 300? For Units 1 - 4
Mathematics VCE Study Design,
accreditation period 2006 - 2009.
Content refers to Operating System 3.0 of the ClassPad 300.
Written by Elena Zema
Former Head of Mathematics - Mildura Senior College.
This resource was proudly funded by the Casio Education Australia in their ongoing efforts to provide the very best support to teachers and students using Casio technology.
Page 3
Contents
Introduction – Key ClassPad 300 features. ................................................ 7 A. Catalogue, Action menu, Interactive menu and 2-D.............................................7
Catalogue and Action menu.........................................................................7 Interactive Menu ..........................................................................................8 The 2-D palette. ..........................................................................................8
B. The basics about variables.....................................................................................9
Section 1 - Main application calculations ................................................. 11 1.1 Basic arithmetic calculations .............................................................................11 1.2 Defining variables to have a numerical value....................................................16 1.3 Defining a list variable using the list editor .......................................................17 1.4 Basic function calculations ................................................................................19 1.5 Working with angles ..........................................................................................20
1.5.1 - To change the default settings to operate in degrees with decimal output...............................................................................................................................20 1.5.2 - Expressing angles in degrees to degrees, minutes and seconds ...............21 1.5.3 - Expressing angles in degrees, minutes and seconds to degrees ...............22 1.5.4 - Convert angles in degrees to radians. .......................................................23 1.5.5 - Convert angles in radians to degrees. .......................................................24
1.8.1 - Random Number Generator......................................................................29 “rand” function...........................................................................................29 “randList” function ....................................................................................30 “RandSeed” command ...............................................................................31
Section 2 – Exploring functions ................................................................. 37 2.1 Create a table of values ......................................................................................37
Customising your plot view .................................................................................38 2.2 Enter & plot functions........................................................................................38
2.2.1 Using the trace function ..............................................................................40 2.3 Finding significant points on a graph.................................................................41
2.3.1 To find the x intercept/s (or root/s): ............................................................41 2.3.2 To find the y intercept/s: .............................................................................42 2.3.3 To find the stationary points: ......................................................................42
Maximum point/s .......................................................................................42 Minimum point/s........................................................................................43 Point/s of inflection....................................................................................43
2.3.4 To find an x-value given a specific y-value:...............................................44 2.3.5 To find a y-value given a specific x-value:.................................................44
2.4 Finding the intersection point/s on a two graphs ...............................................45 2.5 Finding the distance between two points ...........................................................45
Section 3 Navigating/Managing the graph window................................. 46 3.1 Configuring graph view window parameters.....................................................46 3.2 Zooming the graph window...............................................................................47 3.3 Scrolling and panning the graph view window..................................................49
Page 4
Scrolling the graph view window ..............................................................49 Panning the graph view window................................................................49
Section 4 Advanced function graphing options ....................................... 50 4.1 Enter and plot functions using parameters.........................................................50 4.2 Graphing an inequality.......................................................................................51 4.3 Graph functions defined in terms of other functions .........................................52 4.4 Draw the inverse of a function...........................................................................53 4.5 Restrict the domain of a function.......................................................................55
5.2.1 Average rates of change..............................................................................57 5.2.2 Instantaneous rates of change .....................................................................58
5.3 Derivatives .........................................................................................................60 5.3.1 Sketching the derivative function ...............................................................63 5.3.2 Tangent to a curve.......................................................................................64
Section 6 – Statistical Calculations............................................................ 68 6.1 Univariate data ...................................................................................................68
6.1.1 Working with ungrouped univariate data ...................................................68 6.1.2 Working with grouped univariate data .......................................................69 6.1.3 Histogram....................................................................................................70 6.1.4 Box plot.......................................................................................................71
Box plot with outliers.................................................................................72 (Modified box plot)....................................................................................72
6.2 Cumulative frequency curves (or ogives) ..........................................................73 6.4 Bivariate data .....................................................................................................74
6.4.1 Scatter plot ..................................................................................................76 6.4.2 Correlation coefficient, r and coefficient of determination, r2 ...................77 6.4.3 Calculating the Least-squares line ..............................................................78 6.4.4 Sketch Least-squares line............................................................................78 6.4.5 Using the Least-squares line .......................................................................79
Section 7 – Numeric Solver Application ................................................... 80 7.1 Using the numeric solver ...................................................................................80
8.1.1 Matrix calculations......................................................................................83 Addition .....................................................................................................83 Subtraction .................................................................................................83 Multiplication.............................................................................................84 Computing a given power of a matrix. ......................................................84 Inverse........................................................................................................84 Determinant................................................................................................85
8.2 Solving simultaneous equations using matrices ................................................85 8.3 Geometric transformations using matrices ........................................................86 8.4 Transition matrices (Markov chains) .................................................................87
Page 5
Section 9 – Sequences ................................................................................. 88 9.1 Define, tabulate & plot a sequence. ...................................................................89 9.2 Summing of a sequence .....................................................................................90 9.3 Difference equations ..........................................................................................91
Section 10 - Advanced function graphing options ................................... 92 10.1 Graphing hybrid (mixed or piecewise) functions ............................................92 10.2 Graphing reciprocal functions..........................................................................93 10.3 Graphing rational functions .............................................................................94 10.4 Graphing sum and difference functions ...........................................................95 10.5 Graphing absolute value (modulus) functions .................................................96 10.6 Graphing product functions .............................................................................97 10.7 Graphing composite functions .........................................................................98
Section 11 – More on Calculus. ................................................................. 99 11.1 Area between two curves .................................................................................99 11.2 Mean value of a function ...............................................................................100 11.3 Second derivative...........................................................................................101 11.4 Volumes of solids of revolution.....................................................................102 11.5 Direction fields for a differential equation.....................................................103
Section 12 – Probability distributions..................................................... 105 12.1 Discrete probability distributions...................................................................105
12.1.1 Finding probabilities, the mean, variance & standard deviation associated with discrete random variables. .........................................................................105 12.1.2 Finding probabilities, the expected value, the variance & the standard deviation associated with the binomial distribution...........................................109
12.2 Continuous probability distributions..............................................................114 12.2.1 Finding k, graphing and finding the mean and variance. ........................114 12.2.2 Standard normal distribution...................................................................116 12.2.3 Inverse cumulative normal distribution ..................................................117
Section 13 - Graphing relations, circles and ellipses ............................. 118
A. Catalogue, Action menu, Interactive menu and 2-D.
Catalogue and Action menu.
The CP 300 was made to enable the user to enter mathematics as we write it on paper (natural input) and conduct mathematical processes without the use of syntax. Every command the CP 300 possess resides in the
catalogue. Launch the application, raise the soft
keyboard and tap the (alogue tab. Set the form to be all. Locate the lim( command.
Suppose we want to determine the
−→ − 2
1
2
lim
xx. We
would now have to remember what the syntax for this command is: lim(function, variable, variable value, limit
direction). So, if you like syntax, you can use the CP 300 in this way. A shortcut to the catalogue, if you like this way of operating, is the Action menu. It contains many of the most commonly used command from the catalogue.
However, a syntax free way of working exists –
read on…
Page 8
Interactive Menu
The interactive menu contains the same options as the Action menu. However, it is used differently.
In the application, enter the expression
− 2
1
x. Then select it by dragging
across it with the stylus. Then tap the Interactive menu, then Calculation and then lim. You can see that a box appears prompting you to input the required information – no recall of syntax required. Entering the correct inputs and pressing tapping OK returns the result. Note that in some cases the CP 300 displays the input in natural form. In other cases the syntax is included on the screen. (Note that OS 3 allows commands from the Interactive menu to be used without
first entering and highlighting an input. This supplement will continue to use the
‘old’ method.)
The Interactive menu acts like a wizard so you do not have to remember what information the CP 300 needs, it tells you what it needs.
The 2-D palette.
The 2-D palette allows you to enter a lot of the mathematics you deal with as you see it in books and
write it on paper. Raise the soft k and tap the )
tab. This reveals 2-D palette. (Tap and to reveal other options.) We can achieve that seen opposite. Not all processes can be entered in this way. So, you are able to choose the way you want to work. The
) palette removes the need for excessive bracket entry, which has always been a difficulty with electronic technology.
Page 9
B. The basics about variables
You will notice on the hard keypad the keys x y Z. When pressed they input a bold italic letter.
You can also input letters using the panel on the soft keyboard. Note that when doing this the letters are not bold and italic. Note the outputs. CP 300 understands xyz
to be zyx ×× , thus removing the need to enter
multiplication signs all the time. CP 300 understands xyz to be the name of some other variable. If we wanted to, we could enter x×y×z – your choice, but it is easy to forget them sometimes! This feature helps us to enter algebraic expressions as we see and write then, provided we use the bold and italic letters.
We are not restricted to just x y Z. On the 9 and ) palettes of the
soft keyboard the option holds 52 variables for you to use.
Note that it is possible to define a variable to be a numeric value. If this has occurred, it can be annoying when trying to perform symbolic computation. To be sure the variables a to z are not defined to be some numeric value use the Clear All Variables command in the Action menu. This command does not clear capitalized variables. To do this, enter the ‘delvar’ followed by a space and then the variable you want to ‘clear’. Or, retrieve the delvar command from the catalogue.
Page 10
C. Active windows, menus and tool bars The CP 300 has a large screen. It allows us to have two applications visible the same time.
For example we can have the application and the application visible at once.
Launch the application then do as the directions in picture (below) ask.
We now have two windows open, one with a darker boundary – the top one in this case. Tap in the bottom window – what do you notice?
Notice that the menu options and the toolbar change. The menu options and the icons on the tool bar belong to the application whose window is active (has the bolder border). This is an important thing to remember as we proceed.
Page 11
Section 1 - Main application calculations
1.1 Basic arithmetic calculations
This section explains how to carry out basic mathematical operations in the Main application. To launch the Main Application:
Tap M within the menu of on the icon panel.
Or, if an application is already launched, tap M
on the icon panel.
Once launched, the Main application window will be displayed as below:
Menu bar
Toolbar
Work area – input
displayed on left,
output displayed on
the right.
Status bar – displays
current mode settings
Icon panel
Page 12
To change the mode the calculator is operating in, you can simply tap on the
specific mode name in the status bar to change it. Alternatively, tap OOOO on the
menu bar.
Page 13
Note: To switch between outputs as exact values to decimal approximations, put the
cursor in either the input or output line and tap . (located on the tool bar).
Example Demonstration 2
3
5
1) Use either the hard
keyboard or 9 soft keyboard to enter the calculation.
2) The calculation can also be entered using natural
input via the ) soft keyboard.
2
5
1. Use the 9 soft keyboard to enter the calculation.
2. The calculation can also be entered using natural
input via the ) soft keyboard.
5 76
1. Use the hard keyboard
or 9 soft keyboard to enter the calculation. (Tap . to get decimal output.)
2. The calculation can also be entered using natural
input via the ) soft keyboard.
Page 14
( ) ( )47 101.2109.4 ×÷×
1. Use the hard keyboard
or 9 options on the soft keyboard to enter calculation. (Tap . to get decimal output.)
2. The calculation can also be entered using natural
input via the ) soft keyboard.
Is 64 46 < ? The judge( function will judge the validity of an equality or inequality. Use
the (alogue on the soft keyboard to enter judge( or just type it in.
(Use the 9 options on the soft keyboard, select the
tab, in order to enter the inequality sign.)
Find prime factors of 360? 1. Key in the number, then
highlight/select it.
2. Tap the Interactive option on the menu bar, tap Transformation, and then select factor.
Note: The Action menu
or (alogue on the soft keyboard could also have been used for this example.
Page 15
Evaluate 324log10
1. Use the 9 options on the soft keyboard to enter calculation.
2. The calculation can also be entered using natural
format via the ) soft keyboard.
(Tap . to get decimal output.)
Note: Logarithms of bases other than 10 can be computed.
Page 16
1.2 Defining variables to have a numerical value
Use the variable assignment key W, to assign a numerical value to a variable. This
key can be found in the 9 options and the ) options on the soft keyboard.
Example Demonstration
4
11
a) Find the hypotenuse, h.
b) Find the angle, θ . Using Pythagoras’ Theorem, assign the following; a = 11, o = 4,
( )22oah += . The angle,
θ can be found using
= −
hyp
opp1sinθ .
You can to use the 9 options including
the options or the
) options on the soft keyboard. You can then edit the initial inputs and all calculations will be recalculated below where the cursor is placed.
Note that variables need to be clear of defined numeric
values before doing symbolic calculations. See the section
“The basics about variables” to see how to clear the
definitions from within the Main application.
θ
Page 17
1.3 Defining a list variable using the list editor
This list editor makes short work of creating and using list variables (or lists data). The list editor can be accessed from within the Main, Graph and Table, Statistics and eActivity applications. To access the list editor window from the Main application:
• Select n from the tool bar, then (. The list editor window will open, and
occupy the bottom half of the main work area.
• Note that the menu bar, tool bar options and status bar change when the list editor window is active.
To enter data:
• Select a list, key in data and press E after each
entry.
• The list name can also be changed. Simply select the current list name (e.g. list1) and change it to an appropriate name (e.g. time, height etc).
Page 18
• List variables can be used in various calculations, graph applications and so on as variables are globally recognized in the CP 300. Some examples are shown below working in the Main application, the List Editor and the Graph and Table application.
Also in Graph and table,
Page 19
1.4 Basic function calculations
Example Demonstration
Evaluate 222 ++ xx when
4=x . Method 1:
1. Raise the (alogue on the soft keyboard.
2. Find Define, and tap
it twice to input this command. (Alternatively type Define and then a space
using the 0 keypad.) Key in the equation and
press E.
3. Now type in f(4) and
press E.
Method 2: 1. Key in the equation and
highlight. 2. Tap Interactive on
the menu bar, then tap Define.
3. Enter the function name
and variable/s into the Define box. (The Expression should already be entered.) Tap
. 4. Now type in f(4) and
press E.
Page 20
1.5 Working with angles
When working with angles, always begin by checking that the CP 300 is set to compute in the angle units you are working with. Look at the status bar to find out which angle the CP 300 is set to use. The default setting is radians. It is highly desirable (and critical in some cases) to include the units of the angle when you make an input. When the units are displayed in the input, the CP 300 then knows what the units of the input are. If no units are given it will assume the units are the units it is set to compute in. If the units are given, it will consider the input to be those units, regardless of what it is set to compute in. The output will always be in the units to which the CP 300 has been set to compute in.
Note: The ClassPad utilizes r, not c,
as a notation for the units of radians.
It may be useful, when working with degrees (or when you require a decimal
approximation and not an exact value), to set the ClassPad to output the approximate
decimal answer. See below for details.
1.5.1 - To change the default settings to operate in degrees with decimal output.
Key Operation
Alternatively,
1. Tap O on the menu bar, or Settings on the Icon Panel.
2. Select Basic Format.
3. Change the Angle setting (by using the drop box) to Degree.
4. It may be useful, when working with degrees (or when you require a decimal
approximation and not an exact value), to set the ClassPad to output the
approximate decimal answer. Tick the Decimal Calculation box under the Advanced settings.
5. Tap .
To change the angle mode the calculator is operating in, simply tap on the specific angle indicator in the status bar to change it.
Page 21
Notice the status bar has changed, and the ClassPad will now be operating in degrees and return outputs that are decimal approximations.
1.5.2 - Expressing angles in degrees to degrees, minutes and seconds
Example
Express °65.34 in degrees, minutes, seconds.
Key Operation
1. Enter the angle and put a degree ( o ) symbol after it. The degree symbol is on
panel within the 9 panel of the soft keyboard. While this is not absolutely necessary it is a good habit to have – see later sections for the reason.
2. Highlight the angles and then tap Interactive on the menu bar.
3. Select Transformation, then toDMS.
Page 22
1.5.3 - Expressing angles in degrees, minutes and seconds to degrees
Example
Express 9334 ′° as a decimal degree value.
Key Operation 1. From the Interactive menu tap Transformation and then select
dms.
2. Enter in the degree, minute and second values.
3. Tap . 4. Tapping Standard on the status bar will change the settings to Decimal.
Once changed, press E . The last input line will be recalculated.
Page 23
1.5.4 - Convert angles in degrees to radians.
Example
Express 9334 ′° in radians.
Key Operation 1. Change the CP 300 to compute in radians if it is not presently (look at the
status bar). See section 1.5.1 for instructions.
2. Repeat the procedure from Section 1.5.3. Add the degree symbol at the end
(found on the panel of the 9tab of the soft k).
3. To convert from the exact value to the decimal approximation, highlight the
answer and tap .. (Or, tap Standard on the status bar. This will change the settings to Decimal.)
Note:
It is in this situation that the inclusion of the degree symbol is critical. It tells the
CP 300 your input is in degrees. Without this, it would assume the input is in
radians as the CP 300 is set to radian mode.
Page 24
1.5.5 - Convert angles in radians to degrees.
Example
Express c
8
5πin degrees, minutes, seconds.
Key Operation
1. Change the CP 300 to compute in degrees if it is not presently (look at the
status bar). See section 1.5.1 for instructions.
2. Enter the angle, including the radian symbol, then highlight. 3. From the Interactive menu tap Transformation and then select
toDMS.
4. Tap E.
Page 25
1.6 Basic trigonometric calculations
When working with angles, always begin by checking that the CP 300 is set to compute in the angle unit you are working with. Look at the status bar to find out which angle the CP 300 is set to use. The default setting is the radian. It is highly desirable (and critical in some cases) to include the unit of the angle when you make an input. When the unit is displayed in the input, the CP 300 then knows what the unit of the input is. If no unit is given it will assume the unit is the unit it is set to compute in. If the unit is given, it will consider the input to be that unit, regardless of what it is set to compute in. The output will always be in the units to which the CP 300 has been set to compute in.
Note: The ClassPad utilizes r,not c
, as a notation for the units of radians.
It may be useful, when working with degrees (or when you require a decimal
approximation and not an exact value), to set the ClassPad to output the approximate
decimal answer. See section 1.5.1 for instructions. This section assumes you have read all of Section 1.5.
Example Demonstration
Evaluate )'4225sin( ° .
Method :
1. Using the 9 options on the soft keyboard, select the
tab, and tap
. 2. Use the Interactive
menu, tap Transformation and then select dms.
3. Enter the angle and tap .
Evaluate
c
7
5cos
π
1. Use the 9 soft keyboard, select the
tab, in order to view/enter the trigonometric functions.
2. Type in/or use the
) options on the soft keyboard to enter the angle.
Page 26
Evaluate )45.190tan( ° and
sin(c
4
π).
This example illustrates how the CP 300 can be set to compute in radians, but if the input is in degrees, it is respected and vice versa.
Find θ in radians if
3.0sin =θ . 1. Be sure the CP 300 is
set to compute in radians.
2. Using the 9 options on the soft keyboard, select the
tab.
3. Key in expression
(using the ) options if you wish).
4. To convert from exact values to approximate, highlight the answer and tap ..
Find θ in degrees, minutes and seconds if 75.0cos =θ .
Page 27
1.7 Basic statistical calculations
While using the Main application, you can easily access the Statistics application. Key Operation
1. Tap on the tool bar and select (
2. Enter data into list1 (or an empty list)
3. Select Calc then One-Variable.
4. Select the XList using the drop down menu. Tap .
5. The Stat Calculation screen will appear containing a basic statistical summary of the selected listed data.
Page 28
There is another method for computing statistics for a list of data. The list name can be copied and then pasted into the Main application work area. We can then the Interactive menu options as seen below.
It is also possible to enter the data directly into the Main application using the following: { }, see below.
Page 29
1.8 Basic probability calculations
Raise the soft keyboard. Select the options on 9 palette. The factorial, combination and permutation commands are found here.
1.8.1 - Random Number Generator
The random number generator on the ClassPad can generate:
• Non-sequential random numbers.
• Sequential random numbers. The ClassPad has three ‘random’ functions:
• rand – generates random numbers.
• randList – generates a list of random numbers.
• RandSeed – configures settings for random number generation (i.e. switch between non-sequential and sequential). The ClassPad can generate nine different patterns of sequential random numbers – this function is also used to choose a specific pattern.
“rand” function
Example Demonstration
Generate random numbers between 0 and 1. Method : 1. Type in/ or locate
rand( in the
catalogue. Press E.
2. To generate more
random numbers,
simply press E again.
Page 30
Generate random integers between 25 and 50 inclusive. Method : 1. Type in/ or locate
rand( in the catalogue. Enter the start and end values separated with a comma.
2. Press E.
3. To generate more
random numbers using these limits, simply
press E again.
“randList” function
Example Demonstration
Generate 20 random numbers between 0 and 1. Method : 1. Type in/ or locate
randList( in the catalogue. Enter the number of random numbers you wish to find and close with a bracket.
2. Press E.
Alternatively:
1. Open a Stat list editor window.
2. Go to the Cal cell of list1 and type in/ or locate randList( in the catalogue. Enter the number of random numbers you wish to find and close with a bracket.
3. Press E.
Page 31
Generate 20 random integers between 1 and 100 inclusive. Method : 1. Type in/ or locate
rand( in the catalogue. Enter 20,1,100 and close with a bracket.
2. Press E.
3. Or use Stat list editor
application, as in the previous example.
“RandSeed” command
This command requires an integer between 0 and 9 for the argument. RandSeed 0 – non-sequential random number generation. RandSeed (integer from 1 to 9) – uses that particular value as a seed for specification for sequential random number generation.
Example Demonstration
Generate sequential random numbers using 4 as the seed value. Method : 1. Type in RandSeed
(and a space) or locate it in the catalogue. Enter 4, then
press E.
2. To generate random
numbers - Type in/ or locate rand( in the catalogue. Press E.
3. To generate more random numbers,
simply press E again.
Page 32
1.9 Basic symbolic calculations
When entering symbols it is good practice to use the bold italic letters available on the hard keyboard and
on the 9 and) soft keyboards under the
option. See the Introduction for reasons. To achieve answers in the same format as those displayed in the following examples; go to Settings and select Basic Format. Under the Advanced options, tick Descending Order. Tap . Before doing symbolic computation, clear variables of any numeric definitions. To do so for all lower case variables a to z use the Clear All Variables command.
Example Demonstration
axxx ++ 23
( )( )yx
yx
+
− 22
Page 33
)(3 ab −
Expand
a. 2)( yx +
b. 4)( yx +
To complete example b. edit the calculations from example a. Highlight and then drag and drop the first input into a new working line and edit the 2 to be a 4 and then
press E.
Factorise 162 −x
Page 34
Factorise 62 −x : a) over Q – the rational
numbers b) over R – the real
numbers
Divide 15 +x by 2−x .
Is 22 4ba − equal to )2)(2( baba +− ?
Note: You could also have used the expand command.
Page 35
Express yx 3
3
2
5+ as a
single fraction.
Solve ,2
12sin =x
a) for all x.
b) for 20 << x . 1. Enter in the equation
and highlight. 2. Select Interactive,
Equation/Inequality then solve.
3. Then use the “for”
operator, U, to key in the condition.
(The tab, holds the “for” operator.)
Solve the following simultaneous equations:
52
153
+=
=+−
xy
yx
1. Key in the function,
using the template
on the ) palette of the soft keyboard (choose the option).
Note: To enter a system with more than 2 equations,
repeatedly tap the template.
Alternatively, use the solve function and following syntax:
Simultaneous Equations
solve({-x+3y=15,y=2x+5},{x,y})
Calulator output
Page 36
Solve 072 <−x for x.
The ClassPad can display the solution to this inequality both numerically and graphically.
To see the solution
graphically, select the
inequality. Open the graph
application, then “drag and
drop” the inequality from
Main to the Graph window .
The graph window will
illustrate the values of x for
which the inequality is true.
Solve 325
9+= CF for C.
Note: The Interactive menu options have been used in these examples. The
Action menu options, direct typing or accessing commands from the (alogue on the soft keyboard could also have been used for these examples.
Page 37
Section 2 – Exploring functions
2.1 Create a table of values
Method Demonstration
1. Tap m on the icon
panel.
2. Open the W
application.
3. Tap in the working line of y1 (or an empty line). Define y1 to be
53 +x .
4. Press E to complete
the process. Notice the box in front of the function is now ticked.
5. Tap 8 on the tool
bar. This will display the Table Input box.
6. Enter the domain you are interested in as well as the steps within the domain to be displayed, and then tap .
7. Select # on the tool
bar. This will generate a table of values and will be displayed in a Table
window. (Note that the menu bar and tool bar
options change when the table window is active).
Note: This process is a helpful guide to choosing sensible settings for the graph view window.
Page 38
Customising your plot view
1. Tap O on the menu bar,
or Settings on the Icon
Panel.
2. Select Graph Format.
3. Check settings.
4. Tap .
2.2 Enter & plot functions
There are two methods of plotting functions:
1. via the W application.
2. via the M application window.
Method Demonstration
Method 1.
1. Tap m on the icon
panel.
2. Open the W
application. 3. Tap in the working line
of y1 (or an empty line). Define y1 to be
103 +x .
4. Press E to complete
the process. Notice the box in front of the function is now ticked.
5. Check your graph view
window settings by tapping 6 located on the tool bar. If necessary, change your window settings, then tap .
6. Tap $ to have a graph of the function appear. (Note that the menu bar,
tool bar options and status bar change when the graph window is active).
Alternatively,
highlight the function
and drag it into the
graph window to
have the graph of the
function appear.
Page 39
Method 2
1. Tap M on the icon
panel. 2. Input the function. (In
this example,
103 += xy .)
3. Press E.
4. Insert a graph window by selecting $ from the tool bar. A graph window should appear. (Note that the menu bar,
tool bar options and status bar change when the graph window is active).
5. Highlight the entire function and drag it into the graph window. The graph of the function will automatically appear in this window.
Try it out for yourself:
Graph the following functions. Remember to always check your graph view window, and
if necessary change the settings, in order to view the graph of the function.
a) 52 −= xy
b) 1354 =+ yx
c) 23)( 2 −= xxf
d) 3xy =
e) 1644 23 +−−= xxxy
f) xy sin=
g) xy 2=
h) )2(log xy e=
i) x
y1
=
j) ( )2−= xy
k) xy =
Page 40
2.2.1 Using the trace function
The trace function allows you to move along a graph. The coordinates of the position where the cursor is displayed in the graph view window. To operate the trace function, the graph view window needs to be active so the tool
bar is visible. Tap the Analysis option on the menu bar and select Trace. Alternatively, tap p on the tool bar to scroll and view other options. Tap =. The cursor will automatically be placed at x = 0. The cursor can be moved along the graph by pressing the cursor key, left or right, or by tapping the left or right graph controller arrows (on the edges of the graph window). If multiple graphs are sketched, press the cursor key, up or down, (or tap the up or down graph controller arrow) to jump between graphs.
If you wish to move the cursor to a specific x-value, after activating the Trace function, press a number key to display the Enter Value box. Key in the value and tap .
Page 41
2.3 Finding significant points on a graph
At times, you will be required to do the following:
• Find x and y intercepts
• Find stationary points (i.e. Maximum/minimum points, points of inflection)
• Calculate an x-value given a specific y-value or vice versa. The following instruction will assume that you have already drawn a graph of the function.
2.3.1 To find the x intercept/s (or root/s):
The graph window needs to be active in order to use the appropriate tool bar. Tap the Analysis option on the menu bar. Tap G-solve, and then select Root. Alternatively, tap p on the tool bar to scroll and view other options. Tap Y This will locate and display the x intercept. Where there is more than one x intercept to be found, simply use the cursor key (left and right) to allow the next intercept to be located.
Page 42
2.3.2 To find the y intercept/s:
Tap the Analysis option on the menu bar, tap G-solve, and then select y-Intercept. This function will locate and display the y intercept.
Note: While you can graph ( )22 −= yx ) in this application some of the Analysis
options can not be performed. However, in the Conics application C analysis
tools can be used.
2.3.3 To find the stationary points:
Maximum point/s
Tap the Analysis option on the menu bar, tap G-solve, and then select Max. Alternatively, tap p on the tool bar to scroll and view other options. Tap U. This function will locate and display the local maximum point of the function within
the bounds of the screen. Where there is more than one maximum point to be found, simply use the cursor key (left and right) to allow the next maximum point to be located.
n
Page 43
Minimum point/s
Tap the Analysis option on the menu bar, tap G-solve, and then select Min. This function will locate and display the minimum point of the function. Where there is more than one minimum point to be found, simply use the cursor key (left and right) to locate the next minimum point.
Note: Alternatively, tap p on the tool bar to scroll and view other options. Tap I.
Point/s of inflection
Tap the Analysis option on the menu bar, tap G-solve, and then select Inflection. This function will locate and display the point of inflection of the function. Where there is more than one point of inflection to be found, simply use the cursor key (left and right) to locate the next point of inflection.
Page 44
2.3.4 To find an x-value given a specific y-value:
Tap the Analysis option on the menu bar, tap G-solve, and then select x-Cal. This function will locate and display the x and y coordinates. Where there is more than one x-value given for a specific y-value to be found, simply use the cursor key
(left and right) to allow the next x-value to be located.
2.3.5 To find a y-value given a specific x-value:
Tap the Analysis option on the menu bar, tap G-solve, and then select y-Cal. This function will locate and display the x and y coordinates.
Page 45
2.4 Finding the intersection point/s on a two graphs
Tap the Analysis option on the menu bar, tap G-solve, and then select Intersect. This function will locate and display the intersection point of the graphs. Where there is more than one intersection point to be found, simply use the cursor key (left and right) to allow the next intersection point to be located.
Note that if three or more functions are drawn and the intersection of two is required, the CP 300 will flash the cursor on one function. Use the up and down cursor keys to
select the functions you require and press E when the required function is selected.
2.5 Finding the distance between two points
This function will locate and display the distance between two specific points. Tap the Analysis option on the menu bar, tap G-solve, and then select Distance. Press a number key to display the Enter Value box. Key in the coordinates and tap . The coordinates will be displayed in the graph view window and the distance calculated in the message box.
Alternatively, you can use the stylus to tap the two points on the screen.
Page 46
Section 3 Navigating/Managing the graph window
This section assumes that the ClassPad is operating in the W application.
Also, check the Graph Format settings (see page 37 for further details).
3.1 Configuring graph view window parameters
1. Tap 6 located on the tool bar. (Or, tap O, then select View Window.)
This feature displays the View Window dialog box.
2. If necessary, make the appropriate changes, depending on the nature of the
intended graph. Tap . (Note: The menu bar, tool bar options and status bar change when the graph window is active). Brief explanation of View Window parameters (rectangular coordinates):
xmin – minimum value of x-axis ymin – minimum value of y-axis
xmax – maximum value of x-axis ymax – maximum value of y-axis
xscale – marker spacing of x-axis yscale – marker spacing of y-axis
xdot – value of each screen pixel horizontally
ydot – value of each screen pixel vertically
The x/y dot and x/y dot values will change automatically when the x/y maximum and minimum values are changed.
Graph
Editor
Window
Graph
View
Window
Message box
Page 47
A number of View Window configurations are saved in the memory of the CP 300. Tap the Memory drop down menu when the view window setting input box is open. Brief explanation of some of the preset parameters: Initial – square window settings (Default). Undefined – auto-configuration of the view window box. You can also Store and Recall your own settings.
3.2 Zooming the graph window
The ClassPad features an extensive selection of Zoom commands that can be used for either a specific region of a graph or to enlarge and/or reduce an entire graph.
Page 48
Brief explanation of some of the Zoom commands:
Zoom command Demonstration Box Select the Box zoom option and then select a region of the graph you want enlarge with the stylus by dragging a rectangle on the screen. Once the stylus has been taken off the screen, the selected region will be enlarged to fill the entire graph window display. You can also access this command from the tool bar. Tap p on the tool bar to scroll and view other options. Tap Q.
Factor This command allows you to configure the zoom factor settings.
Zoom In
Quick Zoom There are seven of these commands: Quick Initialize Quick Trig Quick log(x) Quick e^x Quick x^2 Quick –x^2 Quick Standard These quick zoom commands will redraw the graph using preset built-in View Window parameters.
Page 49
3.3 Scrolling and panning the graph view window
Scrolling the graph view window
Once a graph has been sketched, it can be scrolled left, right, up or down using the cursor key or the graph controller arrows.
Note: The graph controller arrows will only be active if the Graph Format settings are set with the G-Controller box ticked (see page 37 for further details).
Panning the graph view window
To operate this function, the graph view window needs to be active in order to use the appropriate tool bar. Tap p on the tool bar to scroll and view other options. Tap z. Position the stylus on the graph view window, and drag the window to an appropriate location. Once the stylus is removed, the graph will be redrawn at that particular location.
Page 50
Section 4 Advanced function graphing options
This section four assumes that the ClassPad is operating in the W application.
4.1 Enter and plot functions using parameters
Example Demonstration
Let cbxaxf +−= 2)()( ,
where a, b and c are integers. How is the function transformed under the following conditions?
a. a = 1, b = 0 and c
varies b. a = 1, b varies and c
= 0 c. a varies, b = 0 and c
= 1 Method: a. Key the following into y1:
}2,1,0,1,2{)0(1 2 −−+−x
Use the ) soft
keyboard and tap to enter the parameter list or
just use the { on the 9 palette. b. Key the following into y2:
0})2,1,0,1,2{(1 2 +−−−x
c. Key the following into y3:
1)0}(2,1,0,1,2{ 2 +−−− x
You could also define a list
as a variable and use that
variable.
a)
b) c)
Page 51
4.2 Graphing an inequality
Example Demonstration
Sketch 12 +≤ xy .
Method: If you already have an equality entered, tap on the equal sign, (=). This will display the Type box, enabling you to select the form you wish to graph. Select the appropriate form and tap . Tap $ to graph the region.
Alternatively,
You can use the option available on the tool bar. Tap d or Type in the menu bar.
Sketch the region bounded by the following:
• 12 +≤ xy and
• xy −> 3 and
• 0≥x and
• 0≥y .
Page 52
4.3 Graph functions defined in terms of other functions
Example Demonstration
Sketch 2)( xxf = . Explain
graphically, the outcome of the following transformations
a. )(xf−
b. 2)( +xf
c. )(2 xf
d. )4( −xf
Method: 1. In the Main application
window: Type in, (followed by a space), or locate Define in the catalogue. Key in the function and
press E.
2. Launch the application. Make y1 be
f(x). Press E after
each entry.
3. Tap $ to graph the function. Functions can be sketched simultaneously or individually, depending on whether the check box is ticked.
4. You can also specify the graph line style. Simply tap the line style next to the function and the Graph Plot Type window will appear. Select your desired type and press .
Line style
Page 53
Alternative method.
Make y1 = 2x .
Then define the remaining functions in terms of y1(x).
4.4 Draw the inverse of a function
Example Demonstration
Sketch 2)( xxf = and its
inverse. Method: 1. Enter the function into
y1. Press E.
2. Select the Analysis option, tap Sketch followed by Inverse.
3. The inverse of the function will automatically appear in the Graph View
Window. The inverse function will also be defined in the message box.
Page 54
Alternative method:
1. Key the function into the Main application
window. Highlight the function and the select Interactive on the menu bar, tap Assistant, followed by invert.
2. Select variables you wish to invert in the invert window and press .
3. The function and its inverse of the function will appear on the right hand side of the screen (work area).
4. This can be sketched if required, by opening a graph view window. (Tap $ on the tool
bar.) Select each in turn and “drag and drop” them into the graph view window.
Page 55
4.5 Restrict the domain of a function
Example Demonstration
Sketch xy = , where 0≥x .
Method: Key in the function, then
the “for” operator, U, followed by the restricted domain.
(Using the 9 palette on the soft keyboard, select the
tab, in order to view/enter the “for” and inequality operators.)
Sketch xy = , where
22 <<− x . Method: Key in the function, then
the “for” operator, U, followed by the restricted domain.
(Using the 9 soft
keyboard, select the tab, in order to view/enter the “for” and inequality operators.)
Note that the “for”, U, can be use in conjunction with the solve command to find
solutions within a given domain.
Page 56
Section 5 – Calculus
5.1 Limits
Example Demonstration
Find
a.
∞→ xx
1lim
b.
+→ xx
1lim
0
c.
−→ xx
1lim
0
Method: 1. Key in the limit
statement, using the
limit feature, , on
the ) soft keyboard (choose the option).
2. To enter the direction of the limit, use the + and - operators available on
the 9 soft keyboard, tap to view/select. (Note: you can also use the standard + and – operators.)
Alternatively,
The limit statement could be entered using the Action or Interactive options on the menu bar, or catalogue.
Page 57
5.2 Rates of Change
5.2.1 Average rates of change
Example Demonstration
Calculate the average rate of change for
22)( 2 ++= xxxf on the
intervals: a. x = 3 and x = 3.1 b. x = 3 and x = 3.05 c. x = 3 and x = 3.001 d. x = 3 and x = 3 + h
Method: 1. Define the function,
press E.
2. Using the ) palette on the soft keyboard, enter a fraction template then enter the average rate of change.
3. Calculations can easily performed by selecting the previous input, dragging and dropping it into the next working line and then editing it.
Page 58
5.2.2 Instantaneous rates of change
Example Demonstration
Calculate the instantaneous rate of change where
22)( 2 ++= xxxf at x = 3.
Method: To find the instantaneous rate of change, find the limit (as h approaches 0) of the average rate of change for the interval [3 , 3+h]. 2. Define the function,
press E.
3. Key in the function,
using the limit template,
, on the ) palette of the soft keyboard (choose the
option).
Page 59
Alternative method:
1. Key in and select the function.
2. Tap Interactive, then Calculation, followed by lim.
3. Enter variable, point and direction into the lim box. Tap .
Note that the Direction
input can be -1 if you want
the limit approaching from
the left, 1 for the right and 0
for both.
Page 60
5.3 Derivatives
Example Demonstration
Find a. the derivative of
22)( 2 ++= xxxf
b. )2(f ′
c. )3(−′f
Method for part (a): 1. Define the function,
press E.
2. Key in the problem,
using the derivative
feature, , on the
) soft keyboard (choose the option).
Alternative method for part
(a) See screen captures 1 &
2.:
1. Key in and select the function.
2. Tap Interactive, then Calculation, followed diff.
3. Select differentiation. Enter variable and order into the diff box. Tap
.
Method for part (b): Key in the function, then
the “for” operator, U, followed by the argument.
(Use the 9 soft
keyboard, select the tab, in order to view/enter the “for” operator.)
1. 2.
Page 61
Alternative method for part
(c):
1. Key in and select the function.
2. Tap Interactive, then Calculation, followed diff.
3. Select Derivative at value, then enter variable, order and derivative into the diff box. Tap .
4. This feature helps you to use syntax to solve the task.
Page 62
Using the W application, find )2(f ′
where 22)( 2 ++= xxxf .
Method: 1. In this example the
function has been defined and stored as f(x).
2. Key in the function as f(x) into the graph
editor window. Press
E. Tap $ to graph.
3. Check the Graph
Format settings. [Tap
O, then Graph Format.] Make sure Derivative/Slope is ticked. Tap .
4. With the graph view
window active, tap Analysis, then Trace.
5. Press 2. The Enter Value box appears. Tap .
6. The derivative at that point, along with the coordinates of the function will be displayed in the graph view window.
Page 63
5.3.1 Sketching the derivative function
Example Demonstration
Sketch 22)( 2 ++= xxxf
and its derivative, )(' xf .
Method: 1. In this example the
function has been defined and stored as f(x).
2. Key in the function as f(x) into the graph
editor window. Press
E.
3. Key in the derivative
function, using the derivative template,
, on the ) palette of the soft keyboard (choose the
option). Press
E. Tap $ to graph.
4. You can also specify the
graph line style. Simply tap the line style next to the function and the Graph Plot Type window will appear. Select your desired type and press .
Graph line style
Page 64
5.3.2 Tangent to a curve.
Example Demonstration
Sketch 22)( 2 ++= xxxf
and the tangent at 2−=x . Find the equation of the tangent.
Method: 1. In this example the
function has been defined as f(x).
2. Key in the function into the graph editor
window. Press E.
Tap $ to graph.
3. With the graph view
window active, tap Analysis, then Sketch, followed by Tangent.
4. Enter -2 and the Enter Value box will appear. Tap .
5. Crosshairs will appear at that point. You must
press E in order for
the tangent to appear.
6. The tangent at that
point, along with the coordinates of the function will be displayed in the graph
view window.
The equation of the
tangent appears in
the message box.
Page 65
5.4 Integration
5.4.1 Indefinite integrals
Example Demonstration
Find the integral of
2610)( 34 ++= xxxf .
Method: Key in the function, using
the integral feature, ,
on the ) soft keyboard (choose the option).
Press E.
Note: Do not enter lower & upper terminals for indefinite integrals. Alternative method:
1. Key in and select the function.
2. Tap Interactive, then Calculation,
followed by ∫ (the
integral sign).
3. Select Indefinite integral. Enter the variable you are integrating with respect to into the variable box. Tap .
When working with indefinite integrals, don’t forget you will need to include the constant of integration, c, when writing down your answer.
Page 66
5.4.2 Definite integrals (without a graphical display)
Example Demonstration
Calculate dxex
x
∫ +4
1
25
.
Method: Key in the function, using
the integral template, ,
on the ) soft keyboard (choose the option).
Press E.
Note: Don’t forget to enter lower & upper limits for definite integrals. Alternative method:
1. Key in and select the function.
2. Tap Interactive, then Calculation,
followed by ∫ , the
integral sign.
3. Select Definite integral. Enter the variable you are integrating with respect to, the lower and upper
limits into the ∫ input
box. Tap . Note: This method will
provide an ‘exact’ result if
possible.
Page 67
5.4.3 Definite integrals (with a graphical display)
Using the W application, compute and display and interpretation of
dxex
x
∫ +4
1
25
.
Method: 1. Key the function into
the graph editor. Press
E. Tap $ to graph.
2. With the graph view
window active, tap Analysis, then G-Solve, followed by
∫dx .
3. Press 1 and the Enter
Value box will appear. Key in the lower and upper intervals and tap
.
4. The function, along with the area interpretation of the integral will be displayed in the graph view window. The decimal approximation of the integrals value will be displayed in the message box.
Note that this method will return a decimal approximation for the integral.
Page 68
Section 6 – Statistical Calculations
In this section we use the I application.
6.1 Univariate data
6.1.1 Working with ungrouped univariate data
Example Demonstration
The height of 20, year 11 students from across Australia has been recorded. The results, in centimeters, are: 185, 176, 184, 175, 173, 183, 182, 184, 174, 174, 169, 179, 190, 175, 178, 203, 145, 188, 177, 162. Calculate the five number summary (min, Q1, median, Q3, max.) for the sample and make a histogram. Method:
1. Enter data into list1 (or an empty list).
2. Select Calc then One-Variable.
3. Select the XList using the drop down menu. Tap .
4. The Stat Calculation screen will appear containing a basic statistical summary of the selected listed data.
To draw a histogram of these data use the SetGraph then Setting ...menu.
Page 69
6.1.2 Working with grouped univariate data
Example Demonstration
The following table shows the number of “Smarties” in each of 50 packets .
# of Smarties Frequency
40 1 41 8 42 29 43 7 44 4 45 1
a. Calculate the mean, median and mode.
b. Find the total number of Smarties in 50 packets.
Method: 1. Enter data into list1
and frequency into list2 (or empty lists).
2. Select Calc then One-Variable.
3. Select list1 for the XList and list2 for the Freq using the drop down menu. Tap .
4. The Stat Calculation screen will appear containing a basic statistical summary of the selected listed data.
Note: Name the list before
entering your data. Once
named, the list is
considered to be a
variable.
Page 70
6.1.3 Histogram
Example Demonstration
The frequency table shows the length (l) of 80 fish caught in a fishing competition.
Length (mm) Frequency
295 ≤ l<305 8
305 ≤ l<315 17
315 ≤ l<325 19
325 ≤ l<335 13
335 ≤ l<345 10
345 ≤ l<355 6
355 ≤ l<365 4
365 ≤ l<375 3
Draw a histogram. Method: 1. Enter the midpoints of
each class into list1, frequency into list2.
2. Tap G on the tool
bar. (Or, select SetGraph from the menu bar, then Setting.)
3. Adjust the Set StatGraphs options. Press .
4. Tap y on the tool bar to sketch the curve.
5. The Set Interval box will appear – set HStart to 300 and HStep to 10 (this is critical). Press .
6. The histogram will appear in the StatGraph window. (Press Analysis, then Trace, to display the XList and Freq on the histogram.)
Page 71
6.1.4 Box plot
Example Demonstration
The height of 20 year 11 students from across Australia has been recorded. The results, in centimeters, are: 185, 176, 184, 175, 173, 183, 182, 184, 174, 174, 169, 179, 190, 175, 178, 203, 145, 188, 177, 162.
1) Construct a box plot with this data.
2) Hence, state the five figure summary (min, Q1, median, Q3, max) for the sample.
Method:
1. Enter data into list1 (or an empty list).
2. Tap G on the tool
bar. (Or, select SetGraph from the menu bar, then Setting.)
3. Adjust the Set StatGraphs options. Type: MedBox. Make sure
you do not tick the Show OutliersShow OutliersShow OutliersShow Outliers box. Tap .
4. Tap y on the tool
bar to sketch the boxplot.
5. The box plot will appear in the StatGraph window.
6. Tap Analysis, then Trace. Use the cursor key or graph
controller arrows (left/right) to jump between values.
Page 72
Box plot with outliers
(Modified box plot)
- utilises the IQR×5.1 rule,
which defines limits for “outliers”. To make modified box plot, make sure you tick the Show OutliersShow OutliersShow OutliersShow Outliers box. We have set up StatGraph 2
as a modified box plot and
then drawn both StatGraph
1 and StatGraph 2.
Page 73
6.2 Cumulative frequency curves (or ogives)
Example Demonstration
The frequency table shows the length of 80 fish caught in a fishing competition.
Length (mm) Frequency
300 – 309 8
310 – 319 17
320 – 329 19
330 – 339 13
340 – 349 10
350 – 359 6
360 – 369 4
370 - 379 3
a. Add a cumulative
frequency column to the table.
b. Represent the data using cumulative frequency curve.
Method: 1. Enter length data into
list1, frequency data into list2 and cumulative frequency values into list3.
2. Tap G on the tool
bar. (Or, select SetGraph from the menu bar, then Setting.)
3. Adjust the Set StatGraphs options. Press .
4. Tap y on the tool bar to sketch the curve.
Page 74
6.4 Bivariate data
This section will use the following example to demonstrate bivariate data analysis with the ClassPad. Example: Swimming Pool Attendance and Daily Maximum Temperature The operators of a local swimming pool record the following data:
Day Max. temp C° Attendance
1 18 870
2 17 819
3 30 2168
4 16 714
5 20 1435
6 22 1458
7 16 819
8 12 406
9 14 231
10 15 572
11 16 603
12 17 839
13 15 572
14 15 806
15 18 1218
16 19 1007
17 23 931
18 21 1215
19 19 995
20 21 275
21 25 1894
22 29 2301
23 26 2207
24 24 2109
25 30 2564
Task: a) Calculate the summary statistics
for the two variables. b) Construct a scatter plot to
examine the relationship between attendance and temperature.
c) Calculate Pearson’s product–moment correlation coefficient, r.
d) Calculate the coefficient of determination, r2
. e) Calculate the equation of the least
squares line. f) Sketch the least squares line. g) Use your equation to predict the
attendance on a day of maximum
temperature at C°23 and compare
your result to Day 17.
Page 75
Example Demonstration
Task: a) Calculate the summary statistics for the two variables. Method:
1. Enter temperature into list1 and attendance into list2. to rename the lists.
2. Select Calc then Two-Variable.
3. Select Programs\temp for the XList and Programs\attend for the YList using the drop down menu. Tap .
4. The Stat Calculation screen will appear containing summary statistics of the selected two variable data. Scroll down to see the y variable statistics.
Page 76
6.4.1 Scatter plot
b) Construct a scatter plot to examine the relationship between temperature and attendance.
Method: 1. Tap G on the tool
bar. (Or, select SetGraph from the menu bar, then Setting.)
2. Adjust the Set StatGraphs options. Press .
3. Tap y on the tool bar to view the scatter plot.
Page 77
6.4.2 Correlation coefficient, r and coefficient of determination, r2
c) Calculate Pearson’s product–moment correlation coefficient, r. d) Calculate the coefficient of determination, r2
. Method: These tasks can be performed simultaneously. 1. With the List Editor
window active, select Calc from the menu
bar, followed by Linear Reg.
2. Adjust the Set
Calculation options. Tap .
3. The Stat Calculation screen will appear containing, correlation coefficient, r, and the coefficient of determination, r2. (MSe is the mean square error.
Note: Once the Set Calculation window is
closed by tapping ,
the least squares line will
automatically be sketched
in a Statgraph window.
Note that this information
can also be accessed from
the StatGraph window when
active:
- select Calc from the menu bar, followed by Linear Reg.
Page 78
6.4.3 Calculating the Least-squares line
e) Calculate the least squares regression line. (linear regression) The ouput screen from the previous section also includes the slope and intercept. Note: Once the Set Calculation window is
closed by tapping ,
the least squares line will
automatically be sketched in
a Statgraph window.
6.4.4 Sketch Least-squares line
f) Sketch the least squares line. An alternative to the method
seen above is:
1. To sketch the least squares line tap G on the tool bar. (Or, select SetGraph from the menu bar, then Setting.)
2. Adjust the Set StatGraphs options. Leave StatGraph 1 as is and set up StatGraph 2 as shown. Tap .
3. Tap y on the tool
bar.
Page 79
6.4.5 Using the Least-squares line
g) Use your equation to predict the attendance on a day of maximum
temperature at C°23 and compare your result to Day 17. There are many different
ways to achieve this – here
is one method:
1. With the List Editor window (or StatGraph window) active, select Calc from the menu
bar, followed by Linear Reg.
2. Adjust the Set
Calculation options. Be sure to change the Copy Formula setting to y1. Tap .
3. Tap M on the icon
panel. Key in y1(23).
Press E. Note. When entering y1(23) or
y1(x), be sure to use the ‘y’
from the qwerty keyboard
and not the y from the hard
keyboard that denotes a
variable.
Page 80
Section 7 – Numeric Solver Application
This section assumes that the ClassPad is operating the N application.
Note: While this application can be launched from the Menu and also be from the
Graph Editor, 3D Graph Editor and the Main application. Simply tap O when in
these applications.
Equations can be “dragged and dropped” from the above mentioned applications into the Numeric Solver window.
7.1 Using the numeric solver
Example Demonstration
The volume of a cone, radius r cm and height h
cm, is given by: 3
2hr
Vπ
= .
a) Find the volume of a cone with r = 12 cm and h = 7 cm.
b) Find the radius of a cone if h = 10 cm and V = 1500 cm3.
Method:
1. Key in the Equation: (Use the
) soft keyboard to enter the equation using natural input).
Tap E.
Page 81
2. The list of expression’s variables will appear. Enter the values.
3. Select the variable you want to solve by checking the adjacent button.
4. Tap 1 on the tool
bar.
5. The Result will appear in a dialogue box. Tap .
Note that the Left-Right = 0
refers to the value of the
right hand side of the
equation subtracted from
the left hand side of the
equation of the value of the
variable computed. If this is
0, then we confident the
correct value of the variable
has been computed.
The lower and upper bounds for the solution can also be specified. If the solution is not within the specified range, an error will occur – see below.
Page 82
Section 8 – Matrices
8.1 Inputting matrix data
The examples below use the ) soft keyboard to enter the matrix using natural input.
Example Demonstration
Define the following matrices.
=
34
12A
−
−=
240
112B
−
−
−
=
2
2
1
C
−=
21
12D
Method: Key in the matrix, using the
features, , on
the ) soft keyboard (choose the option).
Page 83
8.1.1 Matrix calculations
This subsection will use the following exercise to demonstrate matrix calculations using the ClassPad 300. It assumes you have defined matrices A to D as shown in the previous section.
Given the following matrices:
=
34
12A
−
−=
240
112B
−
−
−
=
2
2
1
C
−=
21
12D
Addition
a) DA +
Subtraction
b) DA −2
Calculate the following:
a) DA + b) DA −2 c) BC
d) 2A
e) 1−A f) Adet
Page 84
Multiplication
c) BC
Note that using the BC from the Qwerty key board will not give the result we want. B×C will. It is good practice to use the letters on the VAR panel – the bold and italic ones that denote a variable.
Computing a given
power of a matrix.
d) 2A
Inverse
e) 1−A
Page 85
Determinant
f) Adet
Method:
1. Enter A. 2. From the menu bar,
tap Interactive, then Matrix-Calculation, followed by det.
8.2 Solving simultaneous equations using matrices
Example Demonstration
Using matrices, solve
103 =− yx and 152 =+ yx .
Method: We can express the simultaneous equations in matrix form:
=
−
1
10
52
13
y
x
And so BAy
x×=
−1
Enter
−
52
13as A and
1
10as B and then compute
BA ×−1
Page 86
8.3 Geometric transformations using matrices
Example Demonstration
a) Determine the transformation matrix,
yxD , , for the
combination of transformations: a dilation by a factor of 5 parallel to the x axis followed by a dilation by a factor of 3 parallel to the y axis.
b) Find the coordinates of the transformed image of the point (7,9) under
yxD , .
Note that we know that each
point ),( yx is mapped onto
its image ),( yx ′′ by:
dycxy
byaxx
+=′
+=′
Therefore, in matrix form:
=
′
′
y
x
dc
ba
y
x
Page 87
8.4 Transition matrices (Markov chains)
Example Demonstration
Claude has a coffee shop. He sells coffee and biscotti. He realises that if a person buys (and enjoys) a coffee on a particular day, there is a 75% probability that the person will return a buy coffee the next day. In addition, if a person buys biscotti one day then there is a 50% probability that they will purchase biscotti the next day. On Monday, 90% of Claude’s patrons bought coffee and 40% bought biscotti. a) Determine a transition
matrix, T that models this situation.
b) Determine the initial
state matrix, 0S .
c) What is the probability that a patron will purchase a coffee on Tuesday?
d) What is the probability that a patron will purchase a coffee on Friday?
Parts a) and b).
Part c) Part d).
Page 88
Section 9 – Sequences
When you open the H application, the following will be displayed:
Page 89
9.1 Define, tabulate & plot a sequence.
Example Demonstration
Consider the sequence
132 >+= nnnan , .
a) Enter the sequence into the ClassPad.
b) Tabulate the sequence.
c) Plot the sequence. Method: 1. This is an explicit
relationship and so tap the explicit tab.
2. Enter the sequence using the B available on the
menu bar. Press E.
3. To create a table for the sequence, tap 8, to display the Sequence Table Input box. Enter the desired conditions. Tap . Then tap #, to display the table.
4. To plot the sequence, the table window must be
active. Tap ! to plot. (Or, select Graph, then G-Plot on the menu
bar.)
a)
b)
c)
Page 90
9.2 Summing of a sequence
Example Demonstration
Consider the arithmetic
series: ...392613 +++ a) Find the sum of the
first 20 terms. b) Find the sum of the
first n terms. c) What is the term
number that would sum to an answer of at least 4000?
Method: We note that, a = 13, d = 13. Therefore,
)( 11313 −+= nan .
1. Open the Main application.
Use the ) palette on the soft keyboard to enter the sum template
2. Using the expression
found in part (b), we can set it equal to 4000 and solve for n. Obviously, the solution would be a positive number.
Recall, to use the solve function: Select the equation. Tap Interactive menu, then tap Equation/Inequality, and then solve.
Part a) and b)
Part c)
Page 91
9.3 Difference equations
Example Demonstration
Consider the sequence defined by the difference equation:
12 01 =+=+ ttt nn , .
a) Find the first seven terms of the sequence.
b) Find the 25th term. c) Find the sum of the
first 5 terms. d) Plot the sequence.
Method:
1. Enter the difference equation on the recursive form (since it is a recursive relationship) using the B available on the
menu bar. Press E.
2. To tabulate the sequence, tap 8, to display the Sequence Table Input box. Enter the desired conditions. Tap . Then tap #, to display the table.
3. To plot the sequence, the table window must
be active. Tap ! to plot. (Or, select Graph, then G-Plot on the menu bar.)
Part a)
Part b)
Part c)
Page 92
Section 10 - Advanced function graphing options
This section four assumes that the ClassPad is operating in the W application.
10.1 Graphing hybrid (mixed or piecewise) functions
Example Demonstration
Sketch the graph of
−≤+
<<−+
≥
=
2,)2(
02,2
0,
)(2
xx
xx
xx
xf
Method: Key in the function, then
the “with” operator, U, followed by the restricted domain. Tap $ on the tool bar.
(Using the 9 palette on the soft keyboard, select the
tab, in order to view/enter the “with” and inequality operators.) An alternative way to plot a piecewise function is to use the piecewise command, for the syntax see opposite. We have use a nested system for the command: piecewise(condition, value if this condition is true, value if this condition is false) Note: Using this methods sees an (almost) vertical line joining the pieces at
0=x .
piecewise(x≤-2,(x+2)^2,piecewise(-2<x<0,x+2,x)
Page 93
10.2 Graphing reciprocal functions
Example Demonstration
Sketch the graph of 1)( += xxf and the
reciprocal function, )(
1
xf.
Method: Key in the function into y1 and the reciprocal function into y2. Tap $ on the tool bar. You could also define the function 1)( += xxf in the
Main application. Then go to the Graph & Table application to graph the defined function
Page 94
10.3 Graphing rational functions
Example Demonstration
Sketch the graph of
4
65)(
2
−
+−=
x
xxxf ,
showing axial intercepts and asymptotes.
Method: Key in the function into y1. Tap $ on the tool bar. Check your graph view
window settings by tapping 6 located on the tool bar.
If necessary, change your window settings, then tap
. Alternatively, you can use the Zoom commands to resize the graph view. Use the Table function to help you find any asymptotes. Select # on the tool bar. This will generate a table of values and will be displayed in a Table window.
Page 95
10.4 Graphing sum and difference functions
Example Demonstration
Sketch the graph of
xxy
1+= .
Method:
To sketch the graph of the sum (or difference) function, the individual functions are sketched onto the same set of axes. Using the method of addition of ordinates, the sum (or difference) function can then also be sketched. Key in the sum (or difference) function into y1. y2 = x
y3 = x
1
Use the Table function to help you use the method of addition of ordinates. Select # on the tool bar. This will generate a table of values and will be displayed in a Table window. By adding the y-coordinates of y2 and y3 will give the y-coordinate value of the sum function, in this case y1. Graph (and view the table of) all three functions to check your answers.
Page 96
10.5 Graphing absolute value (modulus) functions
Example Demonstration
Sketch the graph of
xy 2sin2= over the
domain [ ]π2,0 .
Method:
Key in the function, using absolute value then
the “with” operator, U, followed by the restricted domain.
(Using the 9 palette on the soft keyboard, for the absolute value function.
Also, to select the tab, in order to view/enter the “with” and inequality operators.) Check your graph view
window settings by tapping 6 located on the tool bar.
If necessary, change your window settings, then tap
. Alternatively, you can use the Zoom commands to resize the graph view.
Page 97
10.6 Graphing product functions
Example Demonstration
Sketch the graphs of i) xxf =)(
ii) xxg sin)( =
iii) )()( xgxf .
Method: Check your graph view
window settings by tapping 6 located on the tool bar.
If necessary, change your window settings, then tap
. Alternatively, you can use the Zoom commands to resize the graph view.
Page 98
10.7 Graphing composite functions
Example Demonstration
For the functions xxf sin)( = and
xxg =)( :
Sketch and state the domain of i) ))(( xgf
ii) ))(( xfg
Method: 1. Define the functions first. This way you can easily key in calculations and/or graph the functions. Check your graph view
window settings by tapping 6 located on the tool bar.
If necessary, change your window settings, then tap
. Alternatively, you can use the Zoom commands to resize the graph view.
Page 99
Section 11 – More on Calculus.
11.1 Area between two curves
Example Demonstration
Find the area between the two curves over the given interval [0, 1]
21)( xxf −=
xxg −= 1)( .
Method: 1. Define the functions in
the Main application. 2. Tap $ to show the
graph view window. 3. ‘Drag and drop’
functions in the graph
view window. The graphs of the functions will automatically appear in this window.
4. Use the sketch to help you determine which function needs to be ‘subtracted’.
5. Tap in and make the Main application window active.
6. Key in and select the function.
7. Tap Interactive, then Calculation,
followed by ∫ , the
integral sign. 8. Select Definite
integral. Enter the variable you are integrating with respect to, the lower and upper
limits into the ∫ input
box. Tap .
Page 100
11.2 Mean value of a function
Example Demonstration
Find the mean value of the
function 26)( xxf = over
the interval [0, 4]. Method: Use the variable assignment
key W, to assign a numerical value to a variable. This key can be
found in the 9 options
and the ) options on the soft keyboard. By using this method, you can easily change the upper and lower limits and/or the function. Simply “highlight”, key in changes
and press E. The final
answer will appear without having to re-input the integral.
Page 101
11.3 Second derivative
Example Demonstration
Find )(xf ′′ if
xxxf 2)( 2
5
+=
Method: 1. Enter the function and
highlight. 2. Tap Interactive,
then Calculation, followed diff.
3. Select differentiation. Enter variable and order (2) into the diff box. Tap .
Page 102
11.4 Volumes of solids of revolution
Example Demonstration
Consider the region bounded by the x-axis and the given lines for:
20;sin
π=== xandxxy .
Find the volume of solid of revolution generated when the region is rotated about the x-axis. Method: 1. Define the function in
the Main application. 2. Tap ! to show the
graph editor window. Enter the function and tap $ to graph.
3. With the graph view
window active, tap Analysis, then G-Solve, followed by
∫ dxxf 2)(π .
4. Key in the lower value (press 0) and the Enter Value box will appear. Key in the lower and upper intervals and tap .
5. The function, along with the volume interpretation of the integral will be displayed in the graph
view window. The decimal approximation of the volume will be displayed in the message box.
Note: To achieve an exact
solution, use the soft keyboard to input the volume of revolution.
Page 103
11.5 Direction fields for a differential equation.
Enter the application. Enter the DE yy 2=′ . Tap to have a slope field
generated. Tap r to have the full screen view.
Now tap r again and tap the IC (Initial Conditions) tab. Set some ICs and then tap
the again. This will plot a path through the slope field, starting at (0,1) in this case. You can also plot the graph of a function to test your conjecture about the solution to the DE.
Page 104
Tapping the 6 icon reveals the View Window settings and allows you to set at will. Note the Steps setting.
Note that the Spreadsheet on the CP 300 has CAS capabilities and so making a spreadsheet to display Euler’s Method numerically and graphically is quite simple. An eActivity that already does this is available from www.casioed.net.au.
Page 105
Section 12 – Probability distributions
12.1 Discrete probability distributions
12.1.1 Finding probabilities, the mean, variance & standard deviation associated with discrete random variables.
As is true in most sections, there are numerous ways to complete the computations outlined in this section. We have chosen methods that keep the user working within
the Main application, M.
Example Demonstration
Suppose a random variable X has distribution: x 0 1 2 p(x)
8
2k
8
4 3k−
2
2 2k−
Find the value(s) of k and the values of p(x) in each case. Method: 1. Define the three
elements in the list as a function p(x).
2. Find the sum of p(x). 3. Then set the sum equal to 1
and solve the resulting equation.
Page 106
Example Demonstration
Note: If the distribution is given in the form: p(x) = 5,3,1),14( =− xxkx
proceed as shown opposite to find P(X>1).
Page 107
Example Demonstration
Find the mean, variance and standard deviation of the discrete random variable with distribution:
p(x) = 5,3,1,91
)14(=
−x
xx
Method: 1. Define p(x). 2. Compute the mean using
the mean formula. Note that two ways are illustrated opposite.
3. Now store the mean value
by defining a variable to have the value attained. Then use the compute the variance.
Note that any letter may be
used in place of µ (mu).
4. Finding the square root of
the variance value returns the standard deviation.
Page 108
Example Demonstration
Find the mean, variance and standard deviation of the discrete random variable with distribution:
,.......3,2,1,4
1)( =
= xxp
x
Method: 1. Define p(x). 2. Compute the mean using
the appropriate formula formula.
3. Compute the variance
using the appropriate formula.
Note: Prior to doing this example we
have chosen to ‘Clear All
Variables’ from the Edit menu.
Also not that the use of the
symbols µ (mu) and σ (sigma)
are not necessary.
Page 109
12.1.2 Finding probabilities, the expected value, the variance & the standard deviation associated with the binomial distribution.
Example Demonstration
Suppose a random variable X has binomial distribution with n = 10 and p = 0.4. Find P(X =4). Method 1: 1. Enter the Statistics
application, I.
2. From the Calc menu, choose Distribution.
3. Then choose the
Binomial PD option and tap .
4. Enter the values for x,
Numtrial and prob. Tap and the probability value for P(X =4) is returned.
Note: A nice plot of the distribution can be made by tapping the graph icon $ in the top left corner. The plot can be traced to compute any other individual probabilities for this distribution.
Page 110
Method 2: This method requires us to use a more functional approach. 1. Enter the Main application
J.
2. Define the function
Bin(n,r,p) as the ‘binomial formula’.
3. We can use function
notation to compute the value of interest.
Page 111
Example Demonstration
Suppose a random variable X has binomial distribution with n = 10 and p = 0.4. Find the P(X >6) Method 1: This method requires us to determine 1-P(X ≤ 6) 1. Enter the Statistics
application, I.
2. From the Calc menu, choose Distribution.
3. Then choose the
Binomial CD option and tap .
4. Enter the values for x,
Numtrial and prob. Tap and the probability value for P(X ≤ 6) is returned.
Note: A nice plot of the distribution can be made by tapping the graph icon $ in the top left corner. The plot can be traced to compute any other cumulative probabilities for this distribution.
Page 112
5. Now return to the Main
application and compute 1 minus the probability value returned. prob can be found in the catalogue, or simply type it in.
Method 2: This method requires us to use a more functional approach. 1. Enter the Main application
J.
2. Define the function
Bin(n,r,p) as the ‘binomial formula’.
3. We can use the (‘sum’ function) to compute the cumulative probability required. Note the two different ways to achieve the result.
Page 113
Example Demonstration
Suppose a random variable X has binomial distribution with n = 10 and p = 0.4. Find the mean, variance and standard deviation of X. Method: 1. Enter the Main application,
J.
2. Define the function
Bin(n,r,p) as the ‘binomial formula’.
3. Now apply the correct
formula for the mean of a binomial distribution, making use of the defined function Bin(n,r,p). Similarly for the variance and then standard deviation.
Note: For a binomial distribution, the mean can be computed by simply multiplying n by p and the variance by finding
)1( ppn −×× .
Page 114
12.2 Continuous probability distributions.
12.2.1 Finding k, graphing and finding the mean and variance.
Example Demonstration
A continuous random variable, X, has distribution described
by 0,)( 2 ≥= − xkexf x . Find k,
draw the distribution and then find the mean, variance and standard deviation. Method: 1. Enter the Main application
J.
2. Define the function f(x). 3. We know that the total area
under this curve is 1 (as it is a probability distribution). So we can find k as seen opposite.
We could now solve for k,
but in this case k is clearly 2.
4. A quick way to graph this
function is to tap the application launcher icon and select $. Then in Main Work Area, enter f(x)|k=2 and press
E.Then ‘drag and drop’
the result into the Graph
View window.
Page 115
5. Then utilise the correct
formulae for the mean and variance of a continuous random variable.
Note:
It is not necessary to use the Greek symbols (followed by the equal sign) in this computation.
Page 116
12.2.2 Standard normal distribution.
Example Demonstration
Find )2Pr( <Z using the
cumulative normal distribution. Method:
1. In the I application, tap Calc then Distribution.
2. Select Normal CD. Tap .
3. Enter the lower and upper intervals, standard deviation and mean. Tap
. 4. The next screen will
give the probability and the option to sketch the probability region (this
is always a very good
idea). 5. Tap $ to sketch the
probability region.
Page 117
12.2.3 Inverse cumulative normal distribution
Example Demonstration
Find the value of c if 9370.0)Pr( =<<− cZc .
Method:
1. In the I application, tap Calc then Distribution.
2. Select Inverse Normal CD. Tap
. 3. Enter the tail setting,
area, standard deviation and mean. Tap .
4. The next screen will give the unknown z values and the option to sketch the probability region (this is always a
very good idea). 5. Tap $ to sketch the
probability region.
Page 118
Section 13 - Graphing relations, circles and ellipses This section explains how to graph circles and ellipses when the ClassPad is operating
in the C application. (You can also use this application to graph parabolas,
hyperbolas and other general conics.)
When you open the C application, the following will be displayed:
The following describes the buttons located on the tool bar while the Conics Editor
window is active.
The following describes the buttons located on the tool bar while the Conics Graph
window is active.
Note: - You can only input one
conics equation at a time in the Conics Editor window.
- This application contains various preset conic formats making equation input efficient.
- Various graph analysis tools can be used when the Conics Graph window is active.
Page 119
Example Demonstration
Sketch the graph of the circle with centre (2, 2) and radius 1. Method: 1. Enter the equation by
soft keyboard input OR using the preset conics form menu – press q.
2. If using the preset menu, select the form you wish to graph. Tap
. 3. The selected form will
be displayed in the Conics Editor window. The equation can now be modified.
4. Tap ^ to graph. Note: Various graph analysis tools can be used when the Conics Graph
window is active.
Page 120
Example Demonstration
Sketch the graph of the ellipse:
( ) ( )1
9
2
4
122
=−
+− yx
.
Method: 1. Enter the equation by
soft keyboard input OR using the preset conics form menu – press q.
2. If using the preset menu, select the form you wish to graph. Tap
. 3. The selected form will
be displayed in the Conics Editor window. The equation can now be modified.
4. Tap ^ to graph. Note: Various graph analysis tools can be used when the Conics Graph
window is active.
Page 121
Section 14 - Complex Numbers To work with complex number calculations, the ClassPad needs to operate in Complex mode.
The Complex Submenu contains commands that can be used in complex number calculations.
Explanation of the commands: arg – will output the argument of a complex number. cong – will output the conjugate complex number. re – will output the real part of a complex number. im – will output the imaginary part of a complex number. cExpand – expands a complex expression to rectangular form. compToPol – converts a complex number into its polar form. compToTrig – converts a complex number into its trigonometric form.
To change the mode the calculator is operating in, you can simply tap on the specific mode name in the status bar to change it. Alternatively, tap
O on the menu bar.
Page 122
Example Demonstration
For iz 31+= , find the
following: a) argument of z over [0, π2 ]. b) conjugate of z. c) real part of z. d) imaginary part of z.
Method: 1. Enter the equation by
soft keyboard input for i.
2. Tap Interactive, then Complex, followed by arg.
3. Continue using the Complex Submenu to complete the complex calculations.
Note: Conversions from Cartesian form to polar form can be made using the compToTrig and compToPol commands. And vice versa using the cExpand command.
Page 123
Section 15 - Financial Calculations - TVM
Enter the TVM application ; you will see it has an amazing array of abilities
Tap Compound Interest. You will see that the variables associated with compound interest (including Annuity calculations) are laid out with input boxes ready to be filled. If you are not sure what they mean, tap into one and then tap Help at the bottom of the screen.
This is the Financial
Application Initial screen. It appears if you have not yet used the application or when you use the Clear All command in the Edit menu while using the application.
To configure the
settings, tap O and then Financial Format.
Page 124
Suppose that we wish to determine the size of the repayments on a loan of $400 000 for which the interest rate is 6% p.a. compounded monthly and the term of the loan is for 30 years. Then we enter, as seen below left, and then simply tap the variable we wish to compute.
Now tap the Calculation menu and note the Amortization option. The appropriate values from our previous problem are carried over and now we can carry out some ‘what if’ exercises. We can do this for any period within the life of the annuity.
PM1 is the number of the first installment in the period being considered and PM2 is the number of the last installment in that period. Above we can see that after the first 10 installments are paid, the annuity has a balance of $39592.72.
Page 125
Section 16 - Vectors
16.1 Viewing vectors.
Enter the Geometry application G. Tap the Draw menu icon drop down box and
select the vector tool. Then tap on the Cartesian Plane in two different spots, the first for the tail of the vector and the second for the head. A vector appears, labeled as r in this case.
Now tap on the selection tool and then on the vector itself. Then tap the “take me around the corner” icon to reveal the measurement bar.
Page 126
You can now edit the components and change the vector.
Now tap on the Cartesian Plane in ‘free space’ to deselect the vector and tap on the point representing the vectors tail. You can then edit its co-ordinate, say to (0,0). Using the Zoom Out option from the View menu completes the task.
Page 127
16.2 Operating with vectors.
Enter the Main application. Bring up the soft keyboard and tap the button on the 2D sheet. Enter a vector by tapping the column matrix template. Tapping it twice will allow you to enter a vector with three dimensions. You can add and subtract as you would expect.
In the Interactive menu you will see a Vector submenu and all of the commands it contains.
Page 128
Most of these uses are self explanatory; the following screen shots illustrate some of the functionality. Enter the vector (s) first, highlight them and then choose Interactive, Vector and the command you require.
16.3 Vectors that are functions of time
Suppose tjtir sin2cos~~~
+= where t is time. What path does this describe?
This path can be plotted by considering this as a function in parametric form, namely:
ty
tx
sin2
cos
=
=
Enter the application. From the Type menu, tap ParamType and enter the x and y components. Tap the graph icon, $.
The path appears to be elliptical.
Page 129
Note that the settings for the values for t can be found in the View Window setting window (scroll to the bottom).
Page 130
Appendices - Text-book cross referencing
Units 1 & 2
A.01 Cambridge Essential Advanced General Mathematics
Text Page Description
How
do I …
Section
How do I
…
Page
8 Matrix calculation 8.1 82
8 Matrix calculation 8.1 82
15 Determinant & inverses for 2x2 matrices 8.1.1 84
16 Determinant & inverses for 2x2 matrices 8.1.1 84
33 Intersection point 2.4 45
46 Solve application 1.9/7.1 32/80
47 Factorise 1.9 32
48 Expand 1.9 32
48 zeros/roots/x-intercepts 2.3.1 41
49 Approximate 1.1 13
49 Common denominator 1.9 32
49 Proper fraction 1.9 32
49 solve 1.7 27
70 Highest common factor 1.1 11
70 Factor 1.1 11
109 Sequence 9.1 89
110 Sequence 9.1 89
132 Fixed point iteration 9.1 89
144 Solve application 1.9/7.1 32/80
145 Solve application 1.9/7.1 32/80
151 Expand – partial fractions 1.9 32
154 Expand – improper fractions 1.9 32
156 Simultaneous equations 2.4 45
157 Simultaneous equations 2.4 45
199 Transformations 4.3 52
200 Transformations 4.3 52
250 Sketch function over a specific domain 4.5 55
251 Sketch function over a specific domain 4.5 55
255 E.g. 12 solving circular function equations 2.4 45
256 E.g. 12 solving circular function equations 1.9 32