Unit Description and General Aims

This unit introduces the student to core mathematical concepts, processes and techniques necessary to support subsequent studies in Engineering. These concepts include, but are not limited to, the properties and engineering applications of linear, quadratic, logarithmic and exponential functions. The unit commences with fundamental concepts of trigonometry, vectors and complex numbers, and goes on to cover basic differential and integral calculus. It then moves to varied subjects including sequences, series, sets, logarithms and exponentials. It rounds off with using matrices to solve equations, followed by an introduction of probability. The topics in this unit are structured in such a manner that the student will be able to solve problems related to engineering applications by using these mathematical techniques.

Learning Outcomes

1. Perform trigonometric calculations and calculations involving complex numbers and apply vector principles
   Bloom's Level 3
2. Apply vector principles
   Bloom's Level 3
3. Apply the principles of differential and integral calculus
   Bloom's Level 3
4. Evaluate concepts related to sequences, series and sets
   Bloom's Level 4
5. Comprehend and apply the basics of functions and logarithms
   Bloom's Level 3
6. Use matrices to solve mathematical problems
   Bloom's Level 3
7. Conduct basic probability analysis.
   Bloom's Level 3

Student assessment

Overall requirements: Students must achieve a result of 50% or above in the exam itself to pass the exam and must pass the exam to be able to pass the unit. An overall final unit score of 50% or above must be achieved to pass the unit once all assessment, including the exam, has been completed.

Prescribed and Recommended Readings

Suggested Textbook

• J. Bird, Higher Engineering Mathematics, 9th ed. Routledge, 2021 - ISBN: 978-0367643737

Reference Materials

• Peer-reviewed Journals
• Knovel library: http://app.knovel.com
• IDC Technologies publications
• Other material and online collections as advised during lectures

Unit Content

One topic is delivered per contact week.

Topic 1

Trigonometric Functions and Formulae

1. Trigonometric graphs
2. Period, amplitude, cycle, frequency
3. Lag and lead (phase displacement)
4. Trigonometric identities and formulae
5. Cartesian and polar coordinates

Topic 2

Vectors and Scalars

1. Vectors and Scalars
2. Vector notation
3. Resolving vectors
4. Relative velocity
5. Vector Definitions and Components
6. Operations with Vectors
7. Vector Applications
8. Laws of Sines and Cosines

Topic 3

Complex numbers

1. Imaginary numbers
2. Arithmetic of complex numbers
3. The Argand diagram and polar form of a complex number
4. The exponential form of a complex number
5. De Moivre's theorem
6. Solving equations and finding roots of complex numbers
7. Phasors

Topic 4

Differentiation 1

1. Domain and range
2. Limits and Continuity
3. Derivatives by Definition
4. Derivatives of Powers of x
5. Sketching curves
6. Gradient and Tangent to a Curve
7. Maxima, Minima and Points of Inflection
8. Mean Value Theorem
9. Functions from Derivatives

Topic 5

Differentiation 2

1. The Product Rule
2. The Chain Rule
3. The Quotient Rule
4. Parametric equations
5. Derivatives of Other Functions
6. Higher Derivatives and Graphs of Derivatives
7. Partial Differentiation

Topic 6

Integration

1. Integration process and Estimation
2. Substitution Method
3. Riemann Sums
4. The Fundamental Theorem of Calculus
5. Definite Integrals
6. Standard Integrals

Topic 7

Sequences and Series

1. Sequences and Series
2. Sums vs sequences
3. Simple series (progression)
4. Arithmetic progression
5. Geometric progression
6. Pascal’s triangle
7. Permutation and combination
8. Binomial theorem
9. Graphing progressions
10. Power series

Topic 8

Sets

1. Sets and subsets
2. Union
3. Intersection
4. Differences
5. Product
6. Algebra
7. Power set

Topic 9

Logarithms and exponentials

1. Logarithmic expression
2. Laws of logarithms
3. Natural (Naperian, hyperbolic) logarithms
4. Exponential functions
5. Graphing exponential functions
6. Logarithmic Equations
7. Application of Logarithms and exponential functions
8. Change of base

Topic 10

Matrices, determinants and multivariable functions 1

1. Introduction to matrices
2. Multiplication of matrices
3. Determinants
4. The inverse of a matrix
5. Multivariable functions
6. Multivariable calculus
7. Vector valued functions
8. Parameterization

Topic 11

Matrices, determinants and multivariable functions 2

1. Matrix form trigonometric identities
2. Cramer's rule
3. Using the inverse matrix to solve simultaneous equations
4. Gaussian elimination

Topic 12

Introduction to Probability

1. Terminology and Definitions
2. Possible outcomes
3. Independent and dependent events
4. Probability Scale
5. Theoretical Probability
6. Probability Rules
7. Factorial
8. Permutations and Combinations
9. Continuous random variables
10. Probability of occurrence and not occurring
11. Probability density function
12. Exam revision

Software/Hardware Used

Software

• Software: Desmos online calculator
• Version: N/A
• Instructions: https://www.desmos.com/scientific
• Additional resources or files: https://www.mathsisfun.com/algebra/vector-calculator.html

Hardware

• Hardware: N/A

Unit Changes Based on Student Feedback