Mathematics N4: FET College Nated, #6

By Efetobo Emede

The book Mathematics N4 FET College Nated is an Introduction to Differential and Integral Calculus. The students are actually prosective Engineering National Diploma proffessionals.

 

The chapters of the textbook are as stated below:

 

Chapter 1  Functions and Graphs

Chapter 2  Functions and Equations

Chapter 3 Determinants (Matrix)

Chapter 4 Trigonometry

Chapter 5 Complex Numbers

Chapter 6 Differentiation

Chapter 7 Integration

 

 

The author of this textbook has been teaching and lecturing Mechanical Engineering (N1 - N6) and Electrical Engineering (N1-N6) since 2011 (FET College Nated).

 

The contactable references of the students who have studied the textbook and passed the National Exams in Mathematics N4 are available for your perusal.

• Language: English
• Publisher: Efetobo Emede
• Release date: Dec 27, 2023
• ISBN: 9798223531487

Efetobo Emede

The author of this book has a Bachelor of Mechanical Engineering from the University of Benin, Benin City, Nigeria. He has been lecturing since 2011 in the Private FET Colleges Nated in Johannesburg Cbd. He has been teaching the following subjects. Mechanical Engineering N1-N6 Electrical Engineering N1-N6

Book preview

In loving memory of my father,

Late Mr. G. M. T. Emede

Preface

The first chapter of this textbook has been written to explain in the detail graphs of various functions and their corresponding defined and undefined regions. 

It is usually very difficult for FET College students to solve equations of functions when the learner in question does not have a fundamental knowledge of the range and domain of a specific function.

The principle of Complex Numbers has been applied to solve Engineering Science and Electrical Engineering problems in Chapter 5 of this book.

The concept of differentiation has also been defined from trigonometric, direct and indirect proportionality perspectives in Chapter 6. This is a very basic concept for students to understand because they will be applying this knowledge in Applications of Differentiation, Engineering Mathematics N5.

The last chapter of this book is Integration. A practical and logical approach has been applied in teaching the limit of integration or definite integrals.

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Content

Chapter 1 Functions and Graphs  1.1 Quadratic Functions  .............................................8  1.1.1 Quadratic Functions: Domain and Range.........8  1.1.2 Classification of Quadratic Roots......................12  1.2 Continuous and Discontinuous Functions...........20  1.3 Inverse Functions....................................................22  1.3.1 Inverse of a parabola...........................................22  1.4 Lines of Symmetry ..................................................23  1.5 The Straight Line......................................................24  1.5.1 Parallel Lines..........................................................25  1.5.2  Perpendicular lines..............................................25  1.5.3 Angle of Inclination...............................................25  1.5.4 Distance between two points..............................26  1.5.5 Inverse of a Straight Line......................................31  1.6 The Circle....................................................................35 1.7 The Ellipse...................................................................42 1.8 The Hyperbolas............................................................47  1.9 Exponential Functions...............................................55  1.9.1 Exponential Functions: Domain and Range........56  1.9.2 Inverse of an Exponential Function.....................57  1.10 Logarithmic Function..............................................59  1.10.1 Logarithmic Functions: Domain and Range......63  1.10.2 Inverse of a Logarithmic Function......................64  1.11 Trigonometric Function...........................................66  1.11.1 Trigonometric Functions: Domain and Range..77  1.11.2 Inverse of a Trigonometric Function..................77  1.12 Cubic Functions...........................................................81 1.12.1 Cubic Functions  : Domain and Range...................83  1.12.2 Classification of Cubic Functions............................84    1.12.23 Inverse of a Cubic Function...................................85

Chapter 2 Equations  2.1 Linear Equations............................................................87  2.1.1 Simultaneous Linear Equations................................87  2.1.2 Word problems...........................................................88  2.2 Simultaneous Equations...............................................91  (3  Equations and 3 unknowns)  2.3  Quadratic Equations.....................................................94  2.4 Exponential and Logarithmic Equations....................104  2.4.1 Exponential Laws.......................................................105  2.4.2 Logarithmic Laws.......................................................107  2.4.3 Logarithmic Equations..............................................107  2.5  Calculations with Formulas........................................110

Chapter 3 Determinants  3.1 Introduction....................................................................115 3.2 Cramer’s Rule: Solving 2 X 2 matrix.............................120  3.3  Cramer’s rule: Solving 3 X 3 matrix.............................122

Chapter 4 Trigonometry  4.1 Introduction.....................................................................126 4.2 CAST Diagram..................................................................128 4.3 Positive and Negative Angles........................................131  4.4 Trigonometric Identities................................................131  4.5 Compound Angles..........................................................132  4.6 Reduction Formulas.......................................................136  4.7 Half Angle Identity..........................................................137  4.8 Double Angles..................................................................139 4.9 Trigonometric Equations................................................142  4.10 Trigonometric Equations: Rotations and Number of Revolutions.......................................... ..................................147  4. 11 General Formula for Trigonometric Equations........149  4.12 Simultaneous Trigonometric Equations.....................151  4.13 Solving Trigonometric Identities.................................154

Chapter 5 Complex Numbers  5.1 Introduction......................................................................157 5.2 Argand Diagram................................................................157  5.3 Rectangular and Polar Complex Numbers....................157  5.4  Quadratic Complex Number...........................................161  5.5 Addition and Subtraction of Complex Numbers...........162  5.6 Multiplication and Division of Complex Numbers  (Rectangular)...........................................................................163 5.7 Multiplication and Division of Complex Numbers  (Polar)........................................................................................165 5.8 De Moivre’s Theorem.......................................................168  5.9 Complex Number Equations............................................169  5.10 Powers of i........................................................................171 5.11 Application of Complex Numbers.................................174

Chapter 6 Differentiation  6.1 Introduction........................................................................181 6.2 Standard Formula for Differentiation.............................181  6.3 Differentiation: Increasing and Decreasing Function....182  6.4 Differentiation : Rates of Change.....................................187  6.5 Differentiation: First Principles.........................................187  6.6 Methods of Differentiation...............................................190  6.6.1 Product Rule....................................................................190 6.6.2  Quotient Rule..................................................................191  6.6.3 Function of a Function....................................................193  6.7 Differentiation of Trigonometric Functions....................198  6.8 Differentiation of Logarithmic and Exponential Functions  ....................................................................................................2016.9 Binomial Theorem.............................................................203  6.10 Applications of Differentiation......................................206  6.10.1 Maximum and Minimum.............................................206  6.10.2  Rates of Change............................................................214  6.10.3 Limits..............................................................................217

Chapter 7 Integration  7.1 Introduction.........................................................................224 7.2 Standard Formulas for Integration...................................224  7.3 Table of  Integrals................................................................225 7.4 Integration by Substitution................................................227  7.5 Applications of Integration................................................228  7.5.1 Constant of Integration...................................................228  7.5.2 Calculation of Areas.........................................................230   

Chapter 1 Functions and Graphs

1.1  Quadratic Functions  The standard form of a quadratic function is  y = ax² + bx + c  c = y intercept of the curve  b/2a = line of symmetry of the curve  The x intercepts or the roots of the curve can be calculated by:  a) Factorizing  b) Applying Quadratic Formula  c) Completing the square  1.1.1  Domain and Range of a Quadratic Function.  Consider the two curves and tables of values as shown below.

Table 1.1  y = x²

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Table 1.2  y² = x 

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Fig 1.1 The graph of y=x²

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Fig 1.2 The graph of x=y²

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For the equation, y² = x, there are two values of y for each given value of  x ( +5 and -5 for x = 25 etc). This equation is a relation.

Considering the curve y = x², there are two values of x for each given value of y (+3 and -3 for y = 9). This curve is a function. Please note that a vertical line will intersect a relation at two points while a vertical line will intersect a function at one point.

The Domain of a curve specifies all the possible values of the x coordinates.

The Range of a curve specifies all the possible values of the y coordinates.

Example 1 State the Domain and Range of the two curves

y² = xdomain = 0 ≤ x ≤ + ∞range = - ∞ ≤ y ≤ + ∞

y = x²domain = - ∞ ≤ x ≤ + ∞range = 0 ≤ y ≤ + ∞

The term 0 ≤ x means that x is greater than or equal to 0. Likewise, x ≤ + ∞ means that x is less than or equal to plus infinity.

1.1.2 Classification of Quadratic Roots  Quadratic curves are also referred to as parabolas. The general formula for a parabola is y = ax² + bx + c.

a = coefficient of x²  b = coefficient of b²  and c = y intercept  c is the value of y when x is zero.

The roots of the equation or the x intercepts of the equation can be evaluated by applying the methods as stated in section 1.1. In this section, we shall apply the quadratic formula as stated below.

Quadratic formula

The line of symmetry of a parabola is given by the formula:  x = - where a = coefficient of x²

And b = coefficient of x and c is the constant term.

Substituting this value of x = - into the original equation will give the corresponding value of y. These values of x and y are the coordinates of the turning points of the parabola.  The above methods of evaluating the roots of x intercepts of a quadratic curve is explained in examples 7 - 9 of Section 2.3.  The Fig. 1.2 shows graphs of :  a) y = x²- 4x + 4 Repeated roots  b) y = 7x²- 23x + 6 Real unequal roots  c) y = x²- 5x + 6 Real unequal roots  d) y = x²- 5x + 15  Quadratic Complex Number

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Fig 1.3 Classification of quadratic roots. y = x²- 4x + 4 Repeated roots 

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Fig 1.4  Classification of quadratic roots. y = 7x²- 23x + 6 Real unequal roots 

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Fig 1.5  Classification of quadratic roots. y = 7x²- 23x + 6 Real unequal roots 

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Fig 1.6 Classification of quadratic roots. y = x²-5x + 15. Quadratic Complex Numbers. 

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Table 1.3

The term, [b² – 4ac] ⁰.⁵for the equation, x²-5x + 15, is the square root of a negative number. This is the reason why our calculators give an undefined or error value. Furthermore, this curve does not have x intercepts or roots.  The equation, y = x²-5x + 15, is a Quadratic Complex number as stated above.

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Example 1: Given the equation y = 6x² + 11x + 3, draw the graph to scale by calculating the following:

y intercept  b) the roots  c)  the turning points

Solution:

when x = 0, y = 6(0)² + 2(0) + 3 = 3  x, y = (0,3)

Factorizing (3x +1)(2x +3) = 0  x = - ; x =  - x,   y = (- , 0 ) ; (- ,0)

X = - = - = -0.917 

y = 6(-0.917)² + 11(-0.917) + 3 = -2.042  x, y = (-0.917, -2.042)

(-0.917, -2.042)

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Fig 1.7  Graph of y = 6x²