Using ClassPad in the VCE Methods Exam. · This book provides the latest and best ways to use the...

Using ClassPad in the VCE Methods Exam. To be first published in 2010. DRAFT EDITION, November, 2009. Questions about this publication should be directed to [email protected] Copyright © 2009 StepsInLogic. ISBN All rights reserved. Except under the conditions specified in the Copyright Act 1968 of Australia and subsequent amendments, no part of this publication may be reproduced, stored in a retrieval system or be broadcast or transmitted in any form or by any means, electronic, mechanical, photocopying recording or otherwise, without the prior written permission of the copyright owners. This publication makes reference to the CASIO ClassPad. This model description is a registered trademark of CASIO COMPUTER CO., LTD. CASIO® is a registered trademark of CASIO COMPUTER CO., LTD.

mailto:[email protected]

AA BB OO UU TT TT HH II SS PP UU BB LL II CC AA TT II OO NN This book provides the latest and best ways to use the CASIO ClassPad software (OS 3.04.4000) for successful use in the VCE Mathematical Methods examination. The book is written by people who are experts in school mathematics and have an intimate knowledge of the CASIO ClassPad. They have been supported by practicing Victorian teachers who are experts in VCE Mathematical Methods and the examination of this subject’s content. CAS enabled calculators are complex devices that offer multiple ways of performing a given calculation. This book provides the most effective approaches to the calculations that may be needed in examination type questions. The content of this book has been solely influenced by the VCE Mathematical Methods CAS examinations, from the year 2009 and previous, and the strengths and limitations of the CASIO ClassPad algorithms. This books aims to provide a minimum set of CASIO ClassPad skills to help with efficiency and accuracy when sitting the VCE Mathematical Methods examination. It is possible that future examinations may require a calculation that is not included in this book.

Draft Edition This is a draft edition of this publication. The first edition is due for release in early 2010. Your critical feedback about every part of this publication would be greatly appreciated. Please forward all feedback, via email, to [email protected] November, 2009.

mailto:[email protected]

CC OO NN TT EE NN TT SS 1 Functions and Graphs 1.1 Defining in M and then graphing. ........................................ 4 1.2 Solving for x – exact.......................................................... 5 1.3 Solving for x – decimals only ................................................ 6 1.4 Intersection – exact .......................................................... 7 1.5 Minimum & maximum values – exact 1.................................... 8 1.6 Minimum & maximum values –exact 2.................................... 10 1.7 Minimum and maximum – decimals only ................................. 11 1.8 Rational function - asymptotes............................................ 13 2 Algebra 2.1 Composite functions ........................................................ 14 2.2 Inverse function.............................................................. 15 2.3 Solving equations – exact 1 ................................................ 16 2.4 Solving equations - exact 2 ................................................ 17 2.5 Simultaneous equations .................................................... 18 2.6 Equations and matrices..................................................... 19 3 Calculus 3.1 Average rate of change..................................................... 20 3.2 Derivative function .......................................................... 21 3.3 Derivative at a point ........................................................ 22 3.4 Sign of the derivative ....................................................... 23 3.5 Tangents and normals....................................................... 24 3.6 The indefinite integral...................................................... 25 3.7 The definite integral ........................................................ 26 3.8 Average value of a function................................................ 27 3.9 Definite integral equations ................................................ 28 4 Probability 4.1 Probability density function ............................................... 29 4.2 Expected value & median .................................................. 30 4.3 Normal distribution – probability......................................... 31 4.4 Normal distribution - inverse .............................................. 32 4.5 Normal distribution – mu & sigma......................................... 33 4.6 Binomial distribution – probability........................................ 34 4.7 Transition matrix - probability ............................................ 35 5 Troubleshooting, hints & tips ................................................... 36 6 My additions 6.1 My additions -1............................................................... 37 6.2 My additions -2............................................................... 38 6.3 My additions -3............................................................... 39 Index ................................................................................. 40

Draft Edition, November 2009 3

11 .. 11 DD EE FF II NN II NN GG II NN M AA NN DD TT HH EE NN GG RR AA PP HH II NN GG .. 1

How do I define a function in MM and then graph it? How do I define a function in M

1 .. 11 DD EE FF II NN II NN GG II NN M AA NN DD TT HH EE NN GG RR AA PP HH II NN GG ..

M and then graph it?

Example:

Draw a graph of 2( 8)( ) 1000(cos 2) 10002

xC x π −⎛ ⎞= + − x⎜ ⎟⎝ ⎠

for 8 16≤ ≤ .

Method Demonstration

Launch the M application. Set ClassPad to Alg, Standard, Real and Rad modes – just tap on the words.

Press k Enter the function, using the options on the mth:TRIG and the 2D keyboard.

Select the function, open the Interactive menu and tap Define. Name the function C, use a ‘text’ C, not a variable to enter the C in the equation. Tap OK.

Tap $ to open the graph window in the bottom half of thcreen.

e

x), take your stylus off and then drag it into

e graph window. Tap 6 to adjust the View Window settings. Use the domain information to set an appropriate xmin and xmax (8 and 16). Tap OK. Tap Zoom: Auto.

(r if you wish).

s Tap, Zoom: Quick Initialize. Select C(the screenth

4 Draft Edition, November 2009

11 .. 22 SS OO LL VV II NN GG FF OOHow do I calculate x given a value

required?

RR XX –– EE XX AA CC TT C(x) in this case) if exact values are for y (or

Example:

If 2( 8)( ) 1000(cos 2) 10002

xC x π −⎛ ⎞= + − 8 1x⎜ ⎟⎝ ⎠

for 6≤ ≤ , find x if ? ( ) 1250C x =

Method Demonstration This section follows on from the section on the previous page.

In the M application enter the equation to be solved: C(x) = 1250.

e

gin the equation.

:

Select C(x) from the first input linand drag it into the second imput line to be Select the equation and then tapInteractive: Equation: solve.

Make sure that the Solve option s checked. This will ensure that ithe ClassPalues fo

ad will return exact r the solution to the

In this case, the general solutions are given (since the function involved is cyclic).

vequation if possible Tap OK.

To calculate the four solutions within the domain we are

interested in, insert | 8 16x≤ ≤

at the end of the equation in the input line and before the ‘,’.

Press E. The four solutions are given with values in exact form. To convert the exact values to decimal approximations, select the output and tap u.

Note By re-dragging C(x) and then 1250 into the graph window we produce a graphical display of the equation and can see there are 4 solutions.


11 .. 33 SS OO LL VV II NN GG FF OO RR XX –– DD EE CC II MM AA OO

tion is acceptable?

LL SS LL SS NN LL YY cimal

NN LL YY cimal How do I calculate x given a value for the function (y or C(x) in this case) if a de

approxima

How do I calculate x given a value for the function (y or C(x) in this case) if a de

approxima

11 .. 33 SS OO LL VV II NN GG FF OO RR XX –– DD EE CC II MM AA OO

tion is acceptable?

Example:

If 2( 8)( ) 1000(cos 2) 10002

xC x π −⎛ ⎞= + − 8 x⎜ ⎟⎝ ⎠

for 16≤ ≤ , find if ? ( ) 1250C x =x

Method Demonstration This section follows on from the section on the previous page. Tap in the graph window to make

it active and then tap r

.

ap Analysis, then G-Solve,

the osition of the left most

for the he point are

t as a decimal ation and not in exact

form. If there is more than one point of

:

Tthen Intersect. The flashing cursor indicatespintersection poin

valuesf t

t, and numericco-ordinates o

, budisplayedapproxim

intersection, tap to move to the next one.

The same process can be achiev

directly from the ged

pp ion.

e graph and proceed as outlined above.

a licat

Define y2 to be 1250, press E Tap $ to draw th


11 .. 44 II NN TT EE RR SS EE CC TT II OOHow do I find the intersecti n ed? How do I find the intersecti n ed?

NN –– EE XX AA CC TT point of two graphs point of two graphsoo if exact values are requir if exact values are requir

EFind where graph of the function the graph of

xample: 2)( 2 −= xexf intersects xexg =)( .


ML

aunch the application.

– just tap n the words.

Press k Use the 2D keyboard for the simultaneous equations template

~. Enter the two functions, and solve for x and y.

Set ClassPad to Alg, Standard, Real and Rad modeso

To convert the exact value output to decimal approximations, select the output and tap u

Alternatively, the intersection point could be found graphically, but this method will return decimal approximations only. Tap $ to open the graph window in the bottom half of the screen Tap, Zoom: Quick Initialize. Then select one function and drag it into the graph window. Repeat for the second function. Tap, Analysis: G-Solve: Intersect.

Note This method returns exact values of the points of intersection if possible.


11 .. 55 MM II NN II MM UU MM && MM AA XX II MM UU MM VV AA LL UU EE SS –– EE XX?

AA CC TT 11 AA CC TT 11 How do I find the range of a function, given its domain and if exact values are requiredHow do I find the range of a function, given its domain and if exact values are required

11 .. 55 MM II NN II MM UU MM && MM AA XX II MM UU MM VV AA LL UU EE SS –– EE XX?

Example:

Find the range of the function 21)2sin(3)(,3

, →⎟⎞π0: =⎠⎢⎣

⎡ xfRf +−x .

Method Demonstratio n

Launch the M tion. et ClassPad Alg Standard,

applica to , S

Real and Rad modes – just tap on the words.

Enter 3 sin(2 ) 1 2x − + using

4the 2D keyboard for and the

ap e graph

tialize.

mth:TRIG keyboard for sin.

$ to open thTwindow in the bottom half of the screen. Tap, Zoom: Quick Ini Select the input line and drag it into the graph window.

Tap 6 to adjust the View Window settings. Use the domaininformation to set an appropriate xmin and xmax

In this case

question. Tap OK. Tap Zoom: Auto.

. choose values slightly

outside the domain given in the

Select 3 sin(2 ) 1 2x − + .

Tap: Interacti

ve: alculation:

e

and tap OK.

CfMin. In the fMin dialoglue box, enterthe endpoints of the domain as thStart and End values

Note Local maximum and minumum values may fall on the edge of a domain, so draw the graph with extended domain so you can see this.



Method Demonstration fMin searches for the minimum value of the function on the set domain, and so the result could be either a local minumum or an end point – in this case it is a local minumum. The maximum value can be found using fMax. Select and Drag the entire fMin input line into the next working line.

Edit, using the abc keyboard, the“in” to be “ax”.

Press E.

As neither value occurs at x =

3π

the range includes both values.

So the range is [ ]5,2 .

Note We use fMin and fMax ONLY when we can see a complete graph of the function for the domain in which we are interested. If the domain is x R∈ , use one of the methods in the next two sections.

Caution fMin and fMax will only calculate a single minimum or maximum value. Therefore, if you have a cyclic function and multiple maximum or mimum values are visible, do not use fMin or fMax. Use one of the methods in the next two sections.

11 .. 66 MM II NN II MM UU MM && MM AA XX II MM UU MM VV AA S

required?

LL UU EE S –– EE XX AA CC TT 22 exact How do I calculate the minimum or maximum value of a function for all real x if

values are

Example:

Find the value of x that minimizes the function 292 x+ 9 ,

5 13xT x

⎛ ⎞− R= +⎜ ∈⎟⎜ ⎟⎝ ⎠

.


Launch the M application. Define the function as T(x) – see revious sections for assistance.

e see there is only one value of x that satisifies this and so we can ow be sure that there is only one

p Tap $ to open the graph

x) into the window.

window. Tap, Zoom: Quick Initialize. Drag T( Use 6 or n to adjust the axesscales.

W

nstationary point – a minimum in this case.

We need to find the value of x when the derivative of T(x) is zero. Calculate the derivative of T(x).

The } template on the CALC tab of the 2D offers a quick way to do this. Set the derivate equal to zero and select the input.

Tap Interactive: Equation:

exact value(s), if le.

inally, find T(

solve. Make sure Solve is checked. This

ill ensurewpossibTap OK.

F54

) to find the

minimum value.

Note We see a local minimum. But, if not sure of the behaviour outside 5 9x− ≤ ≤ , we can use calculus to proceed.

Note We see there is only one value of x that satisifies this equation and so we can now

sure that there is only one oint – a minimum

be stationary p

ase.in this c


11 .. 77 MM II NN II MM UU MM AA NN EEHow do I calculate maximum a mals

DD MM AA XX II MM UU MM –– DDnd minimum values if deci

CC II MM AA LL SS OO NN LL YY are acceptable?

Example:

inimum value of Find the m 27 ( 8)( ) 1000(cos 2) 10006xC x π −⎛ ⎞= + − for ⎜ ⎟

⎝ ⎠8 1x≤ ≤ 6and the corresponding value(s) of x .


Launch the g application. (Alteratively you can do this from

within the M applciation as in previous sections). Set ClassPad to Rad and Real modes – just tap on the words. Enter the function, using options on the mth:TRIG and the 2D keyboards.

Press E. Tap 6 to adjust the View Window settings.

In the View Window dialogue box, use the domain to set the xmin and xmax. Choose values slightly outside the end points of the domain. Tap OK. Tap $. Not much of a graph, but don’t worry …

Tap Zoom, then Auto This automatically sets an appropriate y-scale based on the x-scale (domain) that you have entered. This is ideal when you are given a domain for the function. Note

Tap r (bottom of screen) to enlarge the viewing windows area.

Note Don’t have 4 black arrows on your graph? Tap: Settings (bottom left of screen): Graph Format: then tick G-Controller.



The graph rewith local mi

veals we are dealing nimums. Min will

then G-So

only search for local minimums. Tap Analysis,

l

ve,

then Min.

The flashing cursor indicates the osition of the relevant point, p

and numeric values for the co-ordinates of the point are displayed. If there is more than one minimum

point, tap or press : to move to the next one.

Caution The first minimum showing is not in the required domain. Be vigiliant. We set the xmin and xmax outside the domain as if the min/max point is at the endpoint of the domain and the xmin and xmax are set at the end points, the ClassPad may not display the min/max point.

Note Min – searches for local minimums. Max – searches for local maximums. fMin – searches for the minu um value of the function within the domain visible. fMax - searches for the maximum value of the function within the domain visible.

m


11 .. 88 RR AA TT II OO NN AA LL FF UU PP TT OO TT EE SS How do I change the form asily see calculate its

asymptotes?

NN CC TT II OO NN -- AA SS YY MM of a rational function to more e

Example:

Find the asymptotes of x

xx−−

=2

3)( . Df fR: → ,


Launch the M application.

Press k Use the 2D keyboard to enter the

n for the function.

his input.

expressio Select t

Tap Interactive, then Transformation, then propFrac.

Fr ,

s a o = -1.

om this form of the expressione see that the function harizontal asymptote of

why


22 .. 11 CC OO MM PP OO SS II TT EE FFHow do I calculate a composite functi

UU NN CC TT II OO NN SS on?

Example:

If and

32)( 2 −+= xxxg 32)( += xexf . Find ))(( xgf .



Enter 2 3xe + .

The use of templates on the 2D keyboard allows for the natural input of expressions like this one.

ap Interactive,

.

Select this input. T

en Defineth

Check that the contents of the Define dialoglue box are correct and tap OK. Note If you wish to change the function’s name, it must be done as text, using the abc keyboard. Repeat this process to define g(x).

Enter f(g(x)), being sure the f and g are text taken from the abc keyboard.

Press E.

To obtain simplified output:

select the input line f(g(x)),

tap Interactive,

then Transformation,

then simplify.

Note Using function notation in this way can be very useful in the exam. Some uses include: • For a given f (x) is

d

f (u) + f (v) = f (u + v )?

f (f (x)) = x? • Determining a transforme

function like f (x – 3 ) • Checking properties of

given functions, e.g. is


22 .. 22 II NN VV EE RR SS EE FF UU NN CC TT II OO NN 22How do I calculate an inverse function? How do I calculate an inverse function?

.. 22 II NN VV EE RR SS EE FF UU NN CC TT II OO NN

Example:

For the function xexh −+=1)( , find the inverse function 1−h .

Method Demonstration Exchanging variables gives us


ML

ress k P

Enter 1 yx e−= + . electS this input.

Tap Interactive, then Equation, then solve. In the solve dialogue box change the Variable input to y. Tap OK.

The inverse function is calculated.


22 .. 33 SS OO LL VV II NN GG EE QQ U TT 11 How do I find solutions to equati d?

U AA TT II OO NN SS –– EE XX AA CCons if exact values are require

Example:

Find the solution set for the equa

tion 045 24 =+− xx ee .



Press k Enter the equation to be solved. Select this input.

Ta

e ,

Make sure that Solve is checked as this will ensure the ClassPad will return exact values, if possible. Tap OK.

p Interactive,

n Equation

th then solve. Check that the contents of the solve dialogue box are correct.

In this case exact values are provided.

Note: No CAS is able to give exact values for the solutions of all equations. In some cases the ClassPad will change methods and provide decimal values. Choosing Solve numerically in the solve dialogue box (see above) instead of Solve will force the ClassPad to return decimal values. This method is often very fast.


22 .. 44 SS OO LL VV II NN GG EE QQ UU AA TT II OO NN SS –– EE XX AA CC TT 22 22How do I find exact values for the solutions to equations within a given domainHow do I find exact values for the solutions to equations within a given domain? ?

.. 44 SS OO LL VV II NN GG EE QQ UU AA TT II OO NN SS –– EE XX AA CC TT 22

Example:

Find the exact values of ∈x ),( ππ− such that 1)2cos(2 =x . Method Demonstration

Launch the tion.

ap OPTN so th u can enter

elect the input.

M

applica

Press k Enter the function, using the mth:TRIG keyboard to enter cos. T at yothe given domain as shown. S

Tap Interactive,

t the contents of the ogue box are correct.

ill

le. an

then Equation, then solve. Check thasolve dial Make sure that Solve is checkedas this will ensure the ClassPad wreturn exact values, if possib

d tap OK.

The solution set is displayed with

ues in exact form.

n the right end of the utp to scroll and see the

lutions.

val

Tap the : ouo t line

rest of the so


22 .. 55 SS II MM UU LL TT AA NN EE OO How do I find the simultaneo uations?

UU SS EE QQ UU AA TT II OO NN SSus solution to a system of eq

Example:

Find the solution to the system of linear equations 12 24

3mx yx my m+ =+ =

.



Press k

~

Note Tapping this template repeatedly increases the number of equations that that can be entered.

Use the 2D keyboard and the template to enter the equations.

Use the VAR keyboard to enter the system of equations. Include the variables x , y for which it will be solved.

Press E

.

This same method will solve systems of non-linear equations like:

2

( 1) 5( 1) 12.5

p qp q

− =

− =


22 .. 66 EE QQ U A TI OO NN SS How can I determine when s a unique s

AA NN DD MM AA TT RR II CC EE SS a system of equations ha olution?

U A TI

E

C stem of lin

xample:

onsider the sy ear equations represented by 1 2 0

4 1k x

k y k+⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

For what values of w solutionk ill this system have a unique ? Method Demonstration

L

aunch the M application.

Press k Use the 2D:CALC keyboard to obtain the matrix template. Enter the elements of the matrix, sing the VAR keyboard. u

Select this matrix.

Tap: I

nteractive:

ae

M trix-Calculation: t. d

Select the expression for the determinent of the matrix, tap on it and drag it across to make it input.

Set this determinant equal to zero and solve. Note This tells us when the system of equations does not

have a unique solution.

For all other values of k it will have a unique solution.


33 .. 11 AA GG EE RR AAHow do I calculate the av

AA VV EE RR TT EE OO FF CC HH AA NN GG EE erage rate of change of a function?

Example: Find the average rate of change of the function 1)( 3 +−= xxxf between 0=x and 3=x .



Press k

Enter 3 1x x− + . Select this input line. Tap Interactive, then Define.

Enter the expression

03)0()3(

−− ff

Use the abc keyboard when entering f as it is text not a variable.

Press E to calculate.


33 .. 22 DD EE RR II VV AA TT II VV EE FF UU NN CC TT II OO NN 33How do I calculate a derivative function? How do I calculate a derivative function?

.. 22 DD EE RR II VV AA TT II VV EE FF UU NN CC TT II OO NN

Example: Find the de 3 2 1y x x= − + . rivative of


L aunch the M tion.

sion for the

e,

then Calculation,

then diff(erentiation).

applica

Press k Enter the expres

nction. fu Select this input line. Tap Interactiv

diff(erentiation) allows us to alculate a derivative function

ative at a iation in this

ive

ble (to be ifferentiated with respect to), x

in this case. Enter the order of the differentiation (i.e. 1st derivative, 2nd derivative etc), 1st in this case. Tap OK when these selections have been made.

cor to evaluate the derivalue, check Differentv

case to calcualte the derivatfunction. Nominate the variad

An expression for the derivative

nction is given.

fu

Note: This result could also be calculated using the template provided on the the 2D:CALC keyboard.


33 .. 33 DD EE RR II VV AA TT II VV EE AA TT AA PP OO II NN TT 33How do I calculate the value of a derivativHow do I calculate the value of a derivative at a point? e at a point?

.. 33 DD EE RR II VV AA TT II VV EE AA TT AA PP OO II NN TT

Example:

If ⎟⎠⎞

⎜⎝⎛=

2tan)( xxf , ⎜

⎛′f ⎟⎠⎞

⎝ 2π find .



Press k

r the expression for the

e,

tiation).

Entefunction. Select this input line. Tap Interactiv then Calculation,

th

en diff(eren

diff(erentiation) allows us to: calculate a derivative function or to evaluate the derivative at a value, check this option. Nominate the variable (to be differentiated with respect to), x

differentiation (i.e. 1st derivative, 2nd derivative etc), 1st in this case.

Enter the Value,

in this case. Enter the order of the

2π

in this case.

Tap OK when these selections have een made.

b

he value of the derivative at the T

given p

oint is provided and so

12

f π⎛ ⎞′ =⎜ ⎟⎝ ⎠

.


33 .. 44 SS II GG NN OO FF TT HH EEHow do I calculate when

DD EE RR a derivative

II VV AA TT II VV EE is negative / positive / zero?

Example:

Find when the derivative of 1030274 23 +−+= xxxy is negative.


Launch the M application. Find an expression for the derivative function. Select this output, open a graphing window and and drag the derivative into the graphing area.

e can now see when the

o

Wderivative is less than zero. Select the derivative and drag it tan empty input box.

Set the derivative to be less than zero, using the mth:OPTN keyboard. Select the inequality. Tap Interactive, then Equation, then solve.

The set of x-values, for which the derivative is less than zero, is calculated.

Note A similar method can be used to find when a derivative is equal to zero or greater than zero.


33 .. 55 TT AA NN GG EE NN TT SS AA NN DD NN OO RR the graph of a function?

MM AA LL SS ent or normal to

MM AA LL SS ent or normal toHow do I calculate the equation of a tangHow do I calculate the equation of a tang

33 .. 55 TT AA NN GG EE NN TT SS AA NN DD NN OO RR the graph of a function?

Example:

Find the equation al to the graph of of the norm ⎟⎠⎞

⎜⎝⎛=

2tan xy at the po

2π

=x . int where



Enter tan2x⎛ ⎞

⎜ ⎟ ⎝ ⎠

ap $ to open the graph

indow, drag

T

w tan2x⎛ ⎞

⎜ ⎟⎝ ⎠

into the

raph window. Select the input again.

g

ap

en on,

en

T Interactive, th Calculati

. th normal

xEnter the -value required in the normal dialogue box and tap OK.

The equation of the normal is provided (without the y = ). The equation of the normal must be written as:

12

y x π= − + +

Both graphs can be drawn.

Note In this case do not enter the y =.


33 .. 66 TT HH EE II NN DD EE FF II NNHow do I calculate an indefini

II TT EE II NN TT EE GG RR AA LL te integral?

Example:

3 5xex

− +∫ dx. Evaluate

Demonstration Method


Enter

3 5xex

− +

Select this input.

Tap Interactive, then Calculation, then ∫ .

∫ allows us to calculate either an

indefinite integral or a definite integral (exact values) or a definite integral (decimal values). Check Indefinite integral. Tap OK.

The indefinite integral is computed as shown.

Note This result could also be obtained using P, the definite integral template provided on the the 2D:CALC keyboard.


33 .. 77 TT HH EE DD EE FF II NN II TT EE II NN TT EE GG RR AA LLred?

33How do I calculate a definite integral when an exact value is requiHow do I calculate a definite integral when an exact value is requi

.. 77 TT HH EE DD EE FF II NN II TT EE II NN TT EE GG RR AA LLred?

Example:

Evaluate 1 1

2 x

−

∫ dx. −


L aunch the M application.

Enter 1x

Select this input.

ap Interactive, T

then Calculation,

an

definite integ definite

heck Definite integral, enter the Upper bounds.

Tap OK.

then ∫ .

∫ allows us to calculate either

in ral or a integral (exact values) or a definite integral (decimal values). CLower and

The definite integral is compus shown.

ted a

Note This result could

Palso be obtained using , the definite integral template provided on the the 2D:CALC keyboard.


33 .. 88 AA VV EE RR AA GG EE VV AA LL UU EE OO FF AA FF UU NN CC TT II OO NN How do I find the average value of a function over a given interval? How do I find the average value of a function over a given interval?

33 .. 88 AA VV EE RR AA GG EE VV AA LL UU EE OO FF AA FF UU NN CC TT II OO NN

Example:

Find the average value of on tan(y the functi )2x over the interval 80, π⎡ ⎤⎣ ⎦. =Method Demonstration

L aunch the M application.

To use the formula

( )b

a

f x dx∫b a−

,

enter the structure using the 2D and 2D:CALC keyboards.

Enter the remainder of

and the input

sing mth:TRIG 2D ukeyboards.

Press E.

The result is calculated and displayed in exact form.


33 .. 99 DD EE FF II NN II TT EE II NN TT OOHow do I solve for an unknow de

EE GG RR AA LL EE QQ UU AA TT IIn in an equation involving a

NN SS finite integral?

Example:

Calculate the value of b if 71

∫b

7 dx =x .


Launch the M application. Use the 2D:CALC keyboard

to enter P. You will need the VAR keyboard to enter the b. Complete and select the equation.

Tap Interactive, then Equation/Inequality, then solve. Change the variable to be solved for to b, using the VAR keyboard. Tap OK.

Two solutions in exact form are provided. It may be that only one of the solutions is suitable, depending on the requirements of the question you are answering.

Note After entering the x in dx

press : to ensure the cursor is flashing alongside the whole integral expression (not just the dx) before entering the ‘=7’.


44 .. 11 PP RR OO BB AA BB II LL II TT YY DD EE NN SS II TT YY FF UU NN CC TT II OO NN ty function?

II OO NN ty function? How do I calculate a probability from a piecewise probability densiHow do I calculate a probability from a piecewise probability densi

44 .. 11 PP RR OO BB AA BB II LL II TT YY DD EE NN SS II TT YY FF UU NN CC TT

Example: T , The continuous random variable

time in minutes, has the probability density function

Find the probability that

f with the rule:

25<T . Method Demonstration

Launch the M application. Use the cces

twice to create a template

a domain field blank implies “otherwise”. Select this input.

2D keyboard to a s the piecewise template. Tap itwith three elements. Enter the probability density function as shown. Leaving

Tap Interactive,

ariable to t the Define dialoglue box and tap

OK.

then Define. Change the vin

Use the 2D:CALC keyboard for

P. Enter the definate integral that will calculate the required probability. Remember that the name of the function, f, is text and not variable and so the abc keyboard is needed to enter the f.

oth0 erwise30if2010if

) ≤≤<≤

tt

20)( 1 30()10(

100

1 01

⎪⎩

⎧

−−

tt0⎪

⎨=tf

Note When a function is defined in a piecewise manner, the result of a definite integral is calculated and displayed as a decimal approximation. We might guess that the exact value for this is 0.875

or 78

.


44 .. 22 EE XX PP EE CCHow do I calculate the e

TT EE DD VV x obability density function?

AA LL UU EE && MM EE DD II AA NNpected value and median of a pr

Example:

Find the median and expected value of a random variable X that has a probability

density function 2

)( xxf = for 2 . 0 ≤≤ x


Launch the M application. To find the expected value, use the formula

( ) ∫=b

a

xfxXE )( .

Enter using the P from the 2D:CALC keyboard.

Press E.

To find the median, m, of the probabitity distribution, enter the equation as shown. Tap Interactive, then Equation/Inequality, then solve. Change the variable to m, using the VAR keyboard and tap OK.

Two solutions, in exact form, are given, but clearly only one of them can be disregarded and so the

median of the distribution is 2 .

Note After entering the x in dx

press : to ensure the cursor is flashing alongside the whole integral expression (not just the dx) before entering the ‘=0.5’.


44 .. 33 NN OO RR MM AA LL DD II SS TT RR II BB UU TT II OO NN –– PP RR OO BB AA BB II LL II TT YY ibution?

BB AA BB II LL II TT YY ibution? How do I calculate a proportion/probability associated with a normal distrHow do I calculate a proportion/probability associated with a normal distr

44 .. 33 NN OO RR MM AA LL DD II SS TT RR II BB UU TT II OO NN –– PP RR OO

Example:

The heights of girls under 14 years old in a particular large city are normdistributed with mean 130 cm and standard deviation 2.7

ally m. What proportion of c

these girls are shorter than 125 cm? Method Demonstration


ap Calc,

tion.

L I T hen Distribut

This opens a calculation wizard.

Use the C to select your type of distribution from the drop-down menu, in this case Normal CD.

Tap Next. Enter the inputs as shown. Tap Next.

The proportion / probability has

he z-

ap on $ to see this result presented graphically on the

standard normal distribution. This provides a good way to check that what you have calculated make sense.

been calculated, along with tscores of the lower and upper

s. bound Tre


44 .. 44 NN OO RR MM AA LL DD II SS VVHow do I calculate a boundary giv

distribution?

TT RR II BB UU TT II OO NN -- II NNen a proportion/probability

EE RR SS EE associated with a normal

Example:

Suppose the girls described in the previous park ride. If 35% of the girls fail to mminimum acceptable height.

section are interestedeet the height requirem

in an amusement ent for the ride, find the


Launch the I application. Tap Calc, then Inv. Distribution.

w.

This opens a calculation windo

In this case the default distribution

ttings are as required. Tap Next. Enter the inputs as shown

using the C to choose the Left tail setting. Tap Next.

se

,

The necessary boundary / cut-off has been calculated. Tap on $ to see this result represented on a graph of the normal distribution. This a good way to check that what you have calculated makes sense.


44 .. 55 NN OO RR MM AA LL DD II SS TT RR II BB UU TT II OO NN –– MM UU && SS II GG MM AA al distribution? How do I calculate unknown population parameters for a norm

Example:

The weights of a certain variety of squash are norm than 15 g, find the m

ally distributed. If 5% weigh more 30 g nd 10% weigh les ean and standard deviation of the

distribution of squash weights. than a s


ion find the Z-scores associated

the I

ution.

distribution CD), enter the

put as shown.

The first Z-score is calculated.

Firstly, we use the Z-distributtowith the two statements about the population.

Launch application. Tap Calc, then Inv. Distrib Using the default(Inverse Normal in

Repeat this process to calcualte the Z-score associated with the second statement about the population.


te

Use the Z-score formula to enter two equations in terms of mu and sigma (represented here by m and s, obtained from the VAR keyboard)

Press E. Use ., if required to see the

m.

Use the 2D keyboard for the simulaneous equations templa

~.

answer in decimal for


44 .. 66 BB II NN OO MM II AA LL DD II RRHow do I calculate a probability ass

SS TT RR II BB UU TT II OO NN –– PPociated with a binomial dis

OO BB AA BB II LL II TT YY tribution?

Example: Assume that 30% of emmarried. If 10 employed womleast 7 of them have never been m

ployed women living en are selected at random,

arried.

in a very large city have never been find the probability that at


Launch the I application. Tap Calc, then Distribution. This opens a calculation wizard.

Use the C to select your type of distribution from the drop-down menu, in this case Binomial CD. Tap Next. Enter the inputs as shown. Tap Next. Ticking Help when using this wizard provides you with additional information about the input and output that you are working with.

The required probability has been calculated.


44 .. 77 TT RR AA NN SS II TT II OO NN MM AA TT RR II XX -- PP AA BB II LL II TT YY using a transition matrix?

RR OO BBes

RR OO BBesHow do I calculate non-independent probabilitiHow do I calculate non-independent probabiliti

44 .. 77 TT RR AA NN SS II TT II OO NN MM AA TT RR II XX -- PP AA BB II LL II TT YY using a transition matrix?

Example: If successful on her finext attempt is 0.84. If she is unsuccessful,

rst goal attem netballer Lisa scores on her then the probability that her next attempt

is successful is 0.64. Give alculate the probability that her 8th attemp

pt, the probability of

n that her first attempt wt is successful?

as successful, c


Launch the M application. Use the matrix templates on 2D:CALC

the keyboard to enter the

ture.

transition matrix struc

Press E.

As this calculation was done with the ClassPad in Standard mode, values in exact form are calcuated and displayed. To obtain a decimal approximation, tap on the output and then tap on .. The probabilities of success and failure in the 8th attempt are calcuated.


55 TT RR OO UU BB LL EE SS HH OO OO TT II NN GG ,, HH II NN TT SS && TT II PP SS

ail your suggested inclusions for this page to [email protected]

Please em .


66 .. 11 MM YY AA DD DD II TT II OO NN SS -- 11







II NN DD EE XX A otes 13 Average value (function) 27 A 10 Binomial distribution 34 Composite function 14 Decimal approximation 5, 7, 16, 35 Definite integral 26 Define a function 4, 14 Derivative 10, 21, 23 At a point 22 Determinant 19 diff 21, 22 Domain 5, 8, 11, 17 Equations 5, 19 Expected value 30 fma 9, 12 f 8, 12 F 14 G ne 5 I al 25, 26 Definite 26 28 25 I 6, 7 I 7 I 15 I mal 32 L ximum 8, 10 Matrices 19 max 12 Maximum values 8, 10, 11 Median 30 min 12 Minimum values 8,10,11 Normal to curve 24 Normal distribution 31, 33 Inverse 32 Piecewise 29 Probability density function 29, 30 Expected value 30 Median 30 PropFrac 13 Range 8, 9 Rate of change 20 Rational function 13 Simplify 14 Simultaneous equations 7, 18 Linear 18 Non-linear 18 S 5, 6, 10, 15, 16, 17 Solve numerically 16 Solving equations 16, 17 Stationary point 10 Tangent 24 Transition matrix 35 Unique solution 19 Zoom auto 4, 11 Zoom quick initialize 4, 7

sympt

xes scales

x in m

unction notation ral solutions e

ntegr

Equations Indefinite

ntersect tersection point n

nverse function nverse norocal ma

olve

Date post:	17-Oct-2018
Category:	Documents
Upload:	lamxuyen
View:	220 times
Download:	0 times

Using ClassPad in the VCE Methods Exam. · This book provides the latest and best ways to use the...

Documents