The Free High School Science Texts: Textbooks for High School Students Mathematics

The Free High School Science Texts: Textbooks for High School Students Mathematics
FHSST Authors
The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Mathematics
Grades 10 - 12
Version 0
September 17, 2008
ii
iii
Copyright 2007 “Free High School Science Texts”
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FHSST Core Team
Mark Horner ; Samuel Halliday ; Sarah Blyth ; Rory Adams ; Spencer Wheaton
FHSST Editors
Jaynie Padayachee ; Joanne Boulle ; Diana Mulcahy ; Annette Nell ; René Toerien ; Donovan
Whitfield
FHSST Contributors
Rory Adams ; Prashant Arora ; Richard Baxter ; Dr. Sarah Blyth ; Sebastian Bodenstein ;
Graeme Broster ; Richard Case ; Brett Cocks ; Tim Crombie ; Dr. Anne Dabrowski ; Laura
Daniels ; Sean Dobbs ; Fernando Durrell ; Dr. Dan Dwyer ; Frans van Eeden ; Giovanni
Franzoni ; Ingrid von Glehn ; Tamara von Glehn ; Lindsay Glesener ; Dr. Vanessa Godfrey ; Dr.
Johan Gonzalez ; Hemant Gopal ; Umeshree Govender ; Heather Gray ; Lynn Greeff ; Dr. Tom
Gutierrez ; Brooke Haag ; Kate Hadley ; Dr. Sam Halliday ; Asheena Hanuman ; Neil Hart ;
Nicholas Hatcher ; Dr. Mark Horner ; Mfandaidza Hove ; Robert Hovden ; Jennifer Hsieh ;
Clare Johnson ; Luke Jordan ; Tana Joseph ; Dr. Jennifer Klay ; Lara Kruger ; Sihle Kubheka ;
Andrew Kubik ; Dr. Marco van Leeuwen ; Dr. Anton Machacek ; Dr. Komal Maheshwari ;
Kosma von Maltitz ; Nicole Masureik ; John Mathew ; JoEllen McBride ; Nikolai Meures ;
Riana Meyer ; Jenny Miller ; Abdul Mirza ; Asogan Moodaly ; Jothi Moodley ; Nolene Naidu ;
Tyrone Negus ; Thomas O’Donnell ; Dr. Markus Oldenburg ; Dr. Jaynie Padayachee ;
Nicolette Pekeur ; Sirika Pillay ; Jacques Plaut ; Andrea Prinsloo ; Joseph Raimondo ; Sanya
Rajani ; Prof. Sergey Rakityansky ; Alastair Ramlakan ; Razvan Remsing ; Max Richter ; Sean
Riddle ; Evan Robinson ; Dr. Andrew Rose ; Bianca Ruddy ; Katie Russell ; Duncan Scott ;
Helen Seals ; Ian Sherratt ; Roger Sieloff ; Bradley Smith ; Greg Solomon ; Mike Stringer ;
Shen Tian ; Robert Torregrosa ; Jimmy Tseng ; Helen Waugh ; Dr. Dawn Webber ; Michelle
Wen ; Dr. Alexander Wetzler ; Dr. Spencer Wheaton ; Vivian White ; Dr. Gerald Wigger ;
Harry Wiggins ; Wendy Williams ; Julie Wilson ; Andrew Wood ; Emma Wormauld ; Sahal
Yacoob ; Jean Youssef
Contributors and editors have made a sincere effort to produce an accurate and useful resource.
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v
vi
Contents
I
Basics
1
1 Introduction to Book
1.1
II
3
The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grade 10
3
5
2 Review of Past Work
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
What is a number? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Letters and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.5
Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.6
Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.7
Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.8
Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9
2.8.1
What is a negative number? . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8.2
Working with Negative Numbers . . . . . . . . . . . . . . . . . . . . . . 11
2.8.3
Living Without the Number Line . . . . . . . . . . . . . . . . . . . . . . 12
Rearranging Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.10 Fractions and Decimal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.11 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.14 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.15 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Rational Numbers - Grade 10
23
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
The Big Picture of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vii
CONTENTS
CONTENTS
3.4
Forms of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5
Converting Terminating Decimals into Rational Numbers . . . . . . . . . . . . . 25
3.6
Converting Repeating Decimals into Rational Numbers . . . . . . . . . . . . . . 25
3.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Exponentials - Grade 10
29
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3
Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1
Exponential Law 1: a0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2
Exponential Law 2: am × an = am+n . . . . . . . . . . . . . . . . . . . 30
4.3.3
Exponential Law 3: a−n =
4.3.4
4.4
m
1
an , a
n
6= 0 . . . . . . . . . . . . . . . . . . . . 31
Exponential Law 4: a ÷ a = am−n . . . . . . . . . . . . . . . . . . . 32
4.3.5
Exponential Law 5: (ab)n = an bn . . . . . . . . . . . . . . . . . . . . . 32
4.3.6
Exponential Law 6: (am )n = amn . . . . . . . . . . . . . . . . . . . . . 33
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Estimating Surds - Grade 10
37
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2
Drawing Surds on the Number Line (Optional) . . . . . . . . . . . . . . . . . . 38
5.3
End of Chapter Excercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Irrational Numbers and Rounding Off - Grade 10
41
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2
Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3
Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.4
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Number Patterns - Grade 10
7.1
45
Common Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.1.1
Special Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2
Make your own Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3.1
7.4
Patterns and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Finance - Grade 10
53
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2
Foreign Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3
8.2.1
How much is R1 really worth? . . . . . . . . . . . . . . . . . . . . . . . 53
8.2.2
Cross Currency Exchange Rates
8.2.3
Enrichment: Fluctuating exchange rates . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . 56
Being Interested in Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
viii
CONTENTS
8.4
Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.4.1
8.5
8.6
8.7
CONTENTS
Other Applications of the Simple Interest Formula . . . . . . . . . . . . . 61
Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.5.1
Fractions add up to the Whole . . . . . . . . . . . . . . . . . . . . . . . 65
8.5.2
The Power of Compound Interest . . . . . . . . . . . . . . . . . . . . . . 65
8.5.3
Other Applications of Compound Growth . . . . . . . . . . . . . . . . . 67
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.2
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Products and Factors - Grade 10
71
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2
Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.1
Parts of an Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.2
Product of Two Binomials . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.3
Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.3
More Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4
Factorising a Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.5
Factorisation by Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.6
Simplification of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.7
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10 Equations and Inequalities - Grade 10
83
10.1 Strategy for Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.4 Exponential Equations of the form ka(x+p) = m . . . . . . . . . . . . . . . . . . 93
10.4.1 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.5 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.6 Linear Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.1 Finding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.2 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.3 Solution by Substitution
. . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.7 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.7.3 Application of Mathematical Modelling
. . . . . . . . . . . . . . . . . . 104
10.7.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.8 Introduction to Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . 107
10.9 Functions and Graphs in the Real-World . . . . . . . . . . . . . . . . . . . . . . 107
10.10Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
CONTENTS
CONTENTS
10.10.1 Variables and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.10.2 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.3 The Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.4 Drawing Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.10.5 Notation used for Functions
. . . . . . . . . . . . . . . . . . . . . . . . 110
10.11Characteristics of Functions - All Grades . . . . . . . . . . . . . . . . . . . . . . 112
10.11.1 Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . 112
10.11.2 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.3 Intercepts with the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.4 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.5 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.6 Lines of Symmetry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.7 Intervals on which the Function Increases/Decreases . . . . . . . . . . . 114
10.11.8 Discrete or Continuous Nature of the Graph . . . . . . . . . . . . . . . . 114
10.12Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.12.1 Functions of the form y = ax + q . . . . . . . . . . . . . . . . . . . . . 116
10.12.2 Functions of the Form y = ax2 + q . . . . . . . . . . . . . . . . . . . . . 120
10.12.3 Functions of the Form y =
a
x
+ q . . . . . . . . . . . . . . . . . . . . . . 125
10.12.4 Functions of the Form y = ab(x) + q . . . . . . . . . . . . . . . . . . . . 129
10.13End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11 Average Gradient - Grade 10 Extension
135
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.2 Straight-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.3 Parabolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
12 Geometry Basics
139
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.2 Points and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
12.3.1 Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.2 Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.3 Special Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.3.4 Parallel Lines intersected by Transversal Lines . . . . . . . . . . . . . . . 143
12.4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12.4.3 Other polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.4.4 Extra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.5.1 Challenge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
x
CONTENTS
13 Geometry - Grade 10
CONTENTS
161
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2 Right Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2.1 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
13.2.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
13.3 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.3.1 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.4 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.2 Distance between Two Points . . . . . . . . . . . . . . . . . . . . . . . . 172
13.4.3 Calculation of the Gradient of a Line . . . . . . . . . . . . . . . . . . . . 173
13.4.4 Midpoint of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.1 Translation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.2 Reflection of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14 Trigonometry - Grade 10
189
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
14.2 Where Trigonometry is Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.3 Similarity of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.4 Definition of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 191
14.5 Simple Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . 195
14.5.1 Height and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14.5.2 Maps and Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.6 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.1 Graph of sin θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.2 Functions of the form y = a sin(x) + q . . . . . . . . . . . . . . . . . . . 200
14.6.3 Graph of cos θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.6.4 Functions of the form y = a cos(x) + q
. . . . . . . . . . . . . . . . . . 202
14.6.5 Comparison of Graphs of sin θ and cos θ . . . . . . . . . . . . . . . . . . 204
14.6.6 Graph of tan θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
14.6.7 Functions of the form y = a tan(x) + q . . . . . . . . . . . . . . . . . . 205
14.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
15 Statistics - Grade 10
211
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2 Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.1 Data and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.2 Methods of Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 212
15.2.3 Samples and Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3 Example Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
xi
CONTENTS
CONTENTS
15.3.1 Data Set 1: Tossing a Coin . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.2 Data Set 2: Casting a die . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.3 Data Set 3: Mass of a Loaf of Bread . . . . . . . . . . . . . . . . . . . . 214
15.3.4 Data Set 4: Global Temperature . . . . . . . . . . . . . . . . . . . . . . 214
15.3.5 Data Set 5: Price of Petrol . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4 Grouping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4.1 Exercises - Grouping Data
. . . . . . . . . . . . . . . . . . . . . . . . . 216
15.5 Graphical Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.1 Bar and Compound Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.2 Histograms and Frequency Polygons . . . . . . . . . . . . . . . . . . . . 217
15.5.3 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.5.4 Line and Broken Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . 220
15.5.5 Exercises - Graphical Representation of Data
. . . . . . . . . . . . . . . 221
15.6 Summarising Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.1 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15.6.3 Exercises - Summarising Data
. . . . . . . . . . . . . . . . . . . . . . . 228
15.7 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
15.7.1 Exercises - Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . 230
15.8 Summary of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
16 Probability - Grade 10
235
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2 Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2.1 Sample Space of a Random Experiment . . . . . . . . . . . . . . . . . . 235
16.3 Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
16.3.1 Classical Theory of Probability . . . . . . . . . . . . . . . . . . . . . . . 239
16.4 Relative Frequency vs. Probability . . . . . . . . . . . . . . . . . . . . . . . . . 240
16.5 Project Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.6 Probability Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.7 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
16.8 Complementary Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16.9 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
III
Grade 11
17 Exponents - Grade 11
249
251
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
17.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
√
m
17.2.1 Exponential Law 7: a n = n am . . . . . . . . . . . . . . . . . . . . . . 251
17.3 Exponentials in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
17.4 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
xii
CONTENTS
CONTENTS
18 Surds - Grade 11
18.1 Surd Calculations . . . . . . . . . .
√
√ √
18.1.1 Surd Law 1: n a n b = n ab
√
p
n
a
18.1.2 Surd Law 2: n ab = √
. .
n
b
√
m
18.1.3 Surd Law 3: n am = a n . .
255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.4 Like and Unlike Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.5 Simplest Surd form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
18.1.6 Rationalising Denominators . . . . . . . . . . . . . . . . . . . . . . . . . 258
18.2 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
19 Error Margins - Grade 11
261
20 Quadratic Sequences - Grade 11
265
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.2 What is a quadratic sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.3 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
21 Finance - Grade 11
271
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.2 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.3 Simple Depreciation (it really is simple!) . . . . . . . . . . . . . . . . . . . . . . 271
21.4 Compound Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
21.5 Present Values or Future Values of an Investment or Loan . . . . . . . . . . . . 276
21.5.1 Now or Later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
21.6 Finding i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
21.7 Finding n - Trial and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
21.8 Nominal and Effective Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 280
21.8.1 The General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
21.8.2 De-coding the Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 282
21.9 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
21.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
22 Solving Quadratic Equations - Grade 11
287
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.2 Solution by Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.3 Solution by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 290
22.4 Solution by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 293
22.5 Finding an equation when you know its roots . . . . . . . . . . . . . . . . . . . 296
22.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
xiii
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CONTENTS
23 Solving Quadratic Inequalities - Grade 11
301
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.2 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
24 Solving Simultaneous Equations - Grade 11
307
24.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
24.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
25 Mathematical Models - Grade 11
313
25.1 Real-World Applications: Mathematical Models . . . . . . . . . . . . . . . . . . 313
25.2 End of Chatpter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
26 Quadratic Functions and Graphs - Grade 11
321
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
26.2 Functions of the Form y = a(x + p)2 + q
. . . . . . . . . . . . . . . . . . . . . 321
26.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
26.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
26.2.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
26.2.4 Axes of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
26.2.5 Sketching Graphs of the Form f (x) = a(x + p)2 + q . . . . . . . . . . . 325
26.2.6 Writing an equation of a shifted parabola . . . . . . . . . . . . . . . . . 327
26.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
27 Hyperbolic Functions and Graphs - Grade 11
329
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
27.2 Functions of the Form y =
a
x+p
+q
. . . . . . . . . . . . . . . . . . . . . . . . 329
27.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
27.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
27.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
27.2.4 Sketching Graphs of the Form f (x) =
a
x+p
+ q . . . . . . . . . . . . . . 333
27.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
28 Exponential Functions and Graphs - Grade 11
335
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
28.2 Functions of the Form y = ab(x+p) + q . . . . . . . . . . . . . . . . . . . . . . . 335
28.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
28.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
28.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
28.2.4 Sketching Graphs of the Form f (x) = ab(x+p) + q . . . . . . . . . . . . . 338
28.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
29 Gradient at a Point - Grade 11
341
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.2 Average Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
xiv
CONTENTS
30 Linear Programming - Grade 11
CONTENTS
345
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.1 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.4 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.5 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.3 Example of a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.4 Method of Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5 Skills you will need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.1 Writing Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.2 Writing the Objective Function . . . . . . . . . . . . . . . . . . . . . . . 348
30.5.3 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
30.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
31 Geometry - Grade 11
357
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
31.2 Right Pyramids, Right Cones and Spheres . . . . . . . . . . . . . . . . . . . . . 357
31.3 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
31.4 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.4.1 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.5 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
31.5.1 Equation of a Line between Two Points . . . . . . . . . . . . . . . . . . 368
31.5.2 Equation of a Line through One Point and Parallel or Perpendicular to
Another Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.5.3 Inclination of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.1 Rotation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.2 Enlargement of a Polygon 1 . . . . . . . . . . . . . . . . . . . . . . . . . 376
32 Trigonometry - Grade 11
381
32.1 History of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2.1 Functions of the form y = sin(kθ) . . . . . . . . . . . . . . . . . . . . . 381
32.2.2 Functions of the form y = cos(kθ) . . . . . . . . . . . . . . . . . . . . . 383
32.2.3 Functions of the form y = tan(kθ) . . . . . . . . . . . . . . . . . . . . . 384
32.2.4 Functions of the form y = sin(θ + p) . . . . . . . . . . . . . . . . . . . . 385
32.2.5 Functions of the form y = cos(θ + p) . . . . . . . . . . . . . . . . . . . 386
32.2.6 Functions of the form y = tan(θ + p) . . . . . . . . . . . . . . . . . . . 387
32.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
32.3.1 Deriving Values of Trigonometric Functions for 30◦ , 45◦ and 60◦ . . . . . 389
32.3.2 Alternate Definition for tan θ . . . . . . . . . . . . . . . . . . . . . . . . 391
xv
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CONTENTS
32.3.3 A Trigonometric Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 392
32.3.4 Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
32.4 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
32.4.3 Solution using CAST diagrams . . . . . . . . . . . . . . . . . . . . . . . 403
32.4.4 General Solution Using Periodicity . . . . . . . . . . . . . . . . . . . . . 405
32.4.5 Linear Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 406
32.4.6 Quadratic and Higher Order Trigonometric Equations . . . . . . . . . . . 406
32.4.7 More Complex Trigonometric Equations . . . . . . . . . . . . . . . . . . 407
32.5 Sine and Cosine Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.1 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.2 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
32.5.3 The Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
32.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
33 Statistics - Grade 11
419
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2 Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.2 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
33.2.3 Interpretation and Application . . . . . . . . . . . . . . . . . . . . . . . 423
33.2.4 Relationship between Standard Deviation and the Mean . . . . . . . . . . 424
33.3 Graphical Representation of Measures of Central Tendency and Dispersion . . . . 424
33.3.1 Five Number Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
33.3.2 Box and Whisker Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 425
33.3.3 Cumulative Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
33.4 Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.1 Symmetric and Skewed Data . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.2 Relationship of the Mean, Median, and Mode . . . . . . . . . . . . . . . 428
33.5 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
33.6 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
33.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
34 Independent and Dependent Events - Grade 11
437
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2.1 Identification of Independent and Dependent Events
. . . . . . . . . . . 438
34.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
IV
Grade 12
35 Logarithms - Grade 12
443
445
35.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
xvi
CONTENTS
CONTENTS
35.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
35.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.4 Logarithm Law 1: loga 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.5 Logarithm Law 2: loga (a) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
35.6 Logarithm Law 3: loga (x · y) = loga (x) + loga (y) . . . . . . . . . . . . . . . . . 448
35.7 Logarithm Law 4: loga xy = loga (x) − loga (y) . . . . . . . . . . . . . . . . . 449
35.8 Logarithm Law 5: loga (xb ) = b loga (x) . . . . . . . . . . . . . . . . . . . . . . . 450
√
35.9 Logarithm Law 6: loga ( b x) = logab(x) . . . . . . . . . . . . . . . . . . . . . . . 450
35.10Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
35.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
35.11Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 454
35.11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
35.12End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
36 Sequences and Series - Grade 12
457
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2.1 General Equation for the nth -term of an Arithmetic Sequence . . . . . . 458
36.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.2 General Equation for the nth -term of a Geometric Sequence . . . . . . . 461
36.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
36.4 Recursive Formulae for Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 462
36.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
36.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 466
36.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
36.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
36.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
36.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
36.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
36.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
37 Finance - Grade 12
477
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
37.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 477
37.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
37.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
xvii
CONTENTS
CONTENTS
37.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 479
37.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 484
37.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 485
37.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.2 Exercises - Investments and Loans . . . . . . . . . . . . . . . . . . . . . 489
37.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 489
37.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
37.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
38 Factorising Cubic Polynomials - Grade 12
493
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.2 The Factor Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.3 Factorisation of Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 494
38.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 498
38.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
39 Functions and Graphs - Grade 12
501
39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 503
39.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.3 Inverse Function of y = ax2
. . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.5 Inverse Function of y = ax . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
40 Differential Calculus - Grade 12
509
40.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 509
40.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
40.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 510
40.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 511
40.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
40.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 516
40.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 519
xviii
CONTENTS
CONTENTS
40.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
40.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 522
40.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 523
40.5.1 Finding Equations of Tangents to Curves
. . . . . . . . . . . . . . . . . 523
40.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
40.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 529
40.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 530
40.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 534
40.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
41 Linear Programming - Grade 12
539
41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.3 Linear Programming and the Feasible Region . . . . . . . . . . . . . . . . . . . 540
41.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
42 Geometry - Grade 12
549
42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
42.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 550
42.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 569
42.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
42.4.1 Rotation of a Point about an angle θ . . . . . . . . . . . . . . . . . . . . 571
42.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.4.3 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
43 Trigonometry - Grade 12
577
43.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.1 Derivation of sin(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.2 Derivation of sin(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.3 Derivation of cos(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.4 Derivation of cos(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.5 Derivation of sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.6 Derivation of cos 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 580
43.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 582
43.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 582
xix
CONTENTS
CONTENTS
43.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 584
43.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.3 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
43.3.4 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
43.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
44 Statistics - Grade 12
591
44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
44.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 594
44.4.1 The Method of Least Squares
. . . . . . . . . . . . . . . . . . . . . . . 596
44.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
44.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
44.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
45 Combinations and Permutations - Grade 12
603
45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 604
45.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 606
45.6 Permutations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
45.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
45.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
45.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
V
Exercises
613
46 General Exercises
615
47 Exercises - Not covered in Syllabus
617
A GNU Free Documentation License
619
xx
Chapter 40
Differential Calculus - Grade 12
40.1
Why do I have to learn this stuff?
Calculus is one of the central branches of mathematics and was developed from algebra and
geometry. Calculus is built on the concept of limits, which will be discussed in this chapter.
Calculus consists of two complementary ideas: differential calculus and integral calculus. Only
differential calculus will be studied. Differential calculus is concerned with the instantaneous rate
of change of quantities with respect to other quantities, or more precisely, the local behaviour
of functions. This can be illustrated by the slope of a function’s graph. Examples of typical
differential calculus problems include: finding the acceleration and velocity of a free-falling body
at a particular moment and finding the optimal number of units a company should produce to
maximize its profit.
Calculus is fundamentally different from the mathematics that you have studied previously. Calculus is more dynamic and less static. It is concerned with change and motion. It deals with
quantities that approach other quantities. For that reason it may be useful to have an overview
of the subject before beginning its intensive study.
Calculus is a tool to understand many natural phenomena like how the wind blows, how water
flows, how light travels, how sound travels and how the planets move. However, other human
activities such as economics are also made easier with calculus.
In this section we give a glimpse of some of the main ideas of calculus by showing how limits
arise when we attempt to solve a variety of problems.
Extension: Integral Calculus
Integral calculus is concerned with the accumulation of quantities, such as areas
under a curve, linear distance traveled, or volume displaced. Differential and integral
calculus act inversely to each other. Examples of typical integral calculus problems
include finding areas and volumes, finding the amount of water pumped by a pump
with a set power input but varying conditions of pumping losses and pressure and
finding the amount of rain that fell in a certain area if the rain fell at a specific rate.
teresting Both Isaac Newton (4 January 1643 – 31 March 1727) and Gottfried Liebnitz
Interesting
Fact
Fact
(1 July 1646 – 14 November 1716 (Hanover, Germany)) are credited with the
‘invention’ of calculus. Newton was the first to apply calculus to general physics,
while Liebnitz developed most of the notation that is still in use today.
When Newton and Leibniz first published their results, there was some controversy over whether Leibniz’s work was independent of Newton’s. While Newton
derived his results years before Leibniz, it was only some time after Leibniz published in 1684 that Newton published. Later, Newton would claim that Leibniz
509
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
got the idea from Newton’s notes on the subject; however examination of the
papers of Leibniz and Newton show they arrived at their results independently,
with Leibniz starting first with integration and Newton with differentiation. This
controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of
mathematical analysis. Today, both Newton and Leibniz are given credit for
independently developing calculus. It is Leibniz, however, who is credited with
giving the new discipline the name it is known by today: ”calculus”. Newton’s
name for it was ”the science of fluxions”.
40.2
Limits
40.2.1
A Tale of Achilles and the Tortoise
teresting Zeno (circa 490 BC - circa 430 BC) was a pre-Socratic Greek philosopher of
Interesting
Fact
Fact
southern Italy who is famous for his paradoxes.
One of Zeno’s paradoxes can be summarised by:
Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles
is very fast. So, the tortoise asks Achilles for a head-start. Achilles agrees to give
the tortoise a 1 000 m head start. Does Achilles overtake the tortoise?
We know how to solve this problem. We start by writing:
xA
=
vA t
(40.1)
xt
=
1000 m + vt t
(40.2)
where
xA
vA
t
xt
vt
distance covered by Achilles
Achilles’ speed
time taken by Achilles to overtake tortoise
distance covered by the tortoise
the tortoise’s speed
If we assume that Achilles runs at 2 m·s−1 and the tortoise runs at 0,25 m·s−1 then Achilles
will overtake the tortoise when both of them have covered the same distance. This means that
510
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.2
Achilles overtakes the tortoise at a time calculated as:
xA
=
xt
(40.3)
vA t
−1
=
=
(40.4)
(40.5)
(2 m · s−1 − 0,25 m · s−1 )t
=
1000 + vt t
1000 m + (0,25 m · s−1 )t
t
=
(2 m · s
)t
=
=
=
=
1000 m
1000 m
1 43 m · s−1
1000 m
7
−1
4 m·s
(4)(1000)
s
7
4000
s
7
3
571 s
7
(40.6)
(40.7)
(40.8)
(40.9)
(40.10)
(40.11)
However, Zeno (the Greek philosopher who thought up this problem) looked at it as follows:
Achilles takes
1000
= 500 s
t=
2
to travel the 1 000 m head start that the tortoise had. However, in this 500 s, the tortoise has
travelled a further
x = (500)(0,25) = 125 m.
Achilles then takes another
125
= 62,5 s
2
to travel the 125 m. In this 62,5 s, the tortoise travels a further
t=
x = (62,5)(0,25) = 15,625 m.
Zeno saw that Achilles would always get closer but wouldn’t actually overtake the tortoise.
40.2.2
Sequences, Series and Functions
So what does Zeno, Achilles and the tortoise have to do with calculus?
Well, in Grades 10 and 11 you studied sequences. For the sequence
1 2 3 4
0, , , , , . . .
2 3 4 5
which is defined by the expression
1
n
the terms get closer to 1 as n gets larger. Similarly, for the sequence
an = 1 −
1 1 1 1
1, , , , , . . .
2 3 4 5
which is defined by the expression
1
n
the terms get closer to 0 as n gets larger. We have also seen that the infinite geometric series
has a finite total. The infinite geometric series is
an =
S∞ =
∞
X
i=1
a1 .ri−1 =
a1
1−r
for
−1
where a1 is the first term of the series and r is the common ratio.
511
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
We see that there are some functions where the value of the function gets close to or approaches
a certain value.
Similarly, for the function:
x2 + 4x − 12
x+6
The numerator of the function can be factorised as:
y=
y=
(x + 6)(x − 2)
.
x+6
Then we can cancel the x − 6 from numerator and denominator and we are left with:
y = x − 2.
However, we are only able to cancel the x + 6 term if x 6= −6. If x = −6, then the denominator
becomes 0 and the function is not defined. This means that the domain of the function does
not include x = −6. But we can examine what happens to the values for y as x gets close to -6.
These values are listed in Table 40.1 which shows that as x gets closer to -6, y gets close to 8.
Table 40.1: Values for the function y =
x
y=
-9
-8
-7
-6.5
-6.4
-6.3
-6.2
-6.1
-6.09
-6.08
-6.01
-5.9
-5.8
-5.7
-5.6
-5.5
-5
-4
-3
(x + 6)(x − 2)
as x gets close to -6.
x+6
(x+6)(x−2)
x+6
-11
-10
-9
-8.5
-8.4
-8.3
-8.2
-8.1
-8.09
-8.08
-8.01
-7.9
-7.8
-7.7
-7.6
-7.5
-7
-6
-5
The graph of this function is shown in Figure 40.1. The graph is a straight line with slope 1 and
intercept -2, but with a missing section at x = −6.
Extension: Continuity
We say that a function is continuous if there are no values of the independent variable
for which the function is undefined.
40.2.3
Limits
We can now introduce a new notation. For the function y =
lim
x→−6
(x + 6)(x − 2)
, we can write:
x+6
(x + 6)(x − 2)
= −8.
x+6
512
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
4
3
2
1
1 2 3 4
−9 −8 −7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
bc
Figure 40.1: Graph of y =
This is read: the limit of
(x+6)(x−2)
x+6
(x+6)(x−2)
.
x+6
as x tends to -6 is 8.
Activity :: Investigation : Limits
If f (x) = x + 1, determine:
f(-0.1)
f(-0.05)
f(-0.04)
f(-0.03)
f(-0.02)
f(-0.01)
f(0.00)
f(0.01)
f(0.02)
f(0.03)
f(0.04)
f(0.05)
f(0.1)
What do you notice about the value of f (x) as x gets close to 0.
Worked Example 172: Limits Notation
Question: Summarise the following situation by using limit notation: As x gets
close to 1, the value of the function
y =x+2
gets close to 3.
513
40.2
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Answer
This is written as:
lim x + 2 = 3
x→1
in limit notation.
We can also have the situation where a function has a different value depending on whether x
approaches from the left or the right. An example of this is shown in Figure 40.2.
4
3
2
1
1
−7 −6 −5 −4 −3 −2 −1
−1
2
3
4
5
6
7
−2
−3
−4
Figure 40.2: Graph of y = x1 .
As x → 0 from the left, y = x1 approaches −∞. As x → 0 from the right, y =
+∞. This is written in limits notation as:
lim
x→0−
1
x
approaches
1
= −∞
x
for x approaching zero from the left and
lim
x→0+
1
=∞
x
for x approaching zero from the right. You can calculate the limit of many different functions
using a set method.
Method:
Limits If you are required to calculate a limit like limx→a then:
1. Simplify the expression completely.
2. If it is possible, cancel all common terms.
3. Let x approach the a.
Worked Example 173: Limits
514
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Question: Determine
lim 10
x→1
Answer
Step 1 : Simplify the expression
There is nothing to simplify.
Step 2 : Cancel all common terms
There are no terms to cancel.
Step 3 : Let x → 1 and write final answer
lim 10 = 10
x→1
Worked Example 174: Limits
Question: Determine
lim x
x→2
Answer
Step 1 : Simplify the expression
There is nothing to simplify.
Step 2 : Cancel all common terms
There are no terms to cancel.
Step 3 : Let x → 2 and write final answer
lim x = 2
x→2
Worked Example 175: Limits
Question: Determine
x2 − 100
x→10 x − 10
lim
Answer
Step 1 : Simplify the expression
The numerator can be factorised.
(x + 10)(x − 10)
x2 − 100
=
x − 10
x − 10
Step 2 : Cancel all common terms
x − 10 can be cancelled from the numerator and denominator.
(x + 10)(x − 10)
= x + 10
x − 10
Step 3 : Let x → 1 and write final answer
x2 − 100
= 20
x→10 x − 10
lim
515
40.2
40.2
40.2.4
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Average Gradient and Gradient at a Point
In Grade 10 you learnt about average gradients on a curve. The average gradient between any
two points on a curve is given by the gradient of the straight line that passes through both
points. In Grade 11 you were introduced to the idea of a gradient at a single point on a curve.
We saw that this was the gradient of the tangent to the curve at the given point, but we did
not learn how to determine the gradient of the tangent.
Now let us consider the problem of trying to find the gradient of a tangent t to a curve with
equation y = f (x) at a given point P .
tangent
P
b
f (x)
We know how to calculate the average gradient between two points on a curve, but we need two
points. The problem now is that we only have one point, namely P . To get around the problem
we first consider a secant to the curve that passes through point P and another point on the
curve Q. We can now find the average gradient of the curve between points P and Q.
secant
P
b
f (a)
f (a − h)
b
Q
f (x)
a
a−h
If the x-coordinate of P is a, then the y-coordinate is f (a). Similarly, if the x-coordinate of Q
is a − h, then the y-coordinate is f (a − h). If we choose a as x2 and a − h as x1 , then:
y1 = f (a − h)
y2 = f (a).
We can now calculate the average gradient as:
y2 − y1
x2 − x1
=
=
f (a) − f (a − h)
a − (a − h)
f (a) − f (a − h)
h
(40.12)
(40.13)
Now imagine that Q moves along the curve toward P . The secant line approaches the tangent
line as its limiting position. This means that the average gradient of the secant approaches the
gradient of the tangent to the curve at P . In (40.13) we see that as point Q approaches point
P , h gets closer to 0. When h = 0, points P and Q are equal. We can now use our knowledge
of limits to write this as:
gradient at P = lim
h→0
f (a) − f (a − h)
.
h
(40.14)
and we say that the gradient at point P is the limit of the average gradient as Q approaches P
along the curve.
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CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.2
Activity :: Investigation : Limits
The gradient at a point x on a curve defined by f (x) can also be written as:
lim
h→0
f (x + h) − f (x)
h
(40.15)
Show that this is equivalent to (40.14).
Worked Example 176: Limits
Question: For the function f (x) = 2x2 − 5x, determine the gradient of the tangent
to the curve at the point x = 2.
Answer
Step 1 : Calculating the gradient at a point
We know that the gradient at a point x is given by:
lim
h→0
f (x + h) − f (x)
h
In our case x = 2. It is simpler to substitute x = 2 at the end of the calculation.
Step 2 : Write f (x + h) and simplify
f (x + h) =
=
=
2(x + h)2 − 5(x + h)
2(x2 + 2xh + h2 ) − 5x − 5h
2x2 + 4xh + 2h2 − 5x − 5h
Step 3 : Calculate limit
f (x + h) − f (x)
h→0
h
lim
=
=
=
=
=
=
2x2 + 4xh + 2h2 − 5x − 5h − (2x2 − 5x)
h
2x2 + 4xh + 2h2 − 5x − 5h − 2x2 + 5x
lim
h→0
h
4xh + 2h2 − 5h
lim
h→0
h
h(4x + 2h − 5)
lim
h→0
h
lim 4x + 2h − 5
h→0
4x − 5
Step 4 : Calculate gradient at x = 2
4x − 5 = 4(2) − 5 = 3
Step 5 : Write the final answer
The gradient of the tangent to the curve f (x) = 2x2 − 5x at x = 2 is 3.
517
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Worked Example 177: Limits
Question: For the function f (x) = 5x2 − 4x + 1, determine the gradient of the
tangent to curve at the point x = a.
Answer
Step 1 : Calculating the gradient at a point
We know that the gradient at a point x is given by:
f (x + h) − f (x)
h→0
h
lim
In our case x = a. It is simpler to substitute x = a at the end of the calculation.
Step 2 : Write f (x + h) and simplify
5(x + h)2 − 4(x + h) + 1
f (x + h) =
5(x2 + 2xh + h2 ) − 4x − 4h + 1
5x2 + 10xh + 5h2 − 4x − 4h + 1
=
=
Step 3 : Calculate limit
f (x + h) − f (x)
h→0
h
lim
=
=
=
=
=
=
5x2 + 10xh + 5h2 − 4x − 4h + 1 − (5x2 − 4x + 1)
h
5x2 + 10xh + 5h2 − 4x − 4h + 1 − 5x2 + 4x − 1
lim
h→0
h
2
10xh + 5h − 4h
lim
h→0
h
h(10x + 5h − 4)
lim
h→0
h
lim 10x + 5h − 4
h→0
10x − 4
Step 4 : Calculate gradient at x = a
10x − 4 = 10a − 5
Step 5 : Write the final answer
The gradient of the tangent to the curve f (x) = 5x2 − 4x + 1 at x = 1 is 10a − 5.
Exercise: Limits
Determine the following
1.
x2 − 9
x→3 x + 3
lim
2.
lim
x→3
3.
x+3
x2 + 3x
3x2 − 4x
x→2 3 − x
lim
518
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
4.
40.3
x2 − x − 12
x→4
x−4
lim
5.
lim 3x +
x→2
40.3
1
3x
Differentiation from First Principles
The tangent problem has given rise to the branch of calculus called differential calculus and
the equation:
f (x + h) − f (x)
lim
h→0
h
defines the derivative of the function f (x). Using (40.15) to calculate the derivative is called
finding the derivative from first principles.
Definition: Derivative
The derivative of a function f (x) is written as f ′ (x) and is defined by:
f (x + h) − f (x)
h→0
h
f ′ (x) = lim
(40.16)
There are a few different notations used to refer to derivatives. If we use the traditional notation
y = f (x) to indicate that the dependent variable is y and the independent variable is x, then
some common alternative notations for the derivative are as follows:
f ′ (x) = y ′ =
dy
df
d
=
=
f (x) = Df (x) = Dx f (x)
dx
dx
dx
d
are called differential operators because they indicate the operation of
The symbols D and dx
differentiation, which is the process of calculating a derivative. It is very important that you
learn to identify these different ways of denoting the derivative, and that you are consistent in
your usage of them when answering questions.
dy
is a limit and
Important: Though we choose to use a fractional form of representation, dx
dy
dy
is not a fraction, i.e. dx does not mean dy ÷ dx. dx means y differentiated with respect to
dp
d
means p differentiated with respect to x. The ‘ dx
’ is the “operator”, operating
x. Thus, dx
on some function of x.
Worked Example 178: Derivatives - First Principles
Question: Calculate the derivative of g(x) = x − 1 from first principles.
Answer
Step 1 : Calculating the gradient at a point
We know that the gradient at a point x is given by:
g(x + h) − g(x)
h→0
h
g ′ (x) = lim
Step 2 : Write g(x + h) and simplify
519
40.3
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
g(x + h) = x + h − 1
Step 3 : Calculate limit
g ′ (x)
=
=
=
=
=
g(x + h) − g(x)
h
x + h − 1 − (x − 1)
lim
h→0
h
x+h−1−x+1
lim
h→0
h
h
lim
h→0 h
lim 1
lim
h→0
h→0
= 1
Step 4 : Write the final answer
The derivative g ′ (x) of g(x) = x − 1 is 1.
Worked Example 179: Derivatives - First Principles
Question: Calculate the derivative of h(x) = x2 − 1 from first principles.
Answer
Step 1 : Calculating the gradient at a point
We know that the gradient at a point x is given by:
g(x + h) − g(x)
h→0
h
g ′ (x) = lim
Step 2 : Write g(x + h) and simplify
g(x + h) = x + h − 1
Step 3 : Calculate limit
g ′ (x)
=
=
=
=
=
g(x + h) − g(x)
h
x + h − 1 − (x − 1)
lim
h→0
h
x+h−1−x+1
lim
h→0
h
h
lim
h→0 h
lim 1
lim
h→0
h→0
= 1
Step 4 : Write the final answer
The derivative g ′ (x) of g(x) = x − 1 is 1.
520
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.4
Exercise: Derivatives
1. Given g(x) = −x2
g(x + h) − g(x)
A determine
h
B hence, determine
lim
h→0
g(x + h) − g(x)
h
C explain the meaning of your answer in (b).
2. Find the derivative of f (x) = −2x2 + 3x using first principles.
1
3. Determine the derivative of f (x) =
using first principles.
x−2
4. Determine f ′ (3) from first principles if f (x) = −5x2 .
5. If h(x) = 4x2 − 4x, determine h′ (x) using first principles.
40.4
Rules of Differentiation
Calculating the derivative of a function from first principles is very long, and it is easy to make
mistakes. Fortunately, there are rules which make calculating the derivative simple.
Activity :: Investigation : Rules of Differentiation
From first principles, determine the derivatives of the following:
1. f (x) = b
2. f (x) = x
3. f (x) = x2
4. f (x) = x3
5. f (x) = 1/x
You should have found the following:
f (x)
b
x
x2
x3
1/x = x−1
f ′ (x)
0
1
2x
3x2
−x−2
If we examine these results we see that there is a pattern, which can be summarised by:
d
(xn ) = nxn−1
dx
(40.17)
There are two other rules which make differentiation simpler. For any two functions f (x) and
g(x):
d
[f (x) ± g(x)] = f ′ (x) ± g ′ (x)
(40.18)
dx
521
40.4
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
This means that we differentiate each term separately.
The final rule applies to a function f (x) that is multiplied by a constant k.
d
[k.f (x)] = kf ′ (x)
dx
(40.19)
Worked Example 180: Rules of Differentiation
Question: Determine the derivative of x − 1 using the rules of differentiation.
Answer
Step 1 : Identify the rules that will be needed
We will apply two rules of differentiation:
d
(xn ) = nxn−1
dx
and
d
d
d
[f (x) − g(x)] =
[f (x)] −
[g(x)]
dx
dx
dx
Step 2 : Determine the derivative
In our case f (x) = x and g(x) = 1.
f ′ (x) = 1
and
g ′ (x) = 0
Step 3 : Write the final answer
The derivative of x − 1 is 1 which is the same result as was obtained earlier, from
first principles.
40.4.1
Summary of Differentiation Rules
d
dx b
=0
d
n
dx (x )
= nxn−1
d
dx (kf )
df
= k dx
d
dx (f
+ g) =
Exercise: Rules of Differentiation
x2 − 5x + 6
.
x−2
√
2. Find f ′ (y) if f (y) = y.
1. Find f ′ (x) if f (x) =
3. Find f ′ (z) if f (z) = (z − 1)(z + 1).
√
x3 + 2 x − 3
dy
4. Determine dx if y =
.
x
522
df
dx
+
dg
dx
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
5. Determine the derivative of y =
40.5
√
x3 +
40.5
1
.
3x3
Applying Differentiation to Draw Graphs
Thus far we have learnt about how to differentiate various functions, but I am sure that you are
beginning to ask, What is the point of learning about derivatives? Well, we know one important
fact about a derivative: it is a gradient. So, any problems involving the calculations of gradients
or rates of change can use derivatives. One simple application is to draw graphs of functions by
firstly determine the gradients of straight lines and secondly to determine the turning points of
the graph.
40.5.1
Finding Equations of Tangents to Curves
In section 40.2.4 we saw that finding the gradient of a tangent to a curve is the same as finding
the slope of the same curve at the point of the tangent. We also saw that the gradient of a
function at a point is just its derivative.
Since we have the gradient of the tangent and the point on the curve through which the tangent
passes, we can find the equation of the tangent.
Worked Example 181: Finding the Equation of a Tangent to a Curve
Question: Find the equation of the tangent to the curve y = x2 at the point (1,1)
and draw both functions.
Answer
Step 1 : Determine what is required
We are required to determine the equation of the tangent to the curve defined by
y = x2 at the point (1,1). The tangent is a straight line and we can find the equation
by using derivatives to find the gradient of the straight line. Then we will have the
gradient and one point on the line, so we can find the equation using:
y − y1 = m(x − x1 )
from grade 11 Coordinate Geometry.
Step 2 : Differentiate the function
Using our rules of differentiation we get:
y ′ = 2x
Step 3 : Find the gradient at the point (1,1)
In order to determine the gradient at the point (1,1), we substitute the x-value into
the equation for the derivative. So, y ′ at x = 1 is:
2(1) = 2
Step 4 : Find equation of tangent
y − y1
y−1
y
y
=
=
m(x − x1 )
(2)(x − 1)
= 2x − 2 + 1
= 2x − 1
523
40.5
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Step 5 : Write the final answer
The equation of the tangent to the curve defined by y = x2 at the point (1,1) is
y = 2x − 1.
Step 6 : Sketch both functions
y = x2
4
3
2
1
−4
−3
−2
−1
b (1,1)
1
2
3
4
−1
−2
y = 2x − 1
−3
−4
40.5.2
Curve Sketching
Differentiation can be used to sketch the graphs of functions, by helping determine the turning
points. We know that if a graph is increasing on an interval and reaches a turning point, then
the graph will start decreasing after the turning point. The turning point is also known as a
stationary point because the gradient at a turning point is 0. We can then use this information
to calculate turning points, by calculating the points at which the derivative of a function is 0.
Important: If x = a is a turning point of f (x), then:
f ′ (a) = 0
This means that the derivative is 0 at a turning point.
Take the graph of y = x2 as an example. We know that the graph of this function has a turning
point at (0,0), but we can use the derivative of the function:
y ′ = 2x
and set it equal to 0 to find the x-value for which the graph has a turning point.
2x = 0
x = 0
We then substitute this into the equation of the graph (i.e. y = x2 ) to determine the y-coordinate
of the turning point:
f (0) = (0)2 = 0
This corresponds to the point that we have previously calculated.
524
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.5
Worked Example 182: Calculation of Turning Points
Question: Calculate the turning points of the graph of the function
f (x) = 2x3 − 9x2 + 12x − 15
.
Answer
Step 1 : Determine the derivative of f (x)
Using the rules of differentiation we get:
f ′ (x) = 6x2 − 18x + 12
Step 2 : Set f ′ (x) = 0 and calculate x-coordinate of turning point
6x2 − 18x + 12 =
2
x − 3x + 2 =
(x − 2)(x − 1) =
0
0
0
Therefore, the turning points are at x = 2 and x = 1.
Step 3 : Substitute x-coordinate of turning point into f (x) to determine
y-coordinates
f (2) = 2(2)3 − 9(2)2 + 12(2) − 15
= 16 − 36 + 24 − 15
= −11
f (1) = 2(1)3 − 9(1)2 + 12(1) − 15
= 2 − 9 + 12 − 15
= −10
Step 4 : Write final answer
The turning points of the graph of f (x) = 2x3 − 9x2 + 12x − 15 are (2,-11) and
(1,-10).
We are now ready to sketch graphs of functions.
Method:
Sketching GraphsSuppose we are given that f (x) = ax3 + bx2 + cx + d, then there are five steps
to be followed to sketch the graph of the function:
1. If a > 0, then the graph is increasing from left to right, and has a maximum and then a
minimum. As x increases, so does f (x). If a < 0, then the graph decreasing is from left
to right, and has first a minimum and then a maximum. f (x) decreases as x increases.
2. Determine the value of the y-intercept by substituting x = 0 into f (x)
3. Determine the x-intercepts by factorising ax3 + bx2 + cx + d = 0 and solving for x. First
try to eliminate constant common factors, and to group like terms together so that the
expression is expressed as economically as possible. Use the factor theorem if necessary.
525
40.5
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
4. Find the turning points of the function by working out the derivative
zero, and solving for x.
df
dx
and setting it to
5. Determine the y-coordinates of the turning points by substituting the x values obtained in
the previous step, into the expression for f (x).
6. Draw a neat sketch.
Worked Example 183: Sketching Graphs
Question: Draw the graph of g(x) = x2 − x + 2
Answer
Step 1 : Determine the y-intercept
y-intercept is obtained by setting x = 0.
g(0) = (0)2 − 0 + 2 = 2
Step 2 : Determine the x-intercepts
The x-intercepts are found by setting g(x) = 0.
g(x)
=
0
=
x2 − x + 2
x2 − x + 2
which does not have real roots. Therefore, the graph of g(x) does not have any
x-intercepts.
Step 3 : Find the turning points of the function
dg
Work out the derivative dx
and set it to zero to for the x coordinate of the turning
point.
dg
= 2x − 1
dx
dg
=
dx
2x − 1 =
2x =
x =
0
0
1
1
2
Step 4 : Determine the y-coordinates of the turning points by substituting
the x values obtained in the previous step, into the expression for f (x).
y coordinate of turning point is given by calculating g( 21 ).
1
g( ) =
2
=
=
1
1
( )2 − ( ) + 2
2
2
1 1
− +2
4 2
7
4
The turning point is at ( 12 , 47 )
Step 5 : Draw a neat sketch
526
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.5
y
9
8
7
6
5
4
3
2
1
b
b
(0.5,1.75)
x
−3 −2 −1
1
2
3
4
Worked Example 184: Sketching Graphs
Question: Sketch the graph of g(x) = −x3 + 6x2 − 9x + 4.
Answer
Step 1 : Calculate the turning points
Find the turning points by setting g ′ (x) = 0.
If we use the rules of differentiation we get
g ′ (x) = −3x2 + 12x − 9
g ′ (x)
−3x2 + 12x − 9
=
=
0
0
x2 − 4x + 3 =
(x − 3)(x − 1) =
0
0
The x-coordinates of the turning points are: x = 1 and x = 3.
The y-coordinates of the turning points are calculated as:
g(x)
=
g(1) =
=
=
g(x)
=
g(3) =
=
=
−x3 + 6x2 − 9x + 4
−(1)3 + 6(1)2 − 9(1) + 4
−1 + 6 − 9 + 4
0
−x3 + 6x2 − 9x + 4
−(3)3 + 6(3)2 − 9(3) + 4
−27 + 54 − 27 + 4
4
Therefore the turning points are: (1,0) and (3,4).
527
40.5
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Step 2 : Determine the y-intercepts
We find the y-intercepts by finding the value for g(0).
= −x3 + 6x2 − 9x + 4
g(x)
yint = g(0) = −(0)3 + 6(0)2 − 9(0) + 4
= 4
Step 3 : Determine the x-intercepts
We find the x-intercepts by finding the points for which the function g(x) = 0.
g(x) = −x3 + 6x2 − 9x + 4
Use the factor theorem to confirm that (x − 1) is a factor. If g(1) = 0, then (x − 1)
is a factor.
g(x)
−x3 + 6x2 − 9x + 4
=
−(1)3 + 6(1)2 − 9(1) + 4
−1 + 6 − 9 + 4
g(1) =
=
=
0
Therefore, (x − 1) is a factor.
If we divide g(x) by (x − 1) we are left with:
−x2 + 5x − 4
This has factors
−(x − 4)(x − 1)
Therefore:
g(x) = −(x − 1)(x − 1)(x − 4)
The x-intercepts are: xint = 1, 4
Step 4 : Draw a neat sketch
y
9
8
7
6
5
4
(3,4)
b
b
3
2
1
(1,0)
(4,0)
x
b
−1
−1
1
b
2
528
3
4
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.5
Exercise: Sketching Graphs
1. Given f (x) = x3 + x2 − 5x + 3:
A Show that (x − 1) is a factor of f (x) and hence fatorise f (x) fully.
B Find the coordinates of the intercepts with the axes and the turning points
and sketch the graph
2. Sketch the graph of f (x) = x3 − 4x2 − 11x+ 30 showing all the relative turning
points and intercepts with the axes.
3.
A Sketch the graph of f (x) = x3 − 9x2 + 24x − 20, showing all intercepts
with the axes and turning points.
B Find the equation of the tangent to f (x) at x = 4.
40.5.3
Local minimum, Local maximum and Point of Inflextion
dy
) is zero at a point, the gradient of the tangent at that point is zero. It
If the derivative ( dx
means that a turning point occurs as seen in the previous example.
y
9
8
7
6
5
4
(3;4)
b
b
3
2
1
(1;0)
(4;0)
x
b
−1
−1
1
b
2
3
4
From the drawing the point (1;0) represents a local minimum and the point (3;4) the local
maximum.
A graph has a horizontal point of inflexion where the derivative is zero but the sign of the sign
of the gradient does not change. That means the graph always increases or always decreases.
529
40.6
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
y
b
(3;1)
x
From this drawing, the point (3;1) is a horizontal point of inflexion, because the sign of the
derivative stays positive.
40.6
Using Differential Calculus to Solve Problems
We have seen that differential calculus can be used to determine the stationary points of functions, in order to sketch their graphs. However, determining stationary points also lends itself to
the solution of problems that require some variable to be optimised.
For example, if fuel used by a car is defined by:
f (v) =
3 2
v − 6v + 245
80
(40.20)
where v is the travelling speed, what is the most economical speed (that means the speed that
uses the least fuel)?
If we draw the graph of this function we find that the graph has a minimum. The speed at the
minimum would then give the most economical speed.
fuel consumption (l)
60
50
40
30
20
10
0
0
10
20
30
40
50
60 70 80 90 100 110 120 130 140
speed (km·hr−1 )
We have seen that the coordinates of the turning point can be calculated by differentiating the
function and finding the x-coordinate (speed in the case of the example) for which the derivative
is 0.
Differentiating (40.20), we get:
3
v−6
40
If we set f ′ (v) = 0 we can calculate the speed that corresponds to the turning point.
530
f ′ (v) =
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.6
3
v−6
40
3
v−6
=
40
6 × 40
=
3
= 80
f ′ (v)
=
0
v
This means that the most economical speed is 80 km·hr−1 .
Worked Example 185: Optimisation Problems
Question: The sum of two positive numbers is 10. One of the numbers is multiplied
by the square of the other. If each number is greater than 0, find the numbers that
make this product a maximum.
Answer
Step 1 : Examine the problem and formulate the equations that are required
Let the two numbers be a and b. Then we have:
a + b = 10
(40.21)
We are required to minimise the product of a and b. Call the product P . Then:
P =a·b
(40.22)
We can solve for b from (40.21) to get:
b = 10 − a
(40.23)
Substitute this into (40.22) to write P in terms of a only.
P = a(10 − a) = 10a − a2
Step 2 : Differentiate
The derivative of (40.24) is:
P ′ (a) = 10 − 2a
Step 3 : Find the stationary point
Set P ′ (a) = 0 to find the value of a which makes P a maximum.
P ′ (a) =
0 =
2a =
a
=
a
=
10 − 2a
10 − 2a
10
10
2
5
Substitute into (40.27) to solve for the width.
b
= 10 − a
= 10 − 5
= 5
Step 4 : Write the final answer
The product is maximised if a and b are both equal to 5.
531
(40.24)
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Worked Example 186: Optimisation Problems
Question: Michael wants to start a vegetable garden, which he decides to fence off
in the shape of a rectangle from the rest of the garden. Michael only has 160 m of
fencing, so he decides to use a wall as one border of the vegetable garden. Calculate
the width and length of the garden that corresponds to largest possible area that
Michael can fence off.
wall
garden
length, l
40.6
width, w
Answer
Step 1 : Examine the problem and formulate the equations that are required
The important pieces of information given are related to the area and modified
perimeter of the garden. We know that the area of the garden is:
A= w·l
(40.25)
We are also told that the fence covers only 3 sides and the three sides should add
up to 160 m. This can be written as:
160 = w + l + l
(40.26)
However, we can use (40.26) to write w in terms of l:
w = 160 − 2l
(40.27)
Substitute (40.27) into (40.25) to get:
A = (160 − 2l)l = 160l − 2l2
(40.28)
Step 2 : Differentiate
Since we are interested in maximising the area, we differentiate (40.28) to get:
A′ (l) = 160 − 4l
Step 3 : Find the stationary point
To find the stationary point, we set A′ (l) = 0 and solve for the value of l that
maximises the area.
A′ (l) =
0 =
∴ 4l
=
l
=
l
=
160 − 4l
160 − 4l
160
160
4
40 m
Substitute into (40.27) to solve for the width.
w
=
=
=
=
160 − 2l
160 − 2(40)
160 − 80
80 m
532
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Step 4 : Write the final answer
A width of 80 m and a length of 40 m will yield the maximal area fenced off.
Exercise: Solving Optimisation Problems using Differential Calculus
1. The sum of two positive numbers is 20. One of the numbers is multiplied by
the square of the other. Find the numbers that make this products a maximum.
2. A wooden block is made as shown in the diagram. The ends are right-angled
triangles having sides 3x, 4x and 5x. The length of the block is y. The total
surface area of the block is 3 600 cm2 .
3x
4x
y
300 − x2
.
x
B Find the value of x for which the block will have a maximum volume.
(Volume = area of base × height.)
A Show that y =
3. The diagram shows the plan for a verandah which is to be built on the corner
of a cottage. A railing ABCDE is to be constructed around the four edges of
the verandah.
y
C
D
x
verandah
F
B
A
E
cottage
If AB = DE = x and BC = CD = y, and the length of the railing must be 30
metres, find the values of x and y for which the verandah will have a maximum
area.
533
40.6
40.6
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.6.1
Rate of Change problems
Two concepts were discussed in this chapter: Average rate of change =
f (b)−f (a)
b−a
and Instan-
(x)
limh→0 f (x+h)−f
.
h
taneous rate of change =
When we mention rate of change, the latter
is implied. Instantaneous rate of change is the derivative. When Average rate of change is
required, it will be specifically refer to as average rate of change.
Velocity is one of the most common forms of rate of change. Again, average velocity = average
rate of change and instantaneous velocity = instantaneous rate of change = derivative.
Velocity refers to the increase of distance(s) for a corresponding increade in time (t). The
notation commonly used for this is:
v(t) =
ds
= s′ (t)
dt
Acceleration is the change in velocity for a corersponding increase in time. Therefore, acceleration
is the derivative of velocity
a(t) = v ′ (t)
This implies that acceleration is the second derivative of the distance(s).
Worked Example 187: Rate of Change
Question: The height (in metres) of a golf ball that is hit into the air after t seconds,
is given by h(t) = 20t = 5t2 . Determine
1. the average velocity of the ball during the first two seconds
2. the velocity of the ball after 1,5 seconds
3. when the velocity is zero
4. the velocity at which the ball hits the ground
5. the acceleration of the ball
Answer
Step 1 : Average velocity
Ave velocity
=
=
=
=
h(2) − h(0)
2−0
[20(2) − 5(2)2 ] − [20(0) − 5(0)2 ]
2
40 − 20
2
10 ms−1
Step 2 : Instantaneous Velocity
v(t)
=
=
dh
dt
20 − 10t
Velocity after 1,5 seconds:
v(1,5) = 20 − 10(1,5)
= 5 ms−1
Step 3 : Zero velocity
534
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
v(t)
=
0
20 − 10t =
10t =
t
40.7
0
20
=
2
Therefore the velocity is zero after 2 seconds
Step 4 : Ground velocity
The ball hits the ground when h(t) = 0
20t − 5t2
=
0
=
0
t=0
or
5t(4 − t)
t=4
The ball hits the ground after 4 seconds. The velocity after 4 seconds will be:
v(4)
= h′ (4)
= 20 − 10(4)
= 20 ms−1
The ball hits the gound at a speed of 20ms−1
Step 5 : Acceleration
a
= v ′ (t)
= −10 ms−1
40.7
End of Chapter Exercises
1. Determine f ′ (x) from first principles if:
f (x) = x2 − 6x
f (x) = 2x − x2
2. Given:
f (x) = −x2 + 3x, find f ′ (x) using first principles.
3. Determine
dx
dy
if:
A
y = (2x)2 −
B
1
3x
√
2 x−5
√
y=
x
4. Given: f (x) = x3 − 3x2 + 4
A Calculate f (−1), and hence solve the equationf (x) = 0
B Determine f ′ (x)
C Sketch the graph of f neatly and clearly, showing the co-ordinates of the turning
points as well as the intercepts on both axes.
535
40.7
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
D Determine the co-ordinates of the points on the graph of f where the gradient is 9.
5. Given: f (x) = 2x3 − 5x2 − 4x + 3. The x-intercepts of f are: (-1;0) ( 21 ;0) and (3;0).
A Determine the co-ordinates of the turning points of f .
B Draw a neat sketch graph of f . Clearly indicate the co-ordinates of the intercepts
with the axes, as well as the co-ordinates of the turning points.
C For which values of k will the equation f (x) = k , have exactly two real roots?
D Determine the equation of the tangent to the graph of f (x) = 2x3 − 5x2 − 4x + 3
at the point where x = 1.
6.
A Sketch the graph of f (x) = x3 − 9x2 + 24x − 20, showing all intercepts with the
axes and turning points.
B Find the equation of the tangent to f (x) at x = 4.
7. Calculate:
1 − x3
x→1 1 − x
lim
8. Given:
f (x) = 2x2 − x
A Use the definition of the derivative to calculate f ′ (x).
B Hence, calculate the co-ordinates of the point at which the gradient of the tangent
to the graph of f is 7.
√
9. If xy − 5 = x3 , determine dx
dy
10. Given: g(x) = (x−2 + x2 )2 . Calculate g ′ (2).
11. Given:
f (x) = 2x − 3
A Find:
B Solve:
f −1 (x)
f −1 (x) = 3f ′ (x)
12. Find f ′ (x) for each of the following:
√
5
x3
+ 10
A f (x) =
3
(2x2 − 5)(3x + 2)
B f (x) =
x2
13. Determine the minimum value of the sum of a positive number and its reciprocal.
14. If the displacement s (in metres) of a particle at time t (in seconds) is governed by the
equation s = 21 t3 − 2t, find its acceleration after 2 seconds. (Acceleration is the rate of
change of velocity, and velocity is the rate of change of displacement.)
15.
A After doing some research, a transport company has determined that the rate at
which petrol is consumed by one of its large carriers, travelling at an average speed
of x km per hour, is given by:
P (x) =
55
x
+
2x 200
litres per kilometre
i. Assume that the petrol costs R4,00 per litre and the driver earns R18,00 per
hour (travelling time). Now deduce that the total cost, C, in Rands, for a 2 000
km trip is given by:
256000
+ 40x
C(x) =
x
ii. Hence determine the average speed to be maintained to effect a minimum cost
for a 2 000 km trip.
536
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.7
B During an experiment the temperature T (in degrees Celsius), varies with time t (in
hours), according to the formula:
1
T (t) = 30 + 4t − t2
2
t ∈ [1; 10]
i. Determine an expression for the rate of change of temperature with time.
ii. During which time interval was the temperature dropping?
16. The depth, d, of water in a kettle t minutes after it starts to boil, is given by d =
86 − 81 t − 41 t3 , where d is measured in millimetres.
A How many millimetres of water are there in the kettle just before it starts to boil?
B As the water boils, the level in the kettle drops. Find the rate at which the water
level is decreasing when t = 2 minutes.
C How many minutes after the kettle starts boiling will the water level be dropping at
a rate of 12 81 mm/minute?
537
40.7
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
538
Appendix A
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Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled “History” in the various original
documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled
“Endorsements”.
622
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under
this License, and replace the individual copies of this License in the various documents with a
single copy that is included in the collection, provided that you follow the rules of this License
for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under
this License, provided you insert a copy of this License into the extracted document, and follow
this License in all other respects regarding verbatim copying of that document.
AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate and independent documents
or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the
copyright resulting from the compilation is not used to limit the legal rights of the compilation’s
users beyond what the individual works permit. When the Document is included an aggregate,
this License does not apply to the other works in the aggregate which are not themselves derivative
works of the Document.
If the Cover Text requirement of section A is applicable to these copies of the Document, then if
the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be
placed on covers that bracket the Document within the aggregate, or the electronic equivalent
of covers if the Document is in electronic form. Otherwise they must appear on printed covers
that bracket the whole aggregate.
TRANSLATION
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section A. Replacing Invariant Sections with translations requires
special permission from their copyright holders, but you may include translations of some or
all Invariant Sections in addition to the original versions of these Invariant Sections. You may
include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and
the original versions of those notices and disclaimers. In case of a disagreement between the
translation and the original version of this License or a notice or disclaimer, the original version
will prevail.
If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the
requirement (section A) to Preserve its Title (section A) will typically require changing the actual
title.
TERMINATION
You may not copy, modify, sub-license, or distribute the Document except as expressly provided
for under this License. Any other attempt to copy, modify, sub-license or distribute the Document
is void, and will automatically terminate your rights under this License. However, parties who
have received copies, or rights, from you under this License will not have their licenses terminated
so long as such parties remain in full compliance.
FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation
License from time to time. Such new versions will be similar in spirit to the present version, but
may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
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APPENDIX A. GNU FREE DOCUMENTATION LICENSE
Each version of the License is given a distinguishing version number. If the Document specifies
that a particular numbered version of this License “or any later version” applies to it, you have the
option of following the terms and conditions either of that specified version or of any later version
that has been published (not as a draft) by the Free Software Foundation. If the Document does
not specify a version number of this License, you may choose any version ever published (not as
a draft) by the Free Software Foundation.
ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of the License in the document
and put the following copyright and license notices just after the title page:
c YEAR YOUR NAME. Permission is granted to copy, distribute and/or
Copyright modify this document under the terms of the GNU Free Documentation License,
Version 1.2 or any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the
license is included in the section entitled “GNU Free Documentation License”.
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.”
line with this:
with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST,
and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other combination of the three,
merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these
examples in parallel under your choice of free software license, such as the GNU General Public
License, to permit their use in free software.
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