TI-NSPIRE: Graphing Technology Pre-University Mathematics 9789670924502

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T1-NSPIRE

Graphing Technology Pre-University Mathematics

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T1-NSPIRE

Graphing Technology Pre-University Mathematics

Noraini Idris Nor’ain Mohd Tajudin Raja Lailatul Zuraida Raja Maamor Shah Mohd Hafiszudin Ab Samad Mary Ann Serdina Parrot

PENERBIT UNIVERSITI PENDIDIKAN SULTAN IDRIS TANJONG MALIM, PERAK 2016

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Cetakan Pertama/First Printing 2016 © Universiti Pendidikan Sultan Idris 2016 Hak Cipta Terpelihara. Tiada bahagian daripada terbitan ini boleh diterbitkan semula, disimpan untuk pengeluaran atau ditukarkan ke dalam sebarang bentuk atau dengan sebarang alat juga pun, sama ada dengan cara elektronik, gambar serta rakaman dan sebagainya tanpa kebenaran bertulis daripada Penerbit Universiti Pendidikan Sultan Idris terlebih dahulu. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Penerbit Universiti Pendidikan Sultan Idris. Diterbitkan di Malaysia oleh/Published in Malaysia by Universiti Pendidikan Sultan Idris 35900 Tanjong Malim, Perak Darul Ridzuan, Malaysia Tel: 05-450 6000, Faks: 05-459 5169 Laman Sesawang: www.upsi.edu.my E-mel: [email protected] Atur huruf dan Grafik oleh/Typesetting and Graphic by Pejabat Karang Mengarang (Penerbit UPSI) Universiti Pendidikan Sultan Idris 35900 Tanjong Malim, Perak Darul Ridzuan, Malaysia Dicetak oleh/Printed by REKA CETAK SDN BHD No. 14, Jalan Jemuju Empat 16/13D Seksyen 16, 40200 Shah Alam Selangor Darul Ehsan

Perpustakaan Negara Malaysia

Cataloguing-in-Publication Data

Nor’ain Mohd. Tajudin T1-NSPIRE : Graphing Technology Pre-University Mathematics / Nor’ain Mohd Tajudin, Noraini Idris, Raja Lailatul Zuraida Raja Maamor Shah, Mohd Hafiszudin Ab Samad, Mary Ann Serdina Parrot. Includes index Bibliography: page 215 ISBN 978-967-0924-50-2 1. Programming (Mathematics). 2. Mathematical models. 3. Mathematics. I. Noraini Idris, 1957-. II. Raja Lailatul Zuraida Raja Maamor Shah. III. Mohd. Hafiszudin Ab. Samad. IV. Mary Ann Serdina Parrot. V. Title. 510

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CONTENTS

Preface

Chapter 1

ix

TI-NSPIRE CX EXPLORATION Introduction The Basis of Using Calculator Summary

Chapter 2

QUADRATIC EQUATIONS Introduction Activity 1 A Projectile Activity 2 Width of the Path Activity 3 Path of a Soccer Ball Summary Self assessment

Chapter 3

1 4 11

13 16 22 24 28 28

FUNCTIONS Introduction Activity 1 Office Temperature Activity 2 Absolute Function Activity 3 Rational Function Summary Self assessment

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31 35 39 41 43 43

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vi

TI-Nspire Graphing Technology: Pre-University Mathematics

Chapter 4

MATRICES Introduction Activity 1 Nutritional Content Activity 2 Stamp Activity 3 Cryptography Summary Self assessment

Chapter 5

TRIGONOMETRY Introduction Activity 1 Trigonometry Ratio Activity 2 Height of a Tree Summary Self assessment

Chapter 6

45 49 52 55 59 59

61 65 67 70 70

COMPLEX NUMBER Introduction Activity 1 Rectangular and Polar Form Activity 2 Forces Summary Self assessment

73 76 79 82 82

Chapter 7 VECTOR Introduction Activity 1 Graphical Vector Addition Activity 2 Speed and Direction of Airplane Activity 3 Inclined Plane Summary Self assessment

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85 89 94 99 106 106

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Contents

Chapter 8

SEQUENCES AND SERIES Introduction Activity 1 Exploring Sequences and Series Activity 2 AIDS Cases Activity 3 Bouncing Ball Summary Self assessment

Chapter 9

vii

109 113 123 126 132 132

DIFFERENTIATION Introduction Activity 1 Approaching Limit Activity 2 Parking Lot Activity 3 Pump Station Summary Self assessment

135 140 144 150 153 153

Chapter 10 INTEGRATION Introduction Activity 1 Riemann Sum Activity 2 Airplane Wing Summary Self assessment

155 159 165 171 171

Chapter 11 STATISTICS AND PROBABILITY Introduction Activity 1 Basketball Tournament Activity 2 Wind Speed Activity 3 Two Dice Activity 4 Permutations Summary Self assessment

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173 177 183 188 191 194 194

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viii TI-Nspire Graphing Technology: Pre-University Mathematics

Chapter 12 DIFFERENTIAL EQUATION Introduction Activity 1 Slope Field Activity 2 Carrying Capacity Activity 3 Spread of Computer Virus Summary Self assessment Bibliography Index

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197 200 204 210 213 213 215 219

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PREFACE

T1-Nspire Graphing Technology: Pre-University Mathematics provides a plethora of ideas for mathematics educators and pre-university students in using TI-Nspire graphing technology for mathematics teaching and learning. This book can be used by all undergraduates in Mathematics Education Program as well as mathematics education lecturers to explore idea of using graphing calculator technology in teaching and learning. Activities in this book were designed to help pre-university students to enhance their higher order thinking skills. Through the T1-Nspire-based learning experience, students will develop a greater understanding of mathematical concepts and ideas and enhance their creative thinking, critical thinking and mathematical reasoning, communication, connection and representation. Students were provided with activities that required students to perform mathematical exploration, investigation, problem solving and modelling. Students were given opportunities to use mathematical tools systematically in solving problems and be engaged in investigation and exploration of patterns and relationship to enhance their reasoning abilities and communication skills. The task required students to model real world situation mathematically so that they can develop understanding of the world through the model and of how the model can be used to make further prediction if changes were to occur to this situation. The TI-Nspire, as a mobile and inexpensive device can be of great help in reducing the challenge of developing and enhancing higher order thinking of students in the classroom. Noraini Idris, PhD Nor’ain Mohd Tajudin, PhD Raja Lailatul Zuraida Raja Maamor Shah, PhD Mohd Hafiszudin Ab Samad Mary Ann Serdina Parrot

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CHAPTER

1

TI-NSPIRE CX EXPLORATION

INTRODUCTION

Figure 1.1 TI-Nspire CX

The TI-Nspire CX is the newest update to the TI-Nspire series as shown in Figure 1.1. It has a thinner design with a thickness of 1.57 cm, a 1060 mAh rechargeable battery, a 320 by 240 pixel full colour backlit display (3.2” diagonal) and OS 3.6.0.550 which contain features such as 3D graphing. The TI-Nspire CX handheld is a powerful, handson learning device that convinces math and science curriculum requirements from middle school through college. The TI-Nspire CX handheld’s inventive potentials in sustaining teaching strategies that research has established to speed up understanding of complex mathematic and scientific concepts. Various

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TI-Nspire Graphing Technology: Pre-University Mathematics

representations of expressions in problems are offered simultaneously, allowing students to envision how algebraic, graphical, geometric, numeric and written forms of those expressions connected to one another. Dynamic linking across documents and multiple representations permits students to interrelate directly with the math by witnessing how maneuvering one form simultaneously changes all the others. TI-Nspire CX also composes real-world associations. It can bring in digital images together with your own photos as well as overlay graphs and equations to see math and science in the physical world. The TI - Nspire CX has incorporated mathematics templates that allow students to penetrate expressions and see math symbols, formulae and stacked fractions just as they are written in textbooks and on the board.

d Removes menus or dialog boxes from the screen.

c Turns on the handheld. If the handheld is on, this key displays the home screen.

» Opens the Scratchpad for performing quick calculations and graphing.

~Opens the document menu. b Displays the application or context menu. aplikasi tambahan.

e Moves to the next entry field. / Provides access to the function or character shown above each key. Also enables shortcuts in combination with other keys.

h Deletes the previous character. · Evaluates an expression, executes an instruction, or selects a menu item.

g Makes the next character typed upper-case.



Figure 1.2 Main Keystrokes of TI-Nspire CX Graphing Calculator

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TI-Nspire CX Exploration

3

Table 1.1 TI-Nspire CX’s Applications APPLICATION



Calculator

Graph



Geometry



FUNCTION Adds a page to a document for entering and evaluating math expressions.

Adds a page for graphing and exploring functions.

Adds a page for creating and exploring geometrical shapes.

Adds a page for working with data in tables.

Lists & Spreadsheet



Data & Statistics

Notes



Vernier DataQuestTM

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Adds a page and provides tools use to visualize sets of data in different types of plots and provides tools for manipulating data sets to explore relationships between the sets of data. Provides text editing functions for adding text to TI-Nspire CX documents for use as notes or to share with other users.

Adds a page for collecting and analyzing data from sensors or probes.

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TI-Nspire Graphing Technology: Pre-University Mathematics

The Basis of Using Calculator The following section describes the basis for using this graphing calculator. To facilitate the reader, the screens related to the description are displayed at right. Description

Screen Display

1. Opening the New Document From HOME c button, choose 1: New Document or press 1.

The next screen will list the seven applications in the TI-Nspire CX. Press ¤ £ to choose the application, then press ·.

2. Saving Document You can save your work in “My Document”, press: a. ~ b. 1: File 1 c. 4: Save 4 Type the name of the document. Then, · at “Save”.

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TI-Nspire CX Exploration

Description

5

Screen Display

3. Inserting the New Page When opening an application, you can add a new document. Press: a. /¿ Then, choose your application. At the top of the screen, there are numbers of 1.1 and 1.2. 1.1 is a previous document, while 1.2 is the new document. If you want to add another new document, the document will be generated as 1.3. These numbers mean: a. 1.1 = Problem 1 Page 1. b. 1.2 = Problem 1 Page 2. c. 1.3 = Problem 1 Page 3. The maximum “Problem” that can be used are 30 problems, meanwhile the “Pages” are 50 pages. This means “Problem” and “Pages” starts with 1.1 and ends with 30.50. 4. Adding Application In One Document In a document, you can display four applications simultaneously. You just need to choose the layout provided. For layout, press: a. ~ b. 5: Page Layout 5 c. 2: Select Layout 2 Press 2 until 8 to choose a layout.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Description

Screen Display

For example, press 2. You will see: On the left side: Calculator application. On the right side: Graphs application. Press / e to change your application from the right to the left.

5. Opening the Settings Menu From the Home screen, press 5 or use the Touchpad to select Settings. The Settings menu opens.

5.1 Changing a Preferred Language Complete the following steps to change a preferred language: 1. From the Home screen, press 5 or select Settings to open the menu. 2. From the menu, select Change Language or press 1. The Change Language dialogue box opens. Press ¢ to open the drop-down list. Press ¤ to highlight a language, then press “x” or “·” to select it. Press e to highlight the OK button, then press “x” or “·” to save the language selection.

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TI-Nspire CX Exploration

Description

7

Screen Display

5.2 Changing Default Document Settings Complete the following steps to define default document settings for TI-Nspire CX documents and Scratchpad. a. Save and close any open documents. b. From the Home screen, press: i. 5: Settings 5 ii. 2: Document Settings 2

The Document Settings dialogue box opens.

c.

Press e to move through the list of settings. Press £ to move backward through the list. A bold line around a box indicates it is active. d. Press ¢ to open the drop-down list to view the values for each setting. e. Press the £ and ¤ keys to highlight the desired option, then press x or ·· to select the value. f. Select Make Default. g. Click OK to save the settings as default settings that will be applied to all TI-Nspire CX documents and to Scratchpad.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Description

Screen Display

5.3 Customizing the Handheld Setup Handheld setup options enable you to customize options to suit your needs. a. Font size (small, medium or large). b. Power standby. (1, 3, 5, 10 or 30 minutes). i. Use this option to extend battery life. ii. By default, the handheld automatically powers down after three minutes of inactivity. c. Hibernate (1, 2, 3, 4, 5 days or never). iii. Use this option to extend battery life. iv. By default, the handheld hibernates after four days. v. When hibernating, the handheld saves current work in memory. vi. When you turn the handheld on again, the system reboots and opens saved work. c. Pointer speed (slow, normal or fast). d. Auto dim (30, 60 or 90 seconds, and two or five minutes). e. Enable tapping to click. a.

From the Home screen, press: i. 5: Settings 5 ii. 3: Handheld Setup 2

The Handheld Setup dialogue box opens.

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TI-Nspire CX Exploration

Description

9

Screen Display

b.

Press e until the desired category is highlighted. c. Press ¢ to view the list of possible settings. d. Press ¤ to highlight the desired setting. e. Press x or · to select the new setting. f. When you have changed all the settings to suit your needs, press e until OK is highlighted, then press x or · to apply your changes. Note: Click Restore to return the handheld to the default settings. 6. Viewing Handheld Status The Handheld Status screen provides the following information about the current state of the handheld: a. Battery status of the rechargeable batteries b. Software version c. Available space d. Network (if any) e. Your student login name and whether you are logged in f. About a.

From the Home screen, press: i. 5: Settings 5 ii. 4: Status 2

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TI-Nspire Graphing Technology: Pre-University Mathematics

Description b.

Screen Display

Click OK to close the Handheld Status screen.

7. Replacing TI-Nspire CX Rechargeable Batteries When you replace the battery, complete the following steps to insert the TI-Nspire CX Rechargeable Battery into a handheld. a.

Use a small screwdriver to release the panel from the back of the handheld.

b.

Remove the panel.

c.

Remove the old battery.

d.

Insert the white connector of the new battery into the jack located at the top of the battery compartment.

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TI-Nspire CX Exploration

Description

11

Screen Display

e. Thread the wire into the case to secure it. Insert the rechargeable battery into its compartment. f.

Replace the back panel and fasten the screws with a screwdriver.

SUMMARY This first chapter introduces some background and features of the latest version of graphing calculator, namely the TI-Nspire CX. TINspire is a handheld calculator that offers innovative capabilities in supporting teaching strategies which research has found accelerate understanding of complex mathematic and scientific concepts. The first part of this chapter illustrates the main features of the graphing calculator keyboard which help educators to understand the operations of a TI-Nspire graphing calculator and its application for computation purposes. The second part explains the TI-Nspire CX’s Applications and its function. Finally, this chapter ends with some descriptions on the basis for using this graphing calculator where readers can explore how to open the new document, save document, adding application in one document , opening the setting menu, viewing handheld status and replacing TI-Nspire CX rechargeable batteries.

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CHAPTER

2

QUADRATIC EQUATIONS

INTRODUCTION

Figure 2.1 Brahmagupta

Babylonian mathematicians employed a simple formula, −b ± b ! − 4ac x= , in explaining quadratic equations as 2a

early as 2000 B.C. Their technique of solution was different from ours and was conveyed verbally as a series in stages (with no evidence.) They also unravelled non-linear simultaneous equations that direct in standard algebra to quadratics. For example, x + y = 10, xy = 5. A Hindu mathematician named Brahmagupta as in Figure 2.1, provided the first overt (although still not completely in general) answer of the quadratic equation ax2 + bx = c as follows: “To the

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14

TI-Nspire Graphing Technology: Pre-University Mathematics

absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. In 800 A.D., al-Khwarizmi, a Persian mathematician, produced six new equations concerning roots, squares of roots, and numbers which he employed to solve quadratic equations, and developed the quadratic equation. The algebra utilised by him was purely rhetorical, and he eliminated negative solutions. These findings led Abraham bar Hiyya Ha-Nasi to publish a book with the absolute solution of the quadratic formula. European mathematics achieved resurgence during the 1500s and in 1545, Girolamo Cardano, an Italian mathematician gathered all works related to quadratic equations as well as produced the chances of imaginary solutions. Later in the year 1594, the quadratic formula was first received by Simon Stevin, who was born in Belgium. Everything we know today stems off of these earlier discoveries. René Descartes, the Father of Modern Mathematics, published the quadratic formula as we are aware of it today in his book La Géométrie, −b ± b ! − 4ac x= and this provides the an explanation 2a to a generic quadratic equation of the form ax 2 + bx + c = 0. Quadratic equations are very relevant to everyday life. The best example of a quadratic equation application is the path of a ball being thrown into the air. The following Figure 2.2 represent some examples of the applications quadratic function:

Figure 2.2 Examples of Quadratic Equation Application

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Quadratic Equations

15

NOTES A quadratic equation is any equation that can be written in the form:

𝑎𝑎𝑥𝑥 ! + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0

such that a and b are coefficients and c is a constant. The main factor that identifies these expressions as quadratic is the exponent 2. The first term must always be 𝑎𝑎𝑎𝑎 ! and a cannot be 0.

KEYSTROKE Below are some keystrokes of TI-Nspire CX that are important in learning quadratic equation. 1. Press q to get the power of 2. Example: 5!

2. To find the roots of quadratic equation, press: a. b. c. d.

b 3:Algebra 3 3:Polynomial Tools 3 1:Find Roots of Polynomial 1

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TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 1 Learning Outcome

Question:

A Projectile In this activity, students will solve a projectile problem using quadratic functions.

1

The formula ℎ = − 2 𝑔𝑔𝑡𝑡 ! + 𝑣𝑣! 𝑡𝑡 + ℎ! can be used

to model the height of a projectile, where g is the acceleration due to gravity, which is 9.8 m/s2 on earth, 𝑣𝑣! is the initial vertical velocity, in metres per seconds, and ℎ! is the initial height in metres. a. Create a model for the height of a toy rocket launched upward at 60m/s from the top of a 3m platform. b. How long would the rocket take to fall to earth, rounded to the nearest hundredth of a second? c. What is the maximum height of the rocket, rounded to the nearest meter? d. Over what time interval is the height of the toy rocket greater than 150m? Round to the nearest hundredth of a second. Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press “·” at NO.

Press: a. 1: Add Calculator 1

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Quadratic Equations

Create a model for the height of a toy rocket launched upward at 60m/s from the top of a 3m platform, as in Figure 2.3. Use formula 1 ℎ = − 𝑔𝑔𝑡𝑡 ! + 𝑣𝑣! 𝑡𝑡 + ℎ! 2

g = 9.8, 𝑣𝑣! = 60, ℎ! = 3

1 ℎ = − (9.8)𝑡𝑡 ! + 60𝑡𝑡 + 3 2

How long would the rocket take to fall to Earth, rounded to the nearest hundredth of a second?

17



Figure 2.3

a. Using Calculation Press: i. b ii. 3: Algebra 3 iii. 3: Polynomial Tools 3 iv. 1: Find Roots of Polynomial 1 Press: i. e, twice ii. ·for OK.

Press: i. v 9^8 p 2 e ii. 60 e iii. 3 e iv. · twice. Now, we have two answers. Which answer is true? −0.04980 (4 s.f) or 12.29 (4 s.f)? Explain.

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TI-Nspire Graphing Technology: Pre-University Mathematics

b. Using Graph Press: i. /¿ ii. 2: Add Graphs 2 Press: i. v 9^8 p 2 Xq ii. +60 X iii. +3· iv. b v. 4: Window / Zoom 4 vi. 1: Window Settings... 1 Change window settings as shown: Xmin: -10 Xmax:20 Xscale:1 Ymin:-10 Ymax:200 Yscale:1 Press ·for OK Press: i. b ii. 6: Analyze Graph 6 iii. 1: Zero 1 When the calculator asks for lower bound, you must select before the intersection of graph and x-axis. Then, press ·.

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Quadratic Equations

19

When the calculator asks for upper bound, you must select after the intersection of graph and x-axis. Then, press ·. You will get the intersection point (12.29, 0.00). Activity Summary:

Hence, what does it mean by (12.29, 0.00)?

What is the maximum height of the rocket?

To find the maximum height of the rocket, press: i. b ii. 6: Analyze Graph 6 iii. 3: Maximum 3 When the calculator asks for lower bound, you must select before the peak of the graph. Then, press ·. When the calculator asks for upper bound, you must select after the peak of the graph. Then, press ·.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Over what time interval is the height of the toy rocket greater than 150m?

Your coordinates should be same as shown beside. Hence, what is the maximum height of the rocket? Press: i. e ii. 150 iii. ·

Press: i. b ii. 6: Analyze Graph 6 iii. 4: Intersection 4 When it asks for lower bound, you must select before the intersection of the both graphs. Then, press ·.

When it asks for upper bound, you must select after the intersection of the both graphs. Then, press ·.

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Quadratic Equations

21

Repeat your steps for another intersection.

Activity Summary:

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Hence, over what time interval is the height of the toy rocket greater than 150m? Justify your answer.

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TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 2 Learning Outcome

Question:

Width of the Path In this activity, students will investigate how to maximize the width of the path.

A rectangular park measures 100m by 60m. A path of constant width is to be paved around the perimeter. The mayor wants to be sure that the path does not reduce the area of grass by more than 10%. What is the maximum allowable width of the path, rounded to the nearest tenth of a meter?

Let x represents the width of the path. Write and solve an equation to find x. Student’s Activity Solutions: The dimensions of the park inside the path are 100 − 2𝑥𝑥 by 60 − 2𝑥𝑥 . The original area is100×60 , or 6000 𝑚𝑚!.

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

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Quadratic Equations

The minimum new area is 90% × 6000, or 5400m.

23

Press: a. 1: Add Calculator 1

Press: a. b b. 3: Algebra 3 c. 1: Numerical Solve 1

Press a. (100-2X) b. (60-2X) c. =5400,X d. · From the answer,

Activity Summary:

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𝑥𝑥 = 1.9211

Discuss about the maximum allowable width of the path.

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TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 3 Learning Outcome

Question:

Path of a Soccer Ball In this activity, students will investigate and predict the movement of a soccer ball.

Farhan uses a digital video recorder to record Salman’s kick of a soccer ball. He then chooses an origin and a scale and makes measurements of the height of the soccer ball at several horizontal distances from the television screen during playback.

Table 2.1 Horizontal Distance (m) 2.5 5.0 7.5 10.0 12.5 15.0 Height 3.25 4.80 5.75 6.10 5.85 5.00

a. Use TI-Nspire CX to determine an equation for the quadratic relation. b. Where does the soccer ball hit the ground? Round to the nearest tenth of a metre. Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 4: Add List & Spreadsheet 4

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Quadratic Equations

25

At A’s box, press DISTANCE, then · Repeat your step for HEIGHT at B’s box. Then, at A1’s box, press 2^5·



Fill in your data from Table 2.1

To view your data as a graph, press: a. / ¿ b. 5: Add Data & Statistics 5

At x-axis, press Click to add variable. Choose distance, then ·.

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TI-Nspire Graphing Technology: Pre-University Mathematics

At y-axis, press at Click to add variable. Choose height, then ·.

Press: a. b b. 4: Analyze 4 c. 6: Regression 6 d. 4: Show Quadratic 4 Activity Summary:

Hence, can you determine an equation for the quadratic relation? Explain your answer.

Press:

a. b b. 4: Analyze 4 c. A: Graph Trace A

To find the location of soccer ball hit the ground, the height must be 0.

Press ¢ repeatedly. You will see that the value of height is not 0.

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Quadratic Equations

27

Press:

a. b b. 5: Window/Zoom 5

c. 3: Zoom-In 3

Move your cursor ã to the intersection between the graph and x-axis.

Repeat your steps Graph-Trace and Zoom-In until the value of height is 0.

Activity Summary:

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Hence, can you determine where does the soccer ball hit the ground? Give your justifications.

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TI-Nspire Graphing Technology: Pre-University Mathematics

SUMMARY Chapter 2 discusses about how to use TI-Nspire graphing calculator in solving problems related to Quadratic Equations topic.   Students that are having a firm understanding of the quadratic equation helps increase logical thinking, critical thinking, and number sense. Understanding the quadratic  formula  could help make them a more intelligent and rational individual. There were three activities given in this chapter to enhance students’ understanding on how to use TINspire to solve the problems. The first activity was about solving a projectile problem, secondly was to investigate how to maximize the width of the path around a rectangular park and finally, to investigate and predict the movement of a soccer ball. With the use of TI-Nspire graphing calculator, students will be able to explore, investigate and solve problems easily and at the same time they will be able to make justifications and use their logical thinking while solving the problems.

SELF-ASSESSMENT 1. a. Create a quadratic model for the height of a toy rocket launched upward at 45m/s from a 2m platform. b. How long would the rocket take to fall to earth, rounded to the nearest hundredth of a second? 2. A firework is launched upward at an initial velocity of 49m/s, from a height of 1.5m above the ground. The height of the firework, in meters, after t seconds, is modelled by the equation. a. What is the maximum height of the firework above the ground? b. Over what time interval is the height of the firework greater than 100m above the ground? Round to the nearest hundredth of a second. 3. The length of a rectangle is 16cm greater than its width. The area is 35m2. Find the dimensions of the rectangle, to the nearest hundredth of a metre.

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Quadratic Equations



29

x x + 16

4. A cylindrical can with height 12cm has capacity 600ml. What is its radius, to the nearest millimetre? [Remember that 1ml = 1cm3.] 5. A corner shelf is to be made from a triangular piece of plywood, as shown below. Find the distance x that the shelf will extend along the walls. Assume that the walls are at right angles. Round off the answer to a tenth of a foot.



Source: http://people.ucsc.edu/~miglior/chapter%20pdf/Ch08_SE.pdf

6. Measurements from the flight path of a tennis ball are recorded. Table 2.2 Horizontal Distance (m) Height

6

8

10

12

14

4.4

4.9

5.0

4.7

4.0

a. Use TI-Nspire CX to create a scatter plot of the data and add a curve of best fit. b. Determine the equation of the quadratic relation. 7. A rectangular garden measures 15m by 24m. A larger garden is to be made by increasing each side length by the same amount. The resulting area is to be 1.5 times the original area. Find the dimensions of the new garden, to the nearest tenth of a metre.

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CHAPTER

3

FUNCTIONS

INTRODUCTION In mathematics, a function is a relation between a set of inputs and a set of acceptable outputs with the property that each input is associated to approximately one output. The input to the function is labelled as the independent variable and is also known as the argument of the function. The output of the function is labelled as the dependent variable. Hence, there are always three main components when we talk about functions: 1. The input 2. The relationship 3. The output A function is a relation and it is often essential to agree on its domain and range. Think about a function defined by the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) . The domain of f is the set of all x-values that when substituted into the function, generate a real number. The range of f is the set of all y-values matching to the values of x in the domain. In more advanced level, you’ll find out about far more complex functions. The domain of a function is also the set of possible inputs and the range of a function is the set of corresponding outputs. Let’s say a relation in x and y, we say “y is a function of x” if for every element x in the domain, there

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TI-Nspire Graphing Technology: Pre-University Mathematics

matches exactly one element y in the range. In real world, functions are mathematical building blocks for developing machines, forecasting natural disasters, alleviating diseases, understanding world economies and for keeping aeroplanes in the air. For instance, money is a function of time. We never have more than one amount of money at any time because we can always add everything to give one total amount. By understanding how our money changes over time, we can plan ahead to spend our money wisely. Businesses discover it very functional to plot the graph of their money over time so that they can observe when they are spending too much. Another example is temperature as a function of diverse factors. Temperature is a very complex function because it has so numerous inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, where we are and many more. But the main thing is that there is only one temperature output when you measure it in a specific place. Figure 3.1 shows some application of function in real life.



Figure 3.1 Applications of function in real life

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Functions

33

NOTES Functions

𝑓𝑓 𝑥𝑥 = 𝑥𝑥

𝑦𝑦 = 𝑥𝑥 !

𝑦𝑦 =

𝑥𝑥 !

𝑓𝑓 𝑥𝑥 = 𝑥𝑥

𝑦𝑦 = 𝑥𝑥 1 𝑦𝑦 = 𝑥𝑥

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Figures

Domain and Range

Domain (−∞, ∞)

Range (−∞, ∞)

Domain (−∞,∞) Range [0,∞)

Domain (−∞, ∞) Range (−∞, ∞)

Domain (−∞, ∞)

Range [0, ∞)

Domain (0,∞) Range (0,∞)

Domain (−∞, 0) ∪ (0, ∞)

Range (−∞, 0) ∪ (0, ∞)

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TI-Nspire Graphing Technology: Pre-University Mathematics

KEYSTROKE The following gives some keystrokes of TI-Nspire CX relevant in learning the topic of Function. 1. How to open the graph application? Press: a. 1: New Document 1 b. 2: Add Graph 2

2. How to choose the trigonometry graph? Press µ. Then, select the desired function.

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Functions

ACTIVITY 1 Learning Outcome

Question:

35

Office Temperature In this activity, student will interpret the office temperature data from the graph.

The temperature in an office is controlled by an electronic thermostat. The temperatures vary according to the sinusoidal function: 𝑦𝑦 = 19 + 6 sin

𝜋𝜋 𝑥𝑥 − 11 12

where y is the temperature (ºC) and x is the time in hours past midnight. a. What is the temperature in the office at 9 a.m., when employees come to work? b. What is the maximum and minimum temperatures in the office? Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1:New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 1: New Document 1 b. 1: Add Calculator 1

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TI-Nspire Graphing Technology: Pre-University Mathematics

What is the temperature in the office at 9 a.m., when employees come to work? 𝑦𝑦 = 19 + 6 sin



𝜋𝜋 𝑥𝑥 − 11 12



Press: a. 19+6 b. µ choose sin c. /Ô ¹ choose π d. ¤ 12 e. ¢ (9 -11) f. ·

Press: a. £ 2 times (highlight equation) b. / C to copy.

Press: a. /¿ b. 2: Add Graphs 2

Press: a. /V b. Change 9 to x c. ·

You cannot see the graph. So, you should change the window setting.

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Functions

37

Press: a. b b. 4: Window / Zoom 4 c. 1: Window Settings... 1 Change window settings as shown: Xmin: 0 Xmax:26 Xscale:1.5 Ymin: 0 Ymax:30 Yscale:1 Press ·for OK Press: a. b b. 6: Analyze Graph 6 c. 2: Minimum 2

When the calculator asks for lower bound, you must select before the lower curve of the graph. Then, press ·.

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TI-Nspire Graphing Technology: Pre-University Mathematics

When the calculator asks for upper bound, you must select after the lower curve of the graph. Then, press ·.

Repeat your step for the higher curve of the graph to find the maximum value. Your value should be same as shown.

Activity Summary:

Hence, what does it mean by (5.00, 13.00) and (17.00, 25.00)? Explain.

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Functions

ACTIVITY 2 Learning Outcome

Question:

Absolute Function In this activity, students will explore the relationship between absolute function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑓𝑓(𝑥𝑥)

If 𝑦𝑦 = 𝑥𝑥 ! − 2 . Draw a graph for 𝑦𝑦 = 𝑥𝑥 ! − 2 . Then, find the relationship between the two graphs. Student’s Activity

Solutions:

39

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 2: Add Graphs 2

𝑦𝑦 = 𝑥𝑥 ! − 2

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Press: a. X q-2·

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TI-Nspire Graphing Technology: Pre-University Mathematics

To insert another equation, you have to press e. 𝑦𝑦 = 𝑥𝑥 ! − 2



Press: a. t Choose: · b. X q-2·

Move your cursor | to the label until it changes to ÷, then press / and x to grip the label.

Touch your touchpad x upwards to move the label. Then, press ·.

Activity Summary

What is the difference between 𝑦𝑦 = 𝑥𝑥 ! − 2 and 𝑦𝑦 = 𝑥𝑥 ! − 2 ?

What value of x does the reflection take place? Explain your answer.

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Functions

ACTIVITY 3 Learning Outcome

Question:

41

Rational Function In this activity, students will learn the shape of rational function and find the asymptotes. 3𝑥𝑥 + 1

. Find what Consider the graph of 𝑦𝑦 = 𝑥𝑥 − 2 value of:

a. x is the rational function as y is undefined? b. y is the rational function as x is undefined? Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1:New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 2: Add Graphs 2

Draw a rational graph by pressing: a. / Ô b. 3X+1¤ c. X -2·

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TI-Nspire Graphing Technology: Pre-University Mathematics

Press: a. b b. 7: Table 7 c. 1: Split-screen Table 1 Find what value of: a. x is the rational function as y is undefined?

From the table, can you answer the (a) question? What is the name of this vertical line? Press: a. b b. 2: Table 2 c. 5: Edit Table Settings 5

Change the Table Step to 100. Press · at OK.

Find what value of: a. y is the rational function as x is undefined?

Can you see the pattern? Try to change Table Step again to: a. 1000 b. 10 000 c. 100 000 Hence, what is the answer to the question (b)? Justify your answer.

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Functions

43

SUMMARY This chapter illustrates about how to use TI-Nspire graphing calculator in solving problems related to Function topic. In the real-world, functions are mathematical representations of many input-output situations. In order to master Calculus, mathematical functions are the first thing that students need to understand. As in Chapter 2, there were also three activities given to develop students’ understanding on how to use TI-Nspire to solve the problems. The first activity was about interpreting the office temperature data from the graph. Secondly, students will investigate the relationship between two absolute functions. Finally, students will explore the shape of rational function by finding the asymptotes. The use of TI-Nspire graphing calculator in solving the problems helps students be more motivated and always be reflective thinking persons.

SELF-ASSESSMENT 1. The height (in feet) of a ball that is dropped from an 80-ft building is given by ℎ 𝑡𝑡 = −16𝑡𝑡 ! + 80 , where t is time in seconds after the ball is dropped. a. Find h(1) and h(1.5) b. Interpret the meaning of the function values found in part (a). 2. If Alina rides a bike at an average of 11.5 mph, the distance that she rides can be represented by 𝑑𝑑 𝑡𝑡 = 11.5𝑡𝑡 , where t is the time in hours. a. Find d(2) and d(5) b. Interpret the meaning of the function values found in part (a). 3. Ahmad’s score in an exam is a function of the number of hours he 100𝑥𝑥 ! spends studying. The function defined by 𝑃𝑃 𝑥𝑥 = 50 + 𝑥𝑥 ! (𝑥𝑥 ≥ 0) indicates that he will achieve a score of 𝑃𝑃% if he studies for x hours.

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TI-Nspire Graphing Technology: Pre-University Mathematics

a. Evaluate P(0), P(5), P(10), P(15), P(20), and P(25). (Round to 1 decimal place.) b. Interpret P(25) in the context of this problem. 4. If 𝑦𝑦 = 𝑥𝑥 ! − 4 . Draw a graph for 𝑦𝑦 = 𝑥𝑥 ! − 4 . Then, find the relationship between the two graphs.

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CHAPTER

4

MATRICES

INTRODUCTION A matrix is a rectangular collection of numbers set in rows and columns. The collection of numbers below is an example of a matrix.

21 62 33 93 44 95 66 13 77 38 79 33

The number of rows and columns that a matrix has is known as its dimension or its order. By principle, rows are listed first; and columns, second. Thus, the dimension (or order) of the above matrix is 3 x 4, meaning that it has 3 rows and 4 columns. Numbers that come into view in the rows and columns of a matrix are known as elements of the matrix. In the above matrix, the element in the first column of the first row is 21; the element in the second column of the first row is 62; and so on. The term “matrix” (Latin for “womb”, originated from mater mother) was coined by James Joseph Sylvester in 1850 as in Figure 4.1, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that originated from the original one by deleting columns and rows.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Figure 4.1 James Joseph Sylvester

An English mathematician named Cullis was the first to make use of modern bracket notation for matrices in 1913 and he simultaneously exhibited the first significant use the notation A = [ 𝑎𝑎!,! ] to represent a matrix where 𝑎𝑎!,! refers to the ith row and the jth column. Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen.

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Matrices

47

NOTES 1. The identity matrix for multiplication for the set of all square matrices of order n is the square matrix of order n, denoted by I, with 1’s along the principal diagonal (from upper left corner to lower right corner) and 0’s elsewhere.

1 0 0 1 0 and 0 1 0 0 1 0 0 1

2. Inverse of a Square Matrix - If M is a square matrix of order n and if there exists a matrix 𝑀𝑀!! (read “M inverse”) such that

𝑀𝑀!! 𝑀𝑀 = 𝑀𝑀𝑀𝑀!! = 𝐼𝐼

then 𝑀𝑀!! is called the multiplicative inverse of M or, more simply, the inverse of M.

3. Basic properties of Matrices. Assuming all products and sums are defined for the indicated matrices A, B, C, I, and 0, then Addition Properties Associative: 𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶 = 𝐴𝐴 + (𝐵𝐵 + 𝐶𝐶) Commutative: 𝐴𝐴 + 𝐵𝐵 = 𝐵𝐵 + 𝐴𝐴 Additive Identity: 𝐴𝐴 + 0 = 0 + 𝐴𝐴 = 𝐴𝐴 Additive Inverse: 𝐴𝐴 + −𝐴𝐴 = (−𝐴𝐴) + 𝐴𝐴

Multiplication Properties Associative Property: 𝐴𝐴 𝐵𝐵𝐵𝐵 = 𝐴𝐴𝐴𝐴 𝐶𝐶 Multiplicative Identity: 𝐴𝐴𝐴𝐴 = 𝐼𝐼𝐼𝐼 = 𝐴𝐴 Multiplicative Inverse: If A is a square matrix and 𝐴𝐴!! exists, then 𝐴𝐴𝐴𝐴!! = 𝐴𝐴!! 𝐴𝐴 = 𝐼𝐼 Equality Addition: If 𝐴𝐴 = 𝐵𝐵 , then 𝐴𝐴 + 𝐶𝐶 = 𝐵𝐵 + 𝐶𝐶 Left Multiplication: If 𝐴𝐴 = 𝐵𝐵 then 𝐶𝐶𝐶𝐶 = 𝐶𝐶𝐶𝐶 Right Multiplication: If 𝐴𝐴 = 𝐵𝐵 , then 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵

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TI-Nspire Graphing Technology: Pre-University Mathematics

KEYSTROKE Below are a few keystrokes that will be needed in learning Matrices. 1. To get more application and operation of matrix. Press: a. b b. 7:Matrix & Vector 7

2. To get the template of matrices, a. Press t

b. Press: i. b ii. 7:Matrix & Vector 7 iii. 1: Create 1 iv. 1: Matrix 1

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Matrices

ACTIVITY 1 Learning Outcome

49

Nutritional Content In this activity, student will solve the nutritional content problem by using matrices.

Question:

Ahmad as a nut distributor would like to know the nutritional content of various mixtures of almonds, cashews, and pecans. His supplier has provided the following nutritional information: Table 4.1 Almonds Cashews Pecans Protein (g/cup) 16.5 22.4 12.2 Carbohydrate (g/cup) 30.4 44.3 13.4 Fat (g/cup) 61.8 55.9 75.1

His first mixture, a protein blend, consists of 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. His second mixture, a low fat mix, consists of 3 cups of almonds, 6 cups of cashews, and 1 cup of pecans. His third mixture, a low carb mix consists of 3 cups of almonds, 1 cup of cashews, and 6 cups of pecans. Table 4.2



Protein Low Fat Low Carb Blend Mix Mix Almonds (cups) 6 3 3 Cashews (cups) 3 6 1 Pecans (cups) 1 1 6

Determine the amount of protein, carbs and fats in one cup serving of each of the mixtures.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Student’s Activity Solutions:

Screen Display

Firstly, convert the data in a matrix form.

16.5 22.4 12.2 30.4 44.3 13.4 61.8 55.9 75.1

= matrix A

6 3 3 3 6 1 = matrix B 1 1 6

By using TI-Nspire CX, press: i. HOME button, c ii. 1: New Document 1

If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: i. 1: Add Calculator 1 Insert the data by pressing: i. t ii. Choose : iii. · iv. e 2 times v. ·for OK.

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Matrices

Matrix A 16.5 22.4 12.2 30.4 44.3 13.4 61.8 55.9 75.1



Matrix B 6 3 3 3 6 1 1 1 6



To determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures, we have to divide by 10 (because we have 10 cups before). Activity Summary:

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51

Press: i. 16^5¤ ii. 30^4¤ iii. 61^8 Insert the data (matrix A) until the end. Then, times with matrix B. Press: i. ¢ r ii. t Choose : iii. · iv. e 2 times v. ·for OK. vi. 6¤3¤1¢ vii. 1£6£3¢ viii. 3¤1¤6· Press: i. p 10 ii. · Therefore, could you complete the Table 4.3 below? Explain about the table that you have completed. Table 4.3



Protein (grams) Carbohydrates (grams) Fat (grams)

Protein Low Fat Low Carb Blend Mix Mix

















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TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 2 Learning Outcome

Question:

Stamp In this activity, student will solve stamp problems by using Gauss–Jordan Elimination.

A friend of yours came out of the post office after spending RM14.00 on 15 cent, 20 cent, and 35 cent stamps. If he bought 45 stamps in all, how many of each type did he buy? Student’s Activity

Solutions:

Screen Display

Convert the data to equations and augmented matrix.

𝑥𝑥! = 15 cent stamp 𝑥𝑥! = 20 cent stamp 𝑥𝑥! = 35 cent stamp 𝑥𝑥! + 𝑥𝑥! + 𝑥𝑥! = 45

15𝑥𝑥! + 20𝑥𝑥! + 35𝑥𝑥! = 1400

Augmented matrix 1 1 45 = 1 15 20 35 1400

By using TI-Nspire CX, press: i. HOME button, c ii. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: i. 1: Add Calculator 1

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Matrices

53

Insert the data by pressing: i. t ii. Choose : iii. · iv. Number of rows: 2 v. e vi. Number of columns: 4 vii. e viii. ·for OK. Insert your data: =

1 1 1 45 15 20 35 1400

Store it as “a” by pressing: i. ¢ ii. / Ë iii. A iv. ·

Press: i. b ii. 7: Matrix & Vector 7 iii. 5: Reduce RowEchelon Form 5 Press: i. A ii. ·

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TI-Nspire Graphing Technology: Pre-University Mathematics

From the calculation, we get: 1 0 −3 −100 = 0 1 4 145

𝑥𝑥! − 3𝑥𝑥! = −100 𝑥𝑥! + 4𝑥𝑥! = 145

Let 𝑥𝑥! = 𝑡𝑡 , then

𝑥𝑥! = 3𝑡𝑡 − 100 𝑥𝑥! = −4𝑡𝑡 + 145 𝑥𝑥! = 𝑡𝑡

3𝑡𝑡 − 100 > 0

𝑡𝑡 >

100 3

𝑡𝑡 > 33.33 𝑡𝑡 = 34

−4𝑡𝑡 + 145 > 0

145 > 𝑡𝑡 4

35.25 > 𝑡𝑡 𝑡𝑡 = 35

From the equation below and the values of t, complete the Table 4.4.

𝑥𝑥! = 3𝑡𝑡 − 100 𝑥𝑥! = −4𝑡𝑡 + 145

t

34 35

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𝑥𝑥! 𝑥𝑥! 𝑥𝑥!

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55

Matrices

ACTIVITY 3 Learning Outcome

Question:

Cryptography In this activity, student will learn about cryptography and how to encoding and decoding message using matrices.

Cryptography is the science of writing in secret code and is an ancient art. We can use matrix inverse to provide a simple and effective procedure for encoding and decoding messages. To begin, we assign the numbers 1 to 26 to the letters in the alphabet, as shown in Table 4.5. We assign the number 27 to a blank to provide for space between words. Table 4.5 A 1 K 11 U 21

B 2 L 12 V 22

C 3 M 13 W 23

D 4 N 14 X 24

E 5 O 15 Y 25

F 6 P 16 Z 26

G 7 Q 17 BLANK 27

H 8 R 18

I 9 S 19

J 10 T 20

By using encoding matrix = 4 3 , encode 5 4 the message I LOVE MATH. Then, decode the message to prove your message is true. Student’s Activity Solutions:

Screen Display

The numbers of I LOVE MATH are 9 27 12 15 22 5 27 13 1 20 8 We can encode the message by multiple with encoding matrix.

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TI-Nspire Graphing Technology: Pre-University Mathematics

4 3 5 4

9 12 22 27 1 8 27 15 5 13 20 27

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1

If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 1: New Document 1 b. 1: Add Calculator 1

Press: a. t b. Choose : c. · Insert your data:

4 3 5 4

Press ¢ until blink line is outside the matrix. Then, press: a. r b. t c. Choose : ·

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Matrices

57

d. Number of rows: 2 e e. Number of columns: 6 f. e · Insert your data, then press ·. 9 12 22 27 1 8 27 15 5 13 20 27

Therefore, the coded message I LOVE MATH is 117 153 93 120 103 130 147 187 64 85 113 148 To decode the message I LOVE MATH, we have to find the inverse of encoding matrix. Press: a. t b. Choose : c. · Insert your data:

4 3 5 4

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TI-Nspire Graphing Technology: Pre-University Mathematics

Activity Summary:

Press ¢ until the blink line is outside the matrix. Then, press: a. l b. -1 c. ·

Press: a. r b. £ 3 times (highlight previous answer) c. ·

Press: a. · 2 times

From the activity above, determine whether the answer is the same as the code of I LOVE MATH.

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Matrices

59

SUMMARY Chapter 4 concerns about how to use TI-Nspire graphing calculator in solving problems related to Matrices topic. There are numerous applications of matrices, both in mathematics and other sciences, thus making it an indispensable concept for solving many practical problems. Students should have a firm understanding of the concept of matrices in order to solve problems such as solving systems of equations, transforming shapes and vectors, and representing realworld situations. There were three activities given in this chapter to help students’ understanding on how to integrate the TI-Nspire graphing calculator to solve the problems. The first activity was about solving the nutritional content of various mixtures of almonds, cashews and pecans by using the keystrokes of Matrix and Vector. Secondly, students will solve the stamp problem by using the Reduce Row- Echelon Form menu. Lastly, students were given an example related to concept of cryptography where they can explore the process of encoding and decoding messages using concept of matrices. By integrating the use of TI-Nspire graphing calculator in solving the problems, students will be more investigative, analytical and enjoyable as well as save their time especially in reducing to row-echelon form procedures.

SELF-ASSESSMENT 1. A parking meter accepts only nickels, dimes and quarters. If the meter contains 32 coins with a total value of RM6.80, how many of each type are there? 2. Solve the problems by using Gauss–Jordan Elimination.

a. 2𝑥𝑥! + 4𝑥𝑥! − 10𝑥𝑥! = −2



3𝑥𝑥! + 9𝑥𝑥! − 21𝑥𝑥! = 0 𝑥𝑥! + 5𝑥𝑥! − 12𝑥𝑥! = 1

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60



TI-Nspire Graphing Technology: Pre-University Mathematics

b. 3𝑥𝑥! + 8𝑥𝑥! − 1𝑥𝑥! = −18



2𝑥𝑥! + 𝑥𝑥! + 5𝑥𝑥! = 8 2𝑥𝑥! + 4𝑥𝑥! + 2𝑥𝑥! = −4

3. Encoding matrix A = 3 1 the message:

5 . With the matrix A given, encode 2

1 0 4. Encoding matrix B = 2 0 1

0 1 1 0 1

a. CAT IN THE HAT b. FOX IN SOCKS c. I LOVE MALAYSIA

1 1 1 1 1

0 0 1 0 2

1 3 1 2 1

a. With the matrix B given, encode the message: i. NEGERI SEMBILAN ii. VIRUS DETECTED iii. COCONUT SHAKE

b. The following message was encoded with the matrix B given above. Decode this message: i. 41 84 44 67 ii. 22 15 51 68 iii. 38 47 100

82 44 74 25 56 67 20 54 43 54 89 39 102 86 44 90 68 135 136 81 149 57 5 47 54 58 89 45 84 46 80 87 53 96 116 39 113 68 135 136 81 149 106 22 113 20 17 59 3 49 56 98 102 63

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CHAPTER

5

TRIGONOMETRY

INTRODUCTION

Figure 5.1 Hipparchus

Trigonometry is a study of angles, triangles and trigonometric. The term “trigonometry” is originated from the Greek “trigonometria”, which plainly means “triangle measuring”. Our modern word “sine” is originated from the Latin word sinus, which implies “bay”, “bosom” or “fold”. The first trigonometric table was actually compiled by Hipparchus of Nicaea (180 – 125 BCE) (Figure 5.1), who is now acknowledged as “the father of trigonometry”. Hipparchus, the father of the science of trigonometry was the first to tabulate the corresponding values of arc and chord for a series of angles. Regarding the six trigonometric functions: Aryabhata (470 CE-550 CE) found the sine and

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TI-Nspire Graphing Technology: Pre-University Mathematics

cosine. Muhammad ibn Musa Al Khawarizmi (780 CE-850 CE) found the tangent; Abu al-Wafa’ Buzjani (940 CE-988 CE) found the secant, cotangent and cosecant. A French mathematician, Albert Girard (1595-1693) was the first to exercise the abbreviation sin, cos and tan in a treatise. In real life, trigonometry is commonly utilized in finding the height of the towers and mountains, in the navigation to search the distance of the shore from a point in the sea, in oceanography in calculating the height of tides in the oceans, in finding the distance between celestial bodies and also use by architects to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles. The following Figure 5.2 shows some applications on the use of trigonometry.

𝑓𝑓 𝑥𝑥 = cos 𝑥𝑥 Figure 5.2 Trigonometric Application

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Trigonometry

63

NOTES 1. For any right-angled triangle (SOH – CAH – TOA):

2.

Sin θ =

Opposite Adjacent Opposite , Cos θ = , Tan θ = Hypotenuse Hypotenuse Adjacent

Functions

Graphs

Functions

𝑓𝑓 𝑥𝑥 = sin 𝑥𝑥

𝑓𝑓 𝑥𝑥 = cosec 𝑥𝑥

𝑓𝑓 𝑥𝑥 = cos 𝑥𝑥

𝑓𝑓 𝑥𝑥 = sec 𝑥𝑥

𝑓𝑓 𝑥𝑥 = tan 𝑥𝑥

𝑓𝑓 𝑥𝑥 = cot 𝑥𝑥

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Graphs

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TI-Nspire Graphing Technology: Pre-University Mathematics

KEYSTROKE

The following describes the keystrokes that often used in Trigonometric topic. 1. To change degree to radian and vice versa settings.

Press: a. b b. 7: Matrix & Vector 7 c. 7: Settings & Status 7 d. 2: Document Settings 2 e. Change the angle : Angle or Degree

2. To get trigonometry mode. Press µ. Then, select the desired function.

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Trigonometry

ACTIVITY 1 Learning Outcome

Question:

Trigonometry Ratio In this activity, student will learn how to use Trigonometry Ratio to determine the side lengths of a right-angled triangle.

An ice-cream cone has a diameter of 6cm and a sloping edge of 15cm. Find the angle at the bottom of the cone.

6 cm

6 cm

15 cm

Student’s Activity Solutions:

(O)

(A)

3cm

(H)

15cm

Use SOH

65

15 cm

Screen Display

Go to Home page by pressing c. Press: a. 1: New Document 1 b. 1: Add Calculator 1 Press: a. b b. 3: Algebra 3 c. 1: Numerical Solve 1

𝑂𝑂

Sin 𝜃𝜃 = 𝐻𝐻

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The angle at the bottom of the cone = 2×𝜃𝜃

Press: a. µ, choose sin, · b. k, choose 𝜃𝜃 , · c. )= / Ô d. 3¤ 15 e. ¢ , /k, choose 𝜃𝜃 , ·· To find the angle at the bottom of the cone, press: a. 2 r / Ý ·

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Trigonometry

ACTIVITY 2 Learning Outcome

67

Height of a Tree In this activity, student will solve the height of a tree problem.

Question:

The tallest tree in the world grow in Redwood National Park in California. Its height is longer than a football field. Based on the information given, find the height of this tree. (The 100-feet measurement is accurate to 3 significant digits.) Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 1: Add Calculator 1

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TI-Nspire Graphing Technology: Pre-University Mathematics

B

A C

Label the figure. Find ∡𝐴𝐴𝐴𝐴𝐴𝐴 and ∡𝐴𝐴𝐴𝐴𝐴𝐴

Change setting: a. ~ b. 7: Settings & Status 7 c. 2: Document Settings 2 d. Change the angel: Degree e. · at OK. To find ∡𝐴𝐴𝐴𝐴𝐴𝐴 , press: a. 180¹ Choose: ° · b. -44¹ Choose: ° · c. 0¹ Choose: ′ · d. · To find ∡𝐴𝐴𝐴𝐴𝐴𝐴 , press: a. 180¹ Choose: ° · b. -136¹ Choose: ° · c. -37¹ Choose: ° · d. 10¹ Choose: ′ · e. · To change the answer to degree, minute and second, press: a. £ · b. / Ð Choose: ¢ · c. DMS·

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Trigonometry

B

A C

Use Sine Rule: 𝐴𝐴 𝐵𝐵 = Sin 𝐴𝐴 Sin 𝐵𝐵 𝐵𝐵 𝐴𝐴 = ×Sin 𝐴𝐴 Sin 𝐵𝐵

B

A C

𝐶𝐶 =

D

𝐷𝐷 ×Sin 𝐶𝐶 Sin 𝐷𝐷

Activity Summary:

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69

Press: a. / Ô 100¤ b. µ Choose: sin · c. 6¹ Choose: ° · d. 50¹ Choose: ′ · e. ¢ 2 times f. rµ Choose: sin · g. 37¹ Choose: ° · h. 10¹ Choose: ′ · i. ·

Press: a. / Ô507^ 7554¤ b. µ Choose: sin · c. 90¹ Choose: ° · d. ¢ 2 times e. rµ Choose: sin · f. 44¹ Choose: ° · g. 0¹ Choose: ′ · h. · Thus, the height of the tree is _____?

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SUMMARY This chapter illustrates about how to use TI-Nspire graphing calculator in solving problems related to Trigonometric topic. Generally, Trigonometry is a study of the properties of triangles, one of the simplest geometrical figures, yet they have varied applications. The primary application of trigonometry is found in scientific studies where precise distances need to be measured. In this chapter, there were two activities given to develop students’ understanding on how to use TINspire in solving the trigonometric problems. The first activity was about learning how to integrate graphing calculator in determining the side lengths of a right-angled triangle using trigonometry ratio. The second activity was related to problem on finding the height of a tree problem. The use of TI-Nspire graphing calculator helps students to be more investigative and always use higher order thinking skill in making decision in solving the problems.

SELF-ASSESSMENT 1. The area of the segment of a circle in the Figure 5.6 is given by 1 𝐴𝐴 = 𝑅𝑅! (𝜃𝜃 − sin𝜃𝜃) where 𝜃𝜃 is in radian. Use a graphing utility 2

to find 𝜃𝜃 , to three decimal places if the radius is 8 inches and the area of the segment is 48 inches2.

2. A polarizing filter for a camera contains two parallel plates of polarizing glass, one fixed and the other able to rotate. If 𝜃𝜃 is the angle of rotation from the position of maximum light transmission,

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Trigonometry

71

then the intensity of light leaving the filter is cos ! 𝜃𝜃 times the intensity I of light entering the filter. Find the smallest positive 𝜃𝜃 (in decimal degrees to two decimal places) so that the intensity of light, leaving the filter is 40% of that entering.

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CHAPTER

6

COMPLEX NUMBER

INTRODUCTION A complex number is a number that can be used in the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 , where a and b are real numbers and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers broaden the concept of the onedimensional number line to the two-dimensional complex plane (also called the Argand plane) by using the horizontal axis of the real part and the vertical axis for the imaginary part. The complex number 𝑎𝑎 + 𝑏𝑏𝑏𝑏 can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this manner, the complex numbers consist the ordinary real numbers while widening them in order to solve problems that cannot be solved with real numbers alone. One of the most beautiful applications of complex number theory is conformal mapping. It is a very powerful tool to extend solutions of certain differential equations in simple geometries to complex geometries. One such practical application is flow part an airfoil. In addition, all fractals utilize complex numbers to produce their images. If certain software is accessible to use in the computer area of the classroom, pupils may search how a variety of formulas

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TI-Nspire Graphing Technology: Pre-University Mathematics

produce precise kinds of images such as in the following Figure 6.1.

Figure 6.1 Images using Complex Numbers

NOTES

Figure 6.4

Rectangular Form: A complex number is any number that can be written in the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where a and b are real numbers and i is the imaginary unit. Thus, associated with each complex number 𝑎𝑎 + 𝑏𝑏𝑏𝑏 is a unique ordered pair of real numbers (a,b), and vice versa. For example, 3 − 5𝑖𝑖 correspond to ( 3, −5 ). When complex numbers are associated with points in a rectangular coordinate system, we refer to the x-axis as the real axis and the y-axis as the imaginary axis. The complex number 𝑎𝑎 + 𝑏𝑏𝑏𝑏 is said to be in rectangular form. Polar Form: Using the polar–rectangular relationships,

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Complex Number

75

x = r cos 𝜃𝜃 and y = r sin 𝜃𝜃

we can write the complex number in polar form as follows:

Figure 6.5

𝑧𝑧 = 𝑥𝑥 + 𝑖𝑖𝑖𝑖 = r cos 𝜃𝜃 + ir cos 𝜃𝜃 = r (1) (cos 𝜃𝜃 + i sin 𝜃𝜃 ) In a more advanced subject, the following equation is established.

𝑒𝑒 !" = cos 𝜃𝜃 + i sin 𝜃𝜃

where it obeys all the basic laws of exponents. Thus, equation (1) takes on the form

𝑧𝑧 = 𝑥𝑥 + 𝑖𝑖𝑖𝑖 = 𝑟𝑟 cos 𝜃𝜃 + 𝑖𝑖 sin 𝜃𝜃 = 𝑟𝑟𝑟𝑟 !"

The number r is called the modulus, or absolute value, of z and is denoted by mod z, or 𝑧𝑧 . The polar angle that the line joining z to the origin makes with the polar axis is called the argument of z and is denoted by arg z.

KEYSTROKE The keystroke that always been used in this topic is ¹, where the imaginary “z” is located here. 1. Press ¹ to get 𝜋𝜋, 𝑖𝑖, 𝑒𝑒 and °

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TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 1 Learning Outcome

Question:

Rectangular and Polar Form In this activity, students will learn how to change the unit from rectangular form to polar form and vice versa.

1. Change the rectangular form below to polar form. a. 1 − 𝑖𝑖 b. − 3 + 𝑖𝑖 c. −5 − 2𝑖𝑖

2. Change the polar form below to rectangular form. !!

a. 2𝑒𝑒 ! ! b. 3𝑒𝑒 !!"° ! c. 7.19𝑒𝑒 !!.!"!

Student’s Activity Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 1: Add Calculator 1

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Complex Number

Question 1 a. 1 − 𝑖𝑖

Press: a. 1b. ¹ Choose: i · c. / Ð Choose: ¢ · d. POLAR ·

Question 1 b. − 3 + 𝑖𝑖

Press: a. ( v / Ò3 b. ¢ +¹ Choose: i · c. / Ð Choose: ¢ · d. POLAR ·

Question 1 c. −5 − 2𝑖𝑖

Press: a. ( v 5-2 b. ¹ Choose: i · c. / Ð Choose: ¢ · d. POLAR ·

Question 2

Press: a. 2 b. ¹ Choose: e · c. l(5¹ Choose: 𝜋𝜋 · p6) ¹ d. Choose: i · e. ¢ / Ð Choose: ¢· f. RECT ·

a. 2𝑒𝑒

!! ! !



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TI-Nspire Graphing Technology: Pre-University Mathematics

Question 2 b. 3𝑒𝑒 !!"° !

Because we use in degree (60° ), then we have to change it in radian. Press: a. 60¹ Choose: ° · b. Ð Choose: ¢ · c. RAD · Press: a. 3¹ Choose: e · b. l(1^04 72) c. ¹ Choose: i · d. ¢ / Ð Choose: ¢ · e. RECT ·

Question 2 7.19𝑒𝑒 !!.!"!

Press: a. 7^19¹ Choose: e · b. lv 2^13 c. ¹ Choose: i · d. ¢ / Ð Choose: ¢ ·

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Complex Number

ACTIVITY 2 Learning Outcome

Question:

79

Forces In this activity, students will solve the forces problem by using complex number.

An object is located at the pole, and two forces

𝑢𝑢 and 𝑣𝑣 act on the object. Let the forces be

vectors going from the pole to the complex numbers 20e0°i and 10e60°i, respectively. Force 𝑢𝑢 has a magnitude of 20 pounds in a direction of 0°. Force v has a magnitude of 10 pounds in a direction of 60°. a. Convert and add the polar forms of these complex numbers to rectangular form. b. Convert the sum from part (a) back to polar form. c. The vector going from the pole to the complex number in part (b) is the resultant of the two original forces. What is its magnitude and direction? Student’s Activity

Solutions:

Screen Display

By using TI-Nspire CX, press: a. HOME button, c b. 1: New Document 1 If the message appears as: Do you want to save “Unsaved Document”, press · at NO.

Press: a. 1: Add Calculator 1

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TI-Nspire Graphing Technology: Pre-University Mathematics

Change 0° and 60° to radian. Press: a. 0¹ Choose: ° · b. Ð Choose: ¢ · c. RAD · Repeat your step for 60° Question a. Convert the polar forms of these complex numbers to rectangular form and add.

Press: a. 20¹ Choose: e · b. l0¹ Choose: i · c. ¢ / Ð Choose: ¢· d. RECT ·

Press: a. 10¹ Choose: e · b. l1^0472 ¹ Choose: i c. · ¢ / Ð Choose: ¢ · d. RECT ·

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Complex Number

Question b. Convert the sum from part (a) back to polar form.

Then, find the sum of both answer. Press: a. £ 3 times ( highlight 20) · b. + c. £ (highlight the answer before) d. · 2 times

Question c. The vector going from the pole to the complex number in part (b) is the resultant of the two original forces. What is its magnitude and direction?

Press: a. £ (highlight the answer before) · b. / Ð Choose: ¢ · c. POLAR ·

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81

From the last answer, can you determine what are 0.3335 and 26.4575? Can you answer question (c)?

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TI-Nspire Graphing Technology: Pre-University Mathematics

SUMMARY The use of TI-Nspire graphing calculator in solving problems related to Complex Numbers is explained in this chapter. Complex numbers are important in our daily life. For example, complex numbers are mainly used in electrical engineering techniques all the time as the Fourier transforms are used in understanding oscillations and wave behavior. Two activities were discussed in this chapter to enhance students’ understanding on how to use the TI-Nspire graphing calculator to solve the related problems. The first activity was about learning the basic concepts of complex numbers such as by integrating the use of graphing calculator on how to change the unit from rectangular form to polar form and vice versa. Secondly, the activity was to solve the forces problem by using complex number. In this activity, students are guided to use the graphing calculator to explore menu buttons of polar forms and rectangular forms to determine the magnitude and directions of the resultant forces. With the use of TI-Nspire graphing calculator, students will be able to explore, investigate and solve the problem easily and at the same time they will be able to think critically and creatively while solving the problems.

SELF-ASSESSMENT 1. An object is located at the pole, and two forces 𝑢𝑢 and 𝑣𝑣 act on the object. Let the forces be vectors going from the pole to the complex numbers 8e0°i and 6e30°i, respectively. Force 𝑢𝑢 has a magnitude of 8 pounds in a direction of 0°. Force 𝑣𝑣 has a magnitude of 6 pounds in a direction of 30°. a. Convert and add the polar forms of these complex numbers to rectangular form. b. Convert the sum from part (a) back to polar form. c. The vector going from the pole to the complex number in part (b) is the resultant of the two original forces. What is its magnitude and direction?

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Complex Number

83

2. Convert the following complex numbers into Polar Form, ensuring that the angle obtained is in the correct quadrant. a. b. c. d.

3 + 4j

− 3 + 5j − 6 – 7j 5 − 12j

3. Convert the following complex numbers into Rectangular Form. a. b. c. d.

5 300 15 300 3 2300 3 1300

The answers you obtained in 3 (c) and 3 (d) are the same. Explain why?

4. Three forces have magnitudes 3N (newton), 6N and 15N each making an angle to a datum axis of 30o, 60o and − 240o respectively. Find the resultant of these three forces and sketch this system of forces on an Argand Diagram.

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7

CHAPTER

VECTORS

INTRODUCTION

A plane is flying all along, heading towards north, but there is a wind approaching from the North-West as shown in Figure 7.1. wi

e an air pl

propellor

nd

Figure 7.1 A Flight of a Plane

The two vectors (velocity by the propeller, velocity of the wind) result in a slightly slower ground speed heading a little East of North. If you watched the plane from the ground it would seem to be slipping sideways a little (Figure 7.2).

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TI-Nspire Graphing Technology: Pre-University Mathematics

Figure 7.2 A Plane Watched from the Ground

Have you ever witnessed this happening? Maybe you have watched birds struggling against a strong wind that seem to fly sideways. Vectors assist in explaining this phenomenon. You may use vectors in almost every day activities you do. Examples of everyday activities that engage vectors include: a. Breathing (your diaphragm muscles exert a force that

has a magnitude and direction).

b. Walking (you walk at a velocity of around 6km/h in

the direction of the bathroom). c. Going to school (the bus has a length of about 20m and is headed towards your school). d. Lunch (the displacement from your class room to the canteen is about 40m in a northerly direction). Each vector quantity has a magnitude and a direction.

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Vectors

87

NOTES A vector is a combination of three things: a. a positive number called its magnitude (size) b. a direction in space c. a sense making more precise the idea of direction

Figure 7.3

Typically, a vector is illustrated as a directed straight line. The vector in Figure 7.3 would be written as AB with: a. The direction of the arrow, from point A to the point B, indicating the sense of the vector b. The magnitude, AB , of AB given by the length of AB . The three vectors given by AB , BC and AC shows in Figure 7.4. Make up the sides of a triangle. This is known as the triangle law of addition:

Figure 7.4

AB  BC  AC Vector can be represented as the 2-dimensional component form. Take i to be a vector of length 1 unit (unit vector) parallel to the x-axis and in the positive direction, and j to be a vector of length 1 parallel to the y-axis and in the positive direction.

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TI-Nspire Graphing Technology: Pre-University Mathematics

From Figure 7.5, OA  AC  OC . The vector OA is

parallel to the vector i and four times its length, thus, OA  4i

. Similarly,

AC  3 j . Thus, vector OC may be written as

OC  4i  3 j .

Figure 7.5

KEYSTROKE Below are some keystrokes of TI-Nspire CX that are important in learning vectors. 1. The function for Vector can be found in the menu under Calculator page. Press: a. b b. 7: Matrix & Vector 7 c. C: Vector C

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Vectors

89

ACTIVITY 1

Graphical Vector Addition

Learning Outcome

In this activity, students will add perpendicular vectors graphically and determine the resultant net vector.

Questions:

1. A cab driver drives four blocks north and six blocks east. Use the Graphs & Geometry application to draw the cab driver’s path and his net displacement vector. 2. Two students apply forces to a heavy box. Student A applies a force of 5 Newton eastward. Student B applies a force of 10 Newton southward. Use the Graphs & Geometry application on the next page to determine the net force vector on the box. 3. A fisherman rows a boat across a river. The fisherman rows the boat at an average velocity of 1m/s north. The river flows at an average velocity of 4m/s west. Use the Graphs & Geometry application on the next page to determine the boat’s net velocity vector. Student’s Activity

Solutions:

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Screen Display

Note: Adding vectors are complicated because vectors have both magnitude and direction. Vectors can be added graphically and mathematically. In this activity, you will learn how to add vectors graphically.

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TI-Nspire Graphing Technology: Pre-University Mathematics

Drawing vector for the cab’s driver path

AC is the sum of AB and BC . Study

the diagram above to understand how vectors can be added graphically. In your TI-Nspire CX, press 1: New Document 1 and choose Graph page.

Move the cursor to the axes until the label ‘axes’ is shown.

Hide the axes by pressing: a. / b. b c. 3: Hide/Show 3 d. 1: Hide Axes 1

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Vectors

91

Grid screen will assist in drawing vector for accuracy. Press: a. b b. 2: View 2 c. 6: Grid 6 d. 2: Dot Grid 2 Dot grid should appear on your screen.

Draw a diagonal vector XY anywhere on the screen. Press: a. b b. 8: Geometry 8 c. 1: Points & Lines 1 d. 8: Vector 8

Move the cursor until “point on” will appear and press · or the click button a. Press X to label the first end of the vector.

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Drag the vector diagonally and press ·. Press Y to label the second end of the vector. Be sure to press d to deactivate the vector function. Draw two perpendicular vectors whose sum is XY . Apply the same method to draw another two perpendicular vectors. This will be another option for two perpendicular vectors.

Measure the magnitude of vector

Measure each of the vectors. Move the cursor until “vector” appear on the screen. Then, press: a. / b. b c. 8: Measurement 8 d. 1: Length 1

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Vectors

Activity Summary:

93

The length of vector XY is 8.94 unit. Your measurement might be different from the measurement here. Find out the length of the other two vectors. Explain how do you relate the length of each vector by using Pythagorean Theorem? Try to visualise the other three situations in the questions.

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ACTIVITY 2 Learning Outcome

Question:

Speed and Direction of Airplane In this activity, students will be able to use vectors to find speed and direction.

An airplane is travelling at a speed of 800km per hour with a bearing of 330° at a fixed altitude with a negligible wind velocity. When the airplane reaches a certain point, it encounters a wind velocity of 112km per hour in the direction N 45 E . What are the resultant speed and direction of the airplane? Student’s Activity

Solutions: Draw the airplane’s vector

Screen Display

Use the Graph Application to draw and visualize the two situations above. We are going to draw the vector of the airplane in the Graph page by pressing: a. b b. 8: Geometry 8 c. 1: Points & Lines 1 d. 8: Vector 8 Starting from the origin draw the vector of bearing 330° . Drag the pencil to any part which we predict where the vector should be. Deactivate the vector function by pressing d.

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Vectors

95

Now, we shall check the angle to be 330°. Before that, make sure to check the graphing angle to be in degree. Press: a. b b. 9: Settings 9 c. · at OK Angle:

360  330  30 . Press: a. b b. 8: Geometry 8 c. 3: Measurement 3 d. 4: Angle 4 Select three points which defined by the angle to measure. Point 1: y-axis; Point 2: Origin; Point 3: Vector. It should be selected in correct order. As you can see, you might not get the correct angle. You have to adjust the vector until 30 by grabbing and moving the vector.

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Label this vector as v1. Move the cursor until the word ‘vector’ appear on the screen. Press: a. / b. b c. 2: Label 2 Type “v1” and press ·.

Next, when the wind came in the direction N 45 E . We predict another vector to be closer to the y-axis. Draw another vector and label as “v2”. From the diagram which we visualise previously, it is easier and more understandable to find the velocity of the airplane and velocity of the wind. v1  800 cos 120, sin 120 v2  112 cos 45, sin 45

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97

This can easily be calculated in the Calculator page or by using Scratchpad. Here, we create another page on Calculator. Press: a. / b. ~ c. 1: Add Calculator 1 v1  800 cos 120, sin 120

By using Calculator page, you should get: v1   400, 692.82

v2  112 cos 45, sin 45 By using Calculator page, you should get: v2  79.196, 79.196

Velocity of the airplane (in the wind) is,v:  v1  v 2   400  79.196,692.82  79.196

  320.8,772.02

The resultant speed of the airplane is: v 

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 320.82  772.02 2

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What is the resultant speed of the airplane? Angle of v1 is 120° and q will be the direction angle of the flight path when it encounters wind. You will have: tan  

772.02  2.4065 - 320.8

From here, find the true direction of the airplane (refer to the vector drawn in pg. 1.1 of the graphing calculator).

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Vectors

ACTIVITY 3 Learning Outcome

Question:

Inclined Plane In this activity, students should be able to construct a free body diagram and determine the net force acting on an object resting on an incline plane through geometric approach.

A block weighing 5.4kg rests on an inclined plane, as indicated in figure below. Draw and determine the components of the weight perpendicular to and parallel to the plane.

35°

Student’s Activity Solutions: Draw the inclined plane diagram

99

5.4kg

Screen Display

By using geometrical approach, we are going to construct the diagram in the Geometry page.

Construct the inclined plane with similar measurements. Press: a. b b. 5: Shapes 5 c. 2: Triangle 2

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100 TI-Nspire Graphing Technology: Pre-University Mathematics

Check whether it is a right angled triangle by measuring the 90° angle.

Measure the other 35° angle. Adjust to get as near to 35° angle. You may hide the angle once you satisfy with the measurement. Select the triangle and press / b to pin the triangle to avoid any unwanted movement to the inclined plane.

Construct the block’s weight vector. Press: a. b b. 4: Points & Lines 4 c. 8: Vector 8 Draw the 5.4kg vector.

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Vectors 101

Move the cursor pointing to the vector until the word ‘vector’ appear.

Measure the vector by pressing: a. / b. b c. 8: Measurement 8 d. 1: Length 1 Change the measurement to 5.4 by editing the previous value. Hide the measurement and pin the vector. Draw the three component vectors

Draw the vector which is parallel to the plane. Press: a. b b. 7: Construction 7 c. 2: Parallel 2 Select the inclined plane.

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102 TI-Nspire Graphing Technology: Pre-University Mathematics

Then, select the point.

The parallel line will be your guide to construct the parallel vector. Use the ‘vector tool’ to construct the parallel vector. Draw another component which is perpendicular to the plane or to the parallel vector. Press: a. b b. 7: Construction 7 c. 1: Perpendicular 1 Click on the centre point followed by the parallel vector.

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Vectors 103

The perpendicular line will guide you to draw the perpendicular vector.

Hide the perpendicular and parallel line once you have constructed the perpendicular vector.

Finally, you will have the free body diagram which consist of the weight vector, perpendicular and parallel vector. Hide the triangle diagram. Label the vector as: W: weight vector E: perpendicular vector A: parallel vector

As we know earlier, vector W is 5.4kg, now determine the magnitude of two components vector.

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Try to measure the angle between vector W and vector E. Explain why the angle is similar to one of the angles in the inclined plane. Translate vector A to form right angle triangle consist of vector W, E and A. Press: a. b b. 8: Transformation 8 c. 3: Translation 3 Select vector A.

Translate the vector and select point on vector W.

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Vectors 105

Create a line that will connect point on vector E and point on vector W. This line is parallel to vector A. Press: a. b b. 4: Points & Lines 4 c. 4: Line 4 Select point on vector W and the translated point.

Create a segment from point on vector E and point on vector W. Press: a. b b. 4: Points & Lines 4 c. 5: Segment 5 Hide the line after creating the segment. Measure the magnitude of vectors

Use measurement tool to measure the magnitude of vector E and vector A.

Calculate the magnitude using different approach and compare the solution

Compare the magnitude of both vectors when the magnitude is calculated through algebraic approach.

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106 TI-Nspire Graphing Technology: Pre-University Mathematics

SUMMARY Chapter 7 discusses about how to use TI-Nspire graphing calculator in solving problems related to Vector topic. There were three activities given in this chapter. In the first activity the students used TI-Nspire to add perpendicular vectors and determine the resultant net vector graphically while in the second activity the students used vectors to find speed and direction of airplane, and in third activity the students constructed a free body diagram and determine the net force acting on an object resting on an incline plane through geometric approach. With the use of TI-Nspire graphing calculator, students will be able to explore, investigate and solve the problem easily and at the same time they will be able to make justifications and use their logical thinking while solving the problems.

SELF-ASSESSMENT 1. The heading and air speed of an airplane are 60 and 402km/hour respectively. If the wind is 64km/hour from 150, find the ground speed, the drift angle and the course. 2. Aircraft BAE Hawk takes off at RMAF Chempaka in Penang. It is headed in a direction due West with a speed of 550miles per hour. There is a wind blowing from the south to the north at a speed of 150miles per hour. a. Use vector addition to diagram the two vectors and calculate the resultant vector, which is the jets speed relative to the ground. b. What is the direction of the jet’s velocity vector relative to the ground? 3. On a piece of paper, iron filings are sprinkled to reveal the magnetic field of a bar magnet. At a particular point on the paper, the magnetic field vector is given by: B1 = -15 gauss X + 10 gauss Y

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Vectors 107



On a second piece of paper, the iron filings from a second magnet are revealed using iron filings. At the same point on the paper as for the first magnet, a measurement is made of the magnetic field vector and it is given by B2 = 26 gauss X - 5 gauss Y a. If both magnets were placed under a third piece of paper at the same location, and iron filings were sprinkled on the paper, what would be net sum of the two magnetic fields at the point used in the first two papers? b. What would be the difference in magnetic field strengths between the two magnets at the measurement point? c. Which bar magnet has the strongest magnetic field?

4. A block rests on an inclined plane that makes an angle of 20° with the horizontal. The component of the weight parallel to the plane is 15.4kg. a. Determine the weight of the block. b. Determine the component of the weight perpendicular to the plane.

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CHAPTER

8

SEQUENCES AND SERIES

INTRODUCTION

Figure 8.1 Sequences and series

Sequences and series describe algebraic patterns such as an example shown in Figure 8.1. To solve problems involving sequences and series, it is a good strategy to list the first few terms and look for patterns that give support in obtaining the general term. When the general term is identified, then one can search any term in the sequence without writing all the preceding terms. Sequences are functional in our daily lives as well as in higher mathematics. One of the careers that involve a myriad of sequences and series is the Computer Software Engineer (CSE). A CSE uses mathematical operations to define the parameters for a software program. The sequences and series are used to create base algorithms for the operation

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110 TI-Nspire Graphing Technology: Pre-University Mathematics

of the programs (Figure 8.2), such as when certain events should trigger, when a button should be available to press, when a screen should pop up, well, if all of these operations were specified to trigger at an accumulation of a certain object or event.

Figure 8.2 Base Algorithm

Examples of everyday activities that involve sequence and series include: 1. The architecture of the HP Laser printer requires some sequence codes that has to be embedded into the print job before it could be sent off. 2. Physics use sequences and series to determine gravitation and mechanics. 3. In sports, sequences and series is used in basketball. In basketball the score increases by two, except for when someone shoots a three pointer.   4. In biomedical engineering, they use sequences and series to determine the growth rate of bacteria, which could help them, make medicines to fight against the bacteria.

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Sequences and Series 111

NOTES A list of numbers in particular order that follow some rules for finding later values is called a sequence. Each number in a sequence is called a term, and terms are often denoted by a1, a2, a3, …, an, … Example: Arithmetic progression, Geometric progression A series is formed by adding together the terms of a sequence. The use of sigma notation can simplify the way that series are written. Example: 12  2 2  32  ...  n 2 may be written as

n

i

2

i 1

A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence

a1 ,a2 ,a3 ,a4 ,...,an ,...

is arithmetic if there is a number, d such that

a2 − a1 = a3 − a2 = a4 − a3 =! = d

The number d is the common difference of the arithmetic sequence.

A sequence is geometric if the ratios of consecutive terms are the same. So, the sequence

a1 ,a2 ,a3 ,a4 ,...,an ,...

is geometric if there is a number, r such that

a2 a3 a4 = = =! = r,r ≠ 0 a1 a2 a3 The number r is the common ratio of the sequence.

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112 TI-Nspire Graphing Technology: Pre-University Mathematics

KEYSTROKE The following gives some keystrokes of TI-Nspire CX relevant in learning the topic of Sequences and Series. 1. Finding the nth term of an arithmetic sequence in the Calculator page. Press: a. b b. 3: Algebra 3 c. 1: Numerical Solve 1

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Sequences and Series 113

ACTIVITY 1 Learning Outcome

Question:

Exploring Sequences and Series In this activity, students will investigate sequences and series and to discover interesting patterns without tedious calculations. Besides, they will explore arithmetic and geometric sequences and series as well as the Fibonacci Sequence.

Investigate arithmetic and geometric sequences and series through different approach in Lists and Spreadsheet application. Student’s Activity

Solutions:

Create a sequence in the Lists & Spreadsheet application. Press: a. c b. 1: New Document 1 c. 4: Add Lists & Spreadsheet 4

Creating a sequence

Type number 2 into cell A1 and press ·.

Screen Display

Type number 5 into cell A2 and press ·.

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114 TI-Nspire Graphing Technology: Pre-University Mathematics

Instead of manually filling in all the cells, you can fill the spreadsheet automatically. Place the cursor in cell A1, press g, then press the down arrow, ¤, to highlight cells A1 until A2, and press ·. Press: a. b b. 3: Data 3 c. 3: Fill 3

A dotted border is shown around the highlighted cells.

Press the down arrow, ¤, until cell A4.

Press ·and the cells will be filled up.

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Sequences and Series 115

Is this an arithmetic or a geometric sequence? There is another method to create a sequence that is by using formula. Clear the previous list beforehand. Type number 2 into cell A1 and press ·.

In cell A2, type =a1+3, and press ·.

Use the arrow up, £, to select cell A2. Then, press: a. b b. 3: Data 3 c. 3: Fill 3 Press the down arrow, ¤ until cell A4.

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116 TI-Nspire Graphing Technology: Pre-University Mathematics

Press ·.

Try to scroll over Column A, observe how each term was generated. If cell A4 is selected, examine the bottom of the screen that displays the formula, which is ‘a3+3’. By using all the formula which appear at the bottom, try to generalize to obtain a formula for finding the nth term of the sequence. Formulating the sum of arithmetic sequence

Type number 2 into cell B1, and press ·.

Type =b1+a2 in cell B2 and press ·.

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Sequences and Series 117

Select cell B2 and press: a. b b. 3: Data 3 c. 3: Fill 3 Press · and use the down arrow to highlight until cell B4.

Press · once again to fill the cells. Realize that the sum a1+a2+a3+a4 = 26.

Generalize a formula for an arithmetic series and the sum of arithmetic sequence based on all the information of the formula appear at the bottom of the screen. Formulating the nth term geometric sequence

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Create a new problem by pressing: a. ~ b. 4: Insert 4 c. 1: Problem 1

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118 TI-Nspire Graphing Technology: Pre-University Mathematics

Choose 4: Add Lists & Spreadsheet 4

Type number 3 into cell A1, press ·.

In cell A2, type = a1⋅ 2 , and press ·.

Follow previous step to fill the data up to cell A4.

Generalize the results to obtain the formula, for the nth term of geometric sequence. Formulating the sum of arithmetic sequence

Type number 3 into cell B1, press ·. In cell B2 type = b1+ a2 , and press ·.

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Sequences and Series 119

Use Data Fill function to fill until cell A4. Observe that the sum

a1+ a2 + a3+ a4 = 45.

Generate a formula for the sum of a geometric sequence. Exploring Fibonacci Sequence

Note: Fibonacci sequence is a well–known recursive sequence that produces interesting relationships. It is generated by choosing values for the first two terms. The third term is obtained by adding the first and the second terms. The fourth term is the sum of the second and third terms. Each successive term is the sum of the previous two terms. Create a new problem from ~. Choose List and Spreadsheet application.

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120 TI-Nspire Graphing Technology: Pre-University Mathematics

Type number 1 in cell A1 and A2.

In cell A3, type = a1+ a2 and press ·.

Fill the data until cell A8 by using the Data Fill function.

Column B will be used to examine the ratios between consecutive terms in Column A. In cell B2, type = a 2 a1 and press ·. Use Data Fill and fill Column B until cell B8.

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Sequences and Series 121

Instead of displaying the data in fractions, use decimal approximations by adding a decimal point in the entry in cell A1 and press ·. Fill column A and column B until cell A25 and B25. Observe that as the value of n increases, the ratio between consecutive terms approaches the Golden Ratio, phi

ϕ ≈ 1.61803398875

Try to change the value in A1 and A2 to any non-zero value and observe the changes in the ratio between consecutive terms. Use Column C to examine the ratios between every other term in Column A. Type = a3 a1 in cell C3 and press ·.

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122 TI-Nspire Graphing Technology: Pre-University Mathematics

Fill the data until cell C25.

Activity Summary:

Note that the sum approaches ϕ +1 . Generate other ratio and subsequent ratios to see that the ratios approach 2ϕ +1 = ϕ 3 .

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Sequences and Series 123

ACTIVITY 2 Learning Outcome

Question:

AIDS Cases In this activity, students will be able to apply the concept of geometric sequence in real life situation.

The numbers an (in thousands) of AIDS cases reported from 2000 to 2007 can be approximated by the model

an = 0.0768n 3 − 3.120n 2 + 41.56n −136.4 n = 10, 11, . . . , 17 where n is the year, with n = 10 corresponding to 2000. Explore the graphical representation of the number of AIDS cases. Student’s Activity

Solutions: Creating the sequence of AIDS cases by year

Screen Display

The sequence can be represented in the form of data tablature and graph. Name Column A as “y” to represent year.

Fill in A1 – A8 year starting from 2000 until 2007.

Name Column B as “cases”.

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124 TI-Nspire Graphing Technology: Pre-University Mathematics

In the formula box, type: = 0.0768n 3 − 3.120n 2 + 41.56n −136.4

Press ·.

Choose “All Variable References” and press ‘OK’.

All the list will appear automatically. The value is very large, thus, you can refer to the below of the screen for the number for each term. List all the terms of the finite sequence. What can you interpret on the number of cases based on the sequence?

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Sequences and Series 125

Representing sequence in the form of line graph

To represent the sequence in graph, create a new page on Graphs. Press: a. / b. ~ c. 5: Add Data & Statistics 5 You have to name the variable. The horizontal axis will be the year, while the vertical axis to be the cases.

Connect the point to form a line graph. Press: a. b b. 1: Plot Type 1 c. 6: XY Line Plot 6

Each of the point is connected now.

Activity Summary:

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What does the graph say about reported cases of AIDS?

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126 TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 3 Learning Outcome

Question:

Bouncing Ball In this activity, students will be able to apply the concept of geometric sequence in real life situation.

A tennis ball rebounds to half the height from which it is dropped. Use an infinite geometric series to approximate the total distance the ball travels after being dropped from 1 m above the ground until it comes to rest. Student’s Activity

Solutions: Understand the bouncing ball problem through visualising diagram

Screen Display

To understand more on the situation, sketch a diagram which apply to the above problem. You may use the Geometry application in your graphing calculator.

When you drop a ball, the rebound height becomes smaller after each bounce. In this situation, it bounces half of its previous height each time it hits the ground. This model is an infinite geometric sequence.

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Sequences and Series 127

To approximate the total distance the ball travelled, d = fall + bounce + fall + bounce + ...

fall bounce

1

Creating the sequence in table

fall bounce

fall

2 3 Bounce number

Name Column A as n, represent the number of bounce. Fill in Column A with the bounce number until cell A20.

Name Column B as h f , represent the height when the ball falls.

Type “1.0” in cell B1. This refers to the initial height before the ball being dropped.

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128 TI-Nspire Graphing Technology: Pre-University Mathematics

During the second bounce, the ball fall half the height previously which is 0.5m. Instead of inserting manually, we use a formula to generate a sequence on the height after nth bounce. Type =b1/2 in cell B2 and press ·. To generate the sequence, press: a. b b. 3: Data 3 c. 3: Fill 3 Make sure cell B2 is selected. Press ¤ to select until cell B20.

Press ·.

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Sequences and Series 129

Repeat similar steps to generate the sequence for hb . The different in hb is that during the first bounce the hb is null because the ball were dropped from 1 m above the ground. Represent the sequence graphically

Observe the resulting graph of the bouncing ball when it fall and when it rebounce. Press / ~ to create a new page on Data & Statistics.

Choose number of bounce as the x-axis.

Choose h f as the y-axis. Describe the shape of the plot.

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130 TI-Nspire Graphing Technology: Pre-University Mathematics

Modelling of bouncing ball path

To develop the formula for h f , press: a. b b. 4: Analyze 4 c. 6: Regression 6\ d. 8: Show Exponential 8 The formula is: h f = 2 ⋅ (0.5)n where n = 1, 2, 3, …

Create another Data & Statistics page to plot the graphical representation when the ball rebounce.

What is the formula for hb? What is the first term of this sequence? Approximate the total distance the ball travelled

Distance = 20

20

n=1

n=2

∑ 2 ⋅ (0.5)n + ∑ 2 ⋅ (0.5)n Create Calculator application on another page.

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Sequences and Series 131

Press t to find appropriate summation symbol and type in the mathematical formula.

What is the approximate distance the ball travelled if it bounces 20 and 30 times?

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132 TI-Nspire Graphing Technology: Pre-University Mathematics

SUMMARY Chapter 8 discusses about how to use TI-Nspire graphing calculator in solving problems related to Sequences and Series topic. There were three activities given in this chapter. In the first activity the students used TI-Nspire to explore arithmetic and geometric sequences and series as well as the Fibonacci Sequence, in the second activity the students applied the concept of geometric sequence in AIDS cases and in the third activity the students applied the concept of geometric sequence in approximating the total distance of a bouncing ball. With the use of TI-Nspire graphing calculator, students will be able to explore, investigate and solve the problem easily and at the same time they will be able to make justifications and use their logical thinking while solving the problems.

SELF-ASSESSMENT 1. Drug A is administered once a day. The concentration of the drug in the patient’s blood-stream increases rapidly at first, but each successive dose has less effect than the preceding one. The total amount of the drug (in mg) in the bloodstream after the nth dose is given by n

∑ 55 ⋅ (0.5)

k−1

k=1



Find the amount of the drug in the bloodstream after n = 8 days.

2. If the ball in Activity 3 is dropped from a height of 8m, then 1 s is required for its first complete bounce - from the instant it first touches the ground until it next touches the ground. Each subsequent complete bounce requires as long as the preceding complete bounce. Use an infinite geometric series to estimate the time interval from the instant the ball first touches the ground until it stops bouncing.

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Sequences and Series 133

3. The midpoints of the sides of a square of side 1 are joined to form a new square. This step is repeated for each new square. Find the sum of the areas of all the squares.

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CHAPTER

9

DIFFERENTIATION

INTRODUCTION

Differentiation is one of the important branches of mathematics within calculus; it allows finding rates of change known as derivatives. Derivatives are continuously used in everyday life to aid measure how much something is changing. They are used by the government in population censuses, various types of sciences, and even in economics. Ability on how to use derivatives, when to use them and apply them in everyday life can be a crucial part of any profession, so understanding it at an early stage is always a good thing. The calculus was invented by European mathematicians, Isaac Newton and Gottfried Leibnitz. The following are some examples of the application of differentiation.

Figure 9.1 shows about finding the rate of change in the number of bacteria in a culture after 2 days.

Figure 9.1 Bacteria in a Culture

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136 TI-Nspire Graphing Technology: Pre-University Mathematics

Figure 9.2 illustrates finding average velocity of a falling object.

Figure 9.2 Falling Object

Figure 9.3 Threatened and Endangered Animals

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Figure 9.3 shows some threatened and endangered animals in the world. Determining in which time interval the number of threatened and endangered animals is increasing is one of the examples in the application of differentiation. .

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Differentiation 137

NOTES There are three primary concepts in calculus: limit, derivative, and integral. The most important for applications in science and engineering are the derivative (which describes a rate of change) and the integral (which describes the total of many small parts). But the most basic of the three concepts is the limit, because the derivative and integral are defined as certain limits. Limit

The meaning of the word “limit” in calculus is rather different from the everyday use of the word. In everyday usage, “limit” means some kind of boundary beyond which one cannot go. But in calculus, limit describes the behaviour of a function near a point. Conceptually, the limit of a function f(x) at some point x0 simply means that if your value of x is very close to the value x0, then the function f(x) stays very close to some particular value. Limit of a function can be described both graphically and numerically.

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138 TI-Nspire Graphing Technology: Pre-University Mathematics

NOTES Differentiation How do you find a rate of change, in any context, and express it mathematically? You use differentiation. What is differentiation?

Differentiation is a process of looking at the way a function changes from one point to another. Given any function, f (x), we may need to find out what it looks like when graphed. When illustrating a function on a graph the rate of change is represented by the gradient, which is called derivative (

dy ). This means that the main aim of differentiation is to dx

find the gradient at a specific point on a graph. For example in a quadratic function graph, the gradients at the points a, b and c are different. Differentiation allows us to find the exact gradient at any of these points.

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Differentiation 139

KEYSTROKE Below are a few keystrokes that will be needed in learning Differentiation. 1. To find the derivative of a function at a certain point in the Calculator page, press: a. b b. 4: Calculus 4 c. 1: Numerical Derivative at a Point 1

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140 TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 1 Learning Outcome

Question:

Approaching Limit In this activity, students will explore the basic concept of calculus in the study of limit. They will view graphically and numerically how a function behaves as it approaches to a particular value.

1. Given the following function and consider the function value as the value of x gets close to 2 graphically.

x 2 +1 x−2 2. Given the following function and consider the function value as the value of x gets close to 1 numerically. f (x) =

f (x) =

Student’s Activity Solutions: Creating the function graph by using Graph application

x2 − 2 x −1 Screen Display

Type in the function by using the Graph application page.

x 2 +1 x−2 Press ·.

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Differentiation 141

You might want to view the whole graph. Press: a. b b. 4: Window/Zoom 4 c. 4: Zoom – Out 4 Click on the centre that you wish to zoom.

Explore the graph

Trace towards the value x = 2 from values above 2 and from values below 2. Get as close as you can. You can zoom in and out if necessary. To trace the point: a. b b. 5: Trace 5 c. 1: Graph Trace 1 Explain what happen to the graph when x = 2

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142 TI-Nspire Graphing Technology: Pre-University Mathematics

Representing the function in table form.

For Question 2, we will use the Lists and Spreadsheet Application. Create new page (pg. 1.2). Label Column A as x and Column B as fx. In the formula box of Column B, type in the function and press ·.

f (x) =

x2 − 2 x −1

Change to “All Variable References” and press ‘OK’.

Exploring the function based on the value in the table.

Find the value of fx when x = 0, 1, 2.

Find the value of fx when x = 0.5, 1, 1.5.

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Differentiation 143

Find the value of fx when x = 0.9, 1, 1.9.

Activity Summary:

What conclusion can you draw at this point? Express in limit notation, based on the results from Question 1 and Question 2.

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144 TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 2

Parking Lot

Learning Outcome

In this activity, students will investigate graphically on the concepts of limit and discontinuity related to charges of a parking lot.

Question:

Sharon will be working at Sapura Crest on this coming May. As a new employee, the company could not give her a free parking at the company building basement lot. Edmono Parking Lot is the most strategic spot to park her car because of its location and the charges. Edmono Parking Lot charges RM3 for the first hour or part of an hour and RM2 for each succeeding hour or part of an hour, up to a maximum charge of RM10 for all day. 1. Sharon is going to park her car at the Edmono parking lot, but she cannot predict how long will she park there. Create a table on the charges that she will have to pay at Edmono. 2. If f (x) equals the total parking bill for x hours, form a piecewise function of the parking lot charges for 0 ≤ x ≤ 24. Student’s Activity

Solutions: Organising information in the table

Screen Display

To organize the charges at the parking lot, create a table based on the duration she parked her car at the parking lot (hour). See table below.

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Differentiation 145

Duration (hour) Charges (RM)

Table 9.1 Duration (hour) 0 ≤ h ≤ 1 1< h ≤ 2 2 < h ≤ 3 3< h ≤ 4 4 < h ≤ 24

Charges (RM) 3 5 7 9 10

Her working hour on Saturday is from 9 am till 1 pm. How much she has to spend on the parking charges? Representing information in graphical form

Use the Graph application to create a piecewise function.

In the Function panel, type in the piecewise function accordingly. Press t and choose the appropriate template for piecewise function. (Refer to the screen display) A pop out screen will appear. Change the number of equations to 5 and press ‘OK’.

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146 TI-Nspire Graphing Technology: Pre-University Mathematics

The template is ready. You may type in the function. 3, 0 ≤ 𝑥𝑥 ≤ 1 5,1 < 𝑥𝑥 ≤ 2 𝑓𝑓1 7, 2 < 𝑥𝑥 ≤ 3 9, 3 < 𝑥𝑥 ≤ 4 10, 4 < 𝑥𝑥 ≤ 24

Press · to view the graph. It will take quite some time, be patient while the graph is loading. (x-axis: duration, y-axis: total charges) Zoom in the screen to view the whole graph. What can you conclude from the parking charges graph? Investigate ne-sided limits

Find the following limits if exist, and explain your results: a. lim− f (x) x→1

b. lim+ f (x) x→1

c. lim f (x) x→1

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Differentiation 147

Imagine examining lim+ f (x) by “walking” x→1

from the right along the proper branch of the graph towards the value x=1, and examining lim− f (x) by x→1

walking from the left along the proper branch toward the value x=1. To find the lim f (x), we x→1

are going to trace the step along the graph. To change the Trace Step, press: a. b b. 5: Trace 5 c. 3: Trace Step… 3 Change the step to 0.1 and press ‘OK’.

To trace along the graph, press: a. b b. 5: Trace 5 c. 2: Trace All 2

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148 TI-Nspire Graphing Technology: Pre-University Mathematics

Try to move the white circle by using ¡ ¢ button. Bring over the white circle to the value x = 0.

Finding limit based on the graphical representation

To find the lim− f (x): x→1

Press ¢ to “walk” along the branch towards x = 1. Observe on the y - value of the coordinate as it approaches to 1.

To find the lim+ f (x): x→1

Bring over the white circle to the value x = 1.9. Press ¡ to “walk” along the branch towards x = 1. When it reaches x = 1.1, observe that it will jump to the value as the x approaches from the left. It shows there is a discontinuity at x = 1+. Explain and discuss on the results of lim f (x). x→1

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Differentiation 149

Activity Summary:

Repeat the steps above and find the following limits if exist, and explain your results:

f (x) a. lim x→2

f (x) b. lim x→3

c. lim f (x) x→4

From the result, state the points of discontinuities.

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150 TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 3

Pump Station

Learning Outcome

In this activity, students will apply differentiation concepts which related to the concept of minimum and maximum points to solve problem.

Question:

Town A is 14 kilometres from a river and town B is nine kilometres from that same river. Town B is on the same side of the river as town A. The river flows a straight path between location O and C. The distance from town A to town B is 13 kilometres. A pump station is to be built along the river to supply water to both towns. Where the pump station should be built so that the sum of the distances from the pump station to the two towns is minimum? Student’s Activity Solutions: Understand the situation by sketching the diagram

Screen Display

Based on the above problem, try to sketch a diagram which represent the situation.

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Differentiation 151

Label the measurement of the plan sketched below. O

C

B A

Predict the location of the pump station that will be built. By applying the concept of Pythagoras Theorem, find the distance between OC. Form a function that expresses the sum of the distance from town A to the pump station and the distance from town B to the pump station in terms of x. What is the method that can be used so that the sum of the distances from the pump station to the two towns are minimum? We are going to use TINspire CX to graph the function and determine the minimum distance of the pump station.

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152 TI-Nspire Graphing Technology: Pre-University Mathematics

Determine the minimum distances from the pump station to the two towns

Create a Graph page and type the function that expresses the sum of the two distances in terms of x that was formed earlier. f (x) = 14 2 + x 2 + 9 2 + (12 − x)2

Press ·. No graph seems to be visible initially. Try to zoom in and out to adjust the graph until it fit in the screen. To find the minimum value, choose: a. b b. 6: Analyse Graph 6 c. 2: Minimum 2 Choose the lower bound and upper bound by dragging the dotted line along the graph.

The coordinate represents the value of x and y when the function is minimum. From the minimum value, what is the minimum distance from town A to the pump station and the minimum distance from town B to the pump station?

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Differentiation 153

SUMMARY Chapter 9 discusses about how to use TI-Nspire graphing calculator in exploring the basic concept of limit and solving problems related to Differentiation topic. There were three activities given in this chapter. In the first activity the students used TI-Nspire to visualize graphically and numerically how a function behaves as it approaches to a particular value, in the second activity the students investigated graphically in applying the concepts of limit and discontinuity to solve parking lot charges and in the third activity students applied the differentiation concepts to solve pump station problem as one of the examples of minimum and maximum problems. With the use of TINspire graphing calculator students will be able to explore, investigate, and solve the problem easily and at the same time they will be able to make justifications and use their logical and critical thinking while solving the problems.

SELF-ASSESSMENT 1. Find the limit, if it exists, for each of the following. 2 a. lim x −1 x→−1 x +1 x b. lim 2 −1 x→0 x

c. lim x + 4 − 2 x→0 x 2. Zamrud is a leading mobile phone provider in Malaysia. From 30th April 2013, Zamrud user will enjoy special IDD rates to all mobile and fixed line numbers in Pakistan. Based on the plan, services will be charged RM0.99 for the first 20 minutes or less of a call and RM 0.07 per minute for each additional minute or fraction of it. a. Write a piecewise definition of the charge g(x) for a longdistance call lasting x minutes. b. Sketch the graph of g(x) for 0 10,000. Describe the growth of the species over time. Student’s Activity Solutions:

Screen Display

Note: One way out of the problem of unlimited growth is to modify equation to take into account the fact that any given ecological system can support only some finite number of creatures over the long term.

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Differential Equations 205

This number is called the carrying capacity of the system. We expect that when a population has reached the carrying capacity of the system, the population should neither grow nor shrink. Given that k = 0.15 and K = 10,000. We will examine the differential equation dP P    0.15 P1   dt  10,000 

Visualising the solution of differential equation graphically

Construct the slope field of the differential equation in the Graph page. Press: a. b b. 3: Graph Entry/ Edit 3 c. 7: Diff Eq 7 Similar to Activity 1, we have to temporarily replace the variable P with y and t with x. Type in the differential equation in the graphing calculator:

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206 TI-Nspire Graphing Technology: Pre-University Mathematics

dy y1    0.15 y11   dx 10 , 000  

Press ·.

Next, we will sketch the particular solution curve for the initial condition t = 0 , P = 1000 . Press: a. e b. £ Type in the initial condition, 0 and 1000.

Press ·. You will observe no changes occur to the slope field.

We have to zoom the graph to see the graph clearly. Press: a. b b. 4: Window/Zoom 4 c. 1: Window Settings 1

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Differential Equations 207

Change the settings based on below: XMin: 0 XMax: 40 Xscale: Auto YMin: 0 YMax: 14000 YScale: Auto Press ‘OK’. Observe that the red curve represents the particular solution curve for the initial condition. Find the original population of the rabbit. To find the original population, we have to find the value of P (y-axis) at t = 0 . Use the TRACE function in the graphing calculator. Press: a. b b. 5: Trace 5 c. 1: Graph Trace 1 Use left ¡ and right ¢ arrow until you reach x = 0 .

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208 TI-Nspire Graphing Technology: Pre-University Mathematics

What is the original population of the rabbit? Use your own word to describe the relationship between the population and the rate of growth. Similarly, use TRACE function once again to observe the changes in the graph as the y-value approaches 10,000. What happen as the population gets closer to 10,000? Next, observe what happen under this condition: t = 0, P = 15,000 Deactivate the TRACE function by pressing d. Change the Window Settings again: XMin: 0 XMax: 40 Xscale: Auto YMin: 0 YMax: 16000 YScale: Auto

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Differential Equations 209

Repeat the same method to insert a new condition.

Press ·.

Activity Summary:

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Describe the solution curve in terms of the original population and the rate of population as it approaches 10,000.

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210 TI-Nspire Graphing Technology: Pre-University Mathematics

ACTIVITY 3 Learning Outcome

Question:

Spread of Computer Virus In this activity, students will be able to use Euler’s method to model the spread of a computer virus.

The Internet Data Analysis reports that the Urazyno computer virus infected over 439,000 computers in 12 hours on August 4, 2014. The infection rate in computers per minute, can be modelled by the differential equation

dN = −8.727 ×10 −8 N 2 + 0.0251N + 200 dt where t = 0 corresponds to 1 p.m. on August 4. The model is accurate for t = 0 to t = 240 minutes. That is, the model represents time interval beginning at 1 p.m. on August 4 and ending at 5 p.m. on August 4. Use Euler’s method and a step size 1 to determine how many computers were affected after 2 hours. Initially, 11,500 computers were affected by the virus. Student’s Activity Solutions: Visualising the solution of differential equation graphically

Screen Display

In the Graph Application page, sketch the field of the differential equation solution. Press: a. b b. 3: Graph Entry/ Edit 3 c. 7: Diff Eq 7

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Differential Equations 211

Type in the differential equation. y1' = −8.727 ×10 −8 y12 + 0.0251y1+ 200

Set the initial condition to : x = 0, y = 11,500

Move cursor to ‘Edit Parameters’ and click it.

Use e to move to each field. Edit the Plot Step to 1.

Press ‘OK’.

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212 TI-Nspire Graphing Technology: Pre-University Mathematics

Change the settings of the screen or choose Zoom-Fit based on what suit you most. When you choose Zoom-Fit, it will appear like on the screen display. Other option, you may change in the Window Settings because the graph in the negative quadrant will be useless in this case. Try to choose your settings based on the question. Now, we are going to determine how many computers were affected after 2 hours. So, our t or value on x-axis will be 120. Use TRACE function to locate number of computer when x = 120. From the graph, how many computers were affected after 2 hours?

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Differential Equations 213

SUMMARY Differential equations have a significant ability to predict and forecast the world around us. They are used in a wide range of disciplines, from biology, economics, physics, chemistry and engineering. The activities given in this chapter is designed to help students understand how the TI-Nspire graphing calculator can be used as an aid in solving differential equation problems. There were three activities related to real life situations such as using the slope field concept to analyze the nature of the differential equation solutions, analyzing the carrying capacity of an ecological system and modeling the spread of computer virus. Students are asked to investigate, explore and create the slope field by using the graphing calculator in order to solve the problems. They are also guided to think logically as well as making reasoning, For example, by observing the created curves using graphing calculator, students are asked to describe the solution curve and predict the rate of population as it approaches certain time. These activities can enhance students’ understanding on how to use the graphing calculator to solve the problems as well as understanding the concepts in differential equations.

SELF-ASSESSMENT 1. Use Euler’s method with step size 0.1 to construct a table of approximate values for the solution of the initial-value problem y ' = x + y, y (0) = 1 . n

xn

1

0.1

2

0.2





yn

10

2. Use Euler’s method with step size 0.1 to estimate y (0.5), where y(x) is the solution of the initial-value problem y' = y + xy, y(0) = 1. Repeat this with step size 0.1.

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214 TI-Nspire Graphing Technology: Pre-University Mathematics

3. Draw a direction field for the logistic equation with k = 0.08 and carrying capacity M = 1000. What can you deduce about the solutions? 4. Write the solution of the initial-value problem

⎛ dP P ⎞ = 0.08P ⎜1− ⎟, P(0) = 100 ⎝ 1000 ⎠ dt

and use it to find the population sizes P(40) and P(80) . At what time does the population reach 900?

5. Suppose a population P (t ) satisfies

dP = 0.4P − 0.001P 2 , P(0) = 50 dt

where t is measured in years. a. What is the carrying capacity? b. What is P' (0) ? c. When will the population reach 50% of the carrying capacity?

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BIBLIOGRAPHY

Abu-Naja, M. The influence of graphing calculators on secondary school pupils’ ways of thinking about the topic “Positively and Negatively of Functions”. International Journal for Technology in Mathematics Education, 15(3), 103-117, 2008. Boers-van Oossterum, M. A.M. Understanding of variables and their uses acquired by students in traditional and computer-intensive algebra (University of Maryland College Park, 1990). Dissertation Abstracts International, 51, 1538A, 1990. Crocker, D. A., 1991. A qualitative study of interactions, concept development and problem solving in a calculus class immersed in the Computer Algebra System. (Doctoral Dissertation, Ohio state University, 1991). Dick, T., 1992. Supercalculators: Implications for calculus curriculum, instruction, and assessment. In J. T. Fey (Ed.), Calculators in Mathematics Education: 1992 Yearbook of the National Council of Teachers of Mathematics (pp. 145-157). Reston, VA: NCTM. Dunham, P. H. & Dick, T. P. Research on graphing calculators. Mathematics Teacher, 87, 440-445, 1994. Ellington, A. J., 2004. The calculator’s role in mathematics attitude. Retrieved: July 1, 2009, from Academic Exchange Quarterly, http:// www. Thefreelibrary.com/The calculator role in mathematics attitude-a0121714099 Farrell, A. M. Roles and behaviours in technology-integrated precalculus classrooms. Journal of Mathematics Behaviour, 15, 35-53, 1996. Frick, F. A. A study of the effect of the utilization of calculators and a mathematics curriculum stressing problem-solving techniques on student learning. (The University of Connecticut, 1988). Dissertation Abstract International, 49, 2104A, 1989. Goos, M. E., Galbraith, P. L., Renshaw, P., & Geiger, V. Reshaping teacher and students role in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303-320, 2000.

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216 TI-Nspire Graphing Technology: Pre-University Mathematics

Hembree, R, & Dessart, D. J. Effects of hand-held calculators in precollege mathematics education: A meta-analysis. Journal for Research in Mathematics Education, 17, 83-99, 1986. Jones, K. Graphing calculators in the teaching and learning of mathematics: A research bibliography. Micromath, 21(2), 31-33, 2005. Keller, B., & Russell, C. Effects of the TI-92 on calculus students solving symbolic problems. International Journal of Computer Algebra in Mathematics Education, 4, 77-97, 1997. Keller, B., & Russell, C. Effects of TI-92 on calculus students solving symbolic problems. International Journal of Computer Algebra in Mathematics Education, 4, 77-97, 1997. McClendon, M. A. The development of a graphics calculator study guide for calculus students. (University of Oregon, 1991). Dissertation Abstracts International, 52, 2450A, 1992. Mesa, V. M. The role of the graphing calculator in solving problems on function. (Master’s thesis, University of Georgia, 1996). Masters Abstracts International, 35, 937, 1997. Noraini Idris, Tay B. L., Nilawati, M., Goh, L. S., & Aziah, A., 2003. Pengajaran & pembelajaran matematik dengan kalkulator grafik TI-83 Plus. Selangor: Penerbitan Bangi. Noraini Idris, 2002. Developing self-confidence among Malay students. Usage of graphing calculator. In Douglas Edge & Yeap Ban Har (Eds.), Proceedings of Second East Asia Regional Conference on Mathematics Education and Ninth Southeast Asian Conference on Mathematics Education (Volume 2, Selected Papers) held on 27-31 May 2002 in Singapore (pp 447-451). Nor’ain Mohd Tajuddin, Rohani Ahmad Tarmizi, Mohd Majid Konting & Wan Zah Wan Ali. Instructional Efficiency of the Integration of Graphing Calculators in Teaching and Learning Mathematics. Journal of Learning Instruction, 2(2), 11-30, 2009. Rich, B. The effect of the use of graphing calculator on the learning of function concepts in precalculus mathematics. (University of Iowa, 1990). Dissertation Abstract International, 51835A, 1991. Rizzuti, J. M. Students’ conceptualizations of mathematical functions: The effects of a pedagogical approach involving multiple representations. (Cornell University, 1991). Dissertation Abstract International, 52, 3549A, 1992. Rosihan M. Ali, Daniel L. Seth, Zarita Zainuddin, Suraiya Kassim, Hajar Sulaiman and Hailiza Kamarul Haili. Learning and Teaching mathematics with a Graphic Calculator. Bull, Malaysian Math. Sc. Soc. (Second Series) 25, 53-82, 2002. Runde, D. C., 1997. The effects of using the TI-92 on basic college algebra students’ ability to solve word problems. Research Report (143). Manatee Community College, Florida. (ERIC Document Reproduction Service No. ED409 046).

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Bibliography 217

Sang, Sook Choi-Koh. Effect of a graphing calculator on a 10th-grade students study of trigonometry. Journal of Educational Research, 96(6), 356-369, 2003. Siskind, T. G. The effect of calculator use on mathematics achievement for rural high school students. Rural Educators. 16(2), 1-4, 1995. Slavit, D., 1994. The effect of the graphing calculators on students’ conception of function. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, April 1994. (ERIC Document Reproduction Service No. ED 374 811). Tall, D., 1996. Functions and calculus. In A. J. Bishop et al (Eds.), International handbook of mathematics education (pp 289-325). Dordrecht, Netherlands: Kluwer Academic Publishers. Tiwari, T. K. Computer graphics as an instructional aid in an introductory differential calculus course. International Electronic Journal of Mathematics Education, 2(1), 35-48, 2007. Van Streun, A., Harskamp, E., & Suhre, C. The effect of the graphic calculator on students’ solution approach: A secondary analysis. Hiroshima Journal of Mathematics Education, 8, 27-39, 2000. Wilkins, C. W. The effect of the graphing calculator on student achievement in factoring quadratic equations. (Mississippi State University, 1995). Dissertation Abstracts International, 56, 2159A, 1995.

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INDEX

Area 22, 73, 154, 165 Argand 73, 83 Arithmetic progression 111 Bar chart 175 Batteries 10, 11 Box plot 175, 182, 185 Calculator application 6, 193 Calculus 43, 135, 155 Column 45, 46 Combination 87, 174 Common difference 111 Common ratio 111 Complex number 73, 74, 82 Constant 15, 159, 204 Data & Statistics 3, 25 Derivatives 135, 198 Differentiation 135, 138, 153 Dimension 45, 73 Domain 31 Function 3, 11, 16, 31, 157

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Geometric progression 111 Geometry application 89 Gradient 138 Graph application 34, 94 Histogram 175 Imaginary 14, 73 – 75 Integral 137, 155, 159 Intergration 135, 155 – 157, 165 Inverse 47 Keys 15, 64 Limit 158, 199 Lists & Spreadsheet application 113 Magnitude 82 Matrices 46, 59 Modulus 75 Parallel 70, 87 Permutation 174, 194 Polar form 74, 82

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220 TI-Nspire Graphing Technology: Pre-University Mathematics

Quadratic 13, 14, 28 Range 31, 175, 213 Rates 135, 153 Ratio 70 Real number 31, 73 Rectangular form 74, 82 Regression 175 Row 45, 59 Scatter plot 175 Sequence 109, 110, 113 Series 13, 61, 109 Sigma 111 Statistics 173, 194 Tail 173 Trigonometry 61, 62, 70 Two Dimensional 73 Vector 59, 82, 89 Vernier DataQuest 3 x-axis 74, 175 y-axis 74

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