Exam Vic V 9 11 09 - Exam Vic V 9 11 09 PDF

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Exam Vic V 9 11 09...

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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.

A B O U T TH I S PU B L IC CA T IO ON 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.

NT E N T TS C ON 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

1.1 D EF FI N I N G I N N M A AN D T H E N G R A P H I N NG . How How do do II define define aa function function in in M M and then graph it?

Example:

⎛ π ( x − 8) ⎞ 2 Draw a graph of C ( x ) = 1000(cos ⎜ ⎟ + 2) −1000 for 8 ≤ x ≤ 16 . 2 ⎝ ⎠ Method Launch the

Demonstration

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 the screen. Tap,

Zoom: Quick Initialize. Select C(x), take your stylus off the screen and then drag it into the 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).

4

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O R X –– E X A C T 1 . 2 S O LLV I N G F O How do I calculate x given a value for y (or C(x) in this case) if exact values are required?

Example: ⎛ π ( x − 8) ⎞ 2 If C ( x ) = 1000(cos ⎜ ⎟ + 2) − 1000 for 8 ≤ x ≤ 16 , find x if C ( x) = 1250 ? 2 ⎝ ⎠ 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. Select C(x) from the first input line and drag it into the second imput line to begin the equation. Select the equation and then tap: Interactive: Equation: solve.

Make sure that the Solve option is checked. This will ensure that the ClassPad will return exact values fo r the solution to the equation if possible Tap OK. In this case, the general solutions are given (since the function involved is cyclic). 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.

To calculate the four solutions within the domain we are interested in, insert | 8 ≤ x ≤ 16 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.

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5

O R X –– D E C I M A L S S ONLY 1 . 3 SO L V I N G FO How do I calculate x given a value for the function (y or C(x) in this case) if a de cimal approxima tion is acceptable?

Example: ⎛ π ( x − 8) ⎞ 2 If C ( x ) = 1000(cos ⎜ ⎟ + 2) − 1000 for 8 ≤ x ≤ 16 , find x if C ( x) = 1250 ? 2 ⎝ ⎠ 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.

Tap Analysis, then G-Solve, then Intersect. The flashing cursor indicates the position of the left most intersection poin t, and numeric values for the co-ordinates o f the point are displayed, but as a decimal approximation and not in exact form. If there is more than one point of intersection, tap the next one.

: to move to

The same process can be achieved directly from the app lication.

g

Define y2 to be 1250, press

E

Tap $ to draw the graph and proceed as outlined above.

6

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N T E R S E CT TI O N – E EX A C T 1 . 4 IN How do I find the intersecti on point of two graphs if exact values are required?

E xample: x 2x Find where graph of the function f ( x) = e − 2 intersects the graph of g ( x ) = e .

Method Launch the

M application.

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

Demonstration

k

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

Use the 2D keyboard for the simultaneous equations template

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

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.

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7

NI M U M & & M A X I M U M V A LU UE S – E X A C T 1 1.5 M IN How do I find the range of a function, given its domain and if exact values are required ?

Example: ⎡ π⎞ Find the range of the function f : ⎢ 0, ⎟ → R, f ( x) = 3 sin( 2 x) −1 + 2 . ⎣ 3⎠ Method

Demonstration

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

3 sin(2 x ) − 1 + 2

using

the 2D keyboard for 4 and the mth:TRIG keyboard for sin. Tap $ to open the graph window in the bottom half of the screen. Tap, Zoom: Quick Initialize. Select the input line and drag it into the graph window.

Tap 6 to adjust the View Window settings. Use the domain information to set an appropriate xmin and xmax. In this case choose values slightly outside the domain given in the question. Tap OK. Tap Zoom: Auto. 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.

Select

3 sin(2 x ) − 1 + 2 .

Tap:

Interactive: C alculation: fMin. In the fMin dialoglue box, enter the endpoints of the domain as the Start and End values and tap OK.

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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. 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.

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 =

π 3

the range includes both values. So the range is

[ 2, 5] .

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.

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9

1.6 M IN NI M U M & & M A XIM MU M V A LU UE S – E X A C T 2 How do I calculate the minimum or maximum value of a function for all real x if exact values are required? Example:

⎛ 9 + x2 9 − x ⎞ Find the value of x that minimizes the function T = 2 ⎜ + ⎟, x ∈ R . ⎜ 5 13 ⎟ ⎝ ⎠ Method Launch the

Demonstration

M application.

Define the function as T(x) – see previous sections for assistance. Tap $ to open the graph window. Tap, Zoom: Quick Initialize. Drag T(x) into the window. Use 6 or n to adjust the axes scales. We see there is only one value of x Note We see a local minimum. But, if not sure of the behaviour outside −5 ≤ x ≤ 9 , we can use calculus to p roceed.

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: solve. Make sure Solve is checked. This w ill ensure exact value(s), if possible. Tap OK. Note We see there is only one value of x that satisifies this equation and so we can now be sure that there is only one stationary point – a minimum in this case.

Finally, find T(

5 ) to find the 4

minimum value.

10

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NI M U M A AN D M A X I M U M – D E C I M A L LS O N L Y 1.7 M IN How do I calculate maximum a nd minimum values if decimals are acceptable? Example:

⎛ 7π ( x − 8) ⎞ + 2)2 − 1000 Find the m inimum value of C (x ) = 1000(cos ⎜ for ⎟ 6 ⎝ ⎠ 8 ≤ x ≤ 16 and the corresponding value(s) of x . Method Launch the

Demonstration

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 … Note Don’t have 4 black arrows on your graph? Tap:

Settings (bottom left of screen): Graph Format: then tick G-Controller.

Tap Zoom, then Auto This automatically sets an appropriate y-scale based on the xscale (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.

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11

Method

Demonstration

The graph reveals we are dealing with local minimums. Min will only search for local minimums. Tap Analysis, then G-Solve, then Min.

The flashing cursor indicates the position of the relevant point, and numeric values for the co-ordinates of the point are displayed. If there is more than one minimum point, tap or press the next one.

: to move to

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.

12

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

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T I O N A L F U N C T I O N -- A S Y M P T TO TE ES 1 . 8 R AT How do I change the form of a rational function to more easily see calculate its asymptotes?

Example: Find the asymptotes of f : D → R, f ( x) = Method Launch the Press

x− 3 . 2−x Demonstration

M application.

k

Use the 2D keyboard to enter the expression for the function. Select this input.

Tap Interactive, then Transformation, then propFrac.

From this form of the expression, we see that the function has a ho rizontal asymptote of y = -1.

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13

E F F U N C T I O NS S 2 . 1 C O MP O SI T E How do I calculate a composite functi on? Example:

If g ( x) = x 2 + 2 x − 3 and f (x ) = e 2 x +3 . Find f ( g ( x)) . Method Launch the Enter

e

2 x+ 3

Demonstration

M application. .

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

Interactive,

then Define .

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 f (f (x)) = x? • Determining a transforme d function like f (x – 3 ) • Checking properties of given functions, e.g. is f (u) + f (v) = f (u + v )?

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NV E R S E F U NC CT I O N 2 . 2 IN How do I calculate an inverse function? Example:

For the function h( x) = 1 + e −x , find the inverse function h−1 . Method

Demonstration

Exchanging variables gives us Launch the Press Enter

M application.

k y x = 1 + e− .

Select 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.

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15

QU A T II O N S – E X A C T 1 1 2 . 3 S O L V I N G EQ How do I find solutions to equations if exact values are require d? Example:

Find the solution set for the equa tion e 4 x − 5e 2 x + 4 = 0 . Method Launch the Press

Demonstration

M application.

k

Enter the equation to be solved. Select this input.

Tap Interactive, the n Equation, then solve. Check that the contents of the solve dialogue box are correct. Make sure that Solve is checked as this will ensure the ClassPad will return exact values, if possible. Tap OK.

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.

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2 . 4 S O L V I N G EQ QU A T II O N S – E X A C T 2 2 How do I find exact values for the solutions to equations within a given domain? domain? Example:

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

Demonstration

Launch the M application. Press

k