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Course Code           : PMAT 21272

Title                         : Infinite Series

Pre-requisites         : PMAT 12253

 Learning Outcomes:

At the end of this course, the student should be able to

1. define the meaning of convergence of a real sequence of real numbers
2. use definitions to discuss the behavior of a given sequence
3. describe the nature of the convergence of infinite series and conditions under what differentiation and
integration can be performed
4. demonstrate knowledge on power series representation of a series.
5. use applications of Taylor polynomial.

 Course Contents:

Sequences: Limits and limit theorems for sequences, Monotone sequences and Cauchy sequences, Bounded
sequences, Monotone sequence theorem, Subsequences, Bolzano-Weierstrass theorem

Series: Convergence of Infinite Series, Geometric series, Harmonic Series, the Integral Test and Estimates of Sums,
The Comparison Tests and Estimates of Sums, Alternating Series and estimates of Sums, Absolute and conditional
Convergence, Ratio Test and Root Test

Power Series: Representation of Functions as Power Series, Differentiation and Integration of Power Series, Taylor
and Maclaurin Series, Binomial Series, Applications of Taylor Polynomials

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart. J. (2020) Calculus Early Transcendentals, Cengage Learning.
2. Knopp, K. (1956). Infinite sequences and series. Courier Corporation.
3. Knopp, K. (1990). Theory and application of infinite series. Courier Corporation.
4. Hirschman, I.I. (2014). Infinite series. Courier Corporation.
5. Bonar, D.D. & Khoury Jr., M.J. (2018). Real infinite series (Vol. 56). American Mathematical Soc.
6. Bromwich, T.J.I.A. (2005). An introduction to the theory of infinite series (Vol. 335). American
Mathematical Soc.

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Course code   : PMAT 12203

Title               : Introduction to Calculus

 Learning Outcomes:

At the end of this course, the student should be able to

1. define the basic concepts of limits
2. evaluate the limits of functions
3. find rates of change
4. recognize the continuity of a function
5. explain basic rules of differentiation
6. identify indeterminate forms and methods of integration
7. define an ordinary differential equation
8. solve applications related to the ordinary differential equations appear in biology

Course Contents:

Limits and Derivatives: Intuitive definition of a limit, one sided limits, Calculating limits using limit laws,
Derivatives and Rates of change, Derivative as a function
Continuity: Continuity at a point, continuity on an interval, Intermediate value theorem
Differentiation Rules: Derivatives of polynomials and Exponential functions, Product and quotient rules,
Derivatives of Trigonometric functions, Chain rule, Implicit differentiation, Derivatives of Logarithmic functions
Applications of Differentiation: Maximum and Minimum values, Indeterminate forms and l’Hospital’s rule,
Taylor's formula, Newton's method
Techniques of Integration: Basic integrals, Integration by parts, Trigonometric integrals, Trigonometric
substitution, Integration of rational functions by partial fractions, Improper integrals
Introduction to Differential Equations: Separable equations, Linear equations, Applications related to biology

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on the tutorials, tests, and end of course examination.

Recommended reading: 

1. Ayres, Jr .F. & Mendelson, E., (6^th edition, 2012). Schaum's Outline of Calculus, McGraw-Hill.
2. Arora, S. & Malik, S.C., (5^th edition, 2017). Mathematical Analysis, New Age International.
3. Stewart, J., Clegg, D.K., & Watson, S. (11th Ed., 2020). Calculus: early transcendentals. Cengage
   Learning.

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Course code  :  PMAT 21263

Title               :  Linear Algebra

Pre-requisites :  PMAT 11232

 Learning outcomes:

Upon successful completion of the course students should be able to

1. demonstrate understanding of the concepts of vector space, subspace linear independence, span and basis
2. determine eigenvalues and eigenvectors and solve eigenvalue problems
3. describe algebraic and geometric multiplicities of eigenvalues and linearly independent eigenvectors
4. apply principles of matrix algebra to linear transformations
5. demonstrate an understanding of inner products and associated norms

 Course Contents:

Vector Spaces: Vector Spaces, Subspaces, Spanning Sets and Linear Independence, Basis and Dimension,
Extension Theorem, Coordinates, Change of Basis and Transition Matrix, Similarity, Dimensional Theorem.

Linear transformations: Linear Transformation, Kernel and Range of Linear Transformation, Rank and Nullity
Theorem, Isomorphisms, Matrix Representation of Linear Transformation, Applications of Linear Transformation.

Eigenvalues and Eigenvectors: Characteristic Polynomial, Eigenvalues and Eigenvectors, Eigen Spaces,
Diagonalization, Inner Product Spaces, Gram-Schmidt Orthogonalization Process, Orthogonal Complement,
Orthogonal Projections, Cayley-Hamilton Theorem, Minimum Polynomial of Matrices of Order Three

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended readings:

1. Larson, R. & Falvo, D.C. (2016). Elementary Linear Algebra, Brooks Cole.
2. Andrilli, S. & Hecker, D. (2016). Elementary Linear Algebra, Elsevier Science.
3. DeFranza, J. & Gagliardi, D. (2015). Introduction to Linear Algebra with Applications, Waveland Press.
4. Lay, D.C., Lay, S.R. & McDonald, J.J. (5th Ed., 2015). Linear Algebra and Its Applications, Pearson

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Course code  :  PMAT 12212

Title               :  Mathematics for Computing II

Pre-requisites :  PMAT 11212

Learning Outcomes:

Upon successful completion of the course unit, the student should be able to:

1. demonstrate knowledge in matrices
2. solve systems of linear equations
3. compute eigenvectors and eigenvalues
4. solve real world problems using counting techniques
5. demonstrate knowledge in basic graph theory.

Course Contents:

Matrices - Matrix notations, Algebra of matrices, Inverse of a matrix, Elementary row operations and
calculation of inverse matrix, Determinant of a matrix, Minor and cofactor, Properties of determinants, Adjoint
of a matrix and calculation of inverse matrix.
Systems of Linear Equations - Matrix form of a system of equations and conditions for unique solutions,
augmented matrix and echelon form, Conditions for existence of a unique solution, infinitely many solutions
and no-solution.
Vector Spaces- Linear Independence, Basis and Rank, Linear Mappings, Eigenvalues and Eigenvectors
Counting Techniques - Factorial notation, Sum rule and product rule, Pigeonhole principle, Permutations and
combinations, Binomial expansion.
Graphs - Graph notation and basic definitions, Complete and bipartite graphs, Euler and Hamiltonian circuits,
Directed and undirected graphs, Planar graphs, Graph colouring, Trees and their basic properties

Method of Teaching and Learning: Lectures, interactive classroom sessions, and case discussions 

Assessment: End of course unit examination, group assignment, mid-term examination, class attendance

Recommended Reading:

1. Johnsonbaugh, R. (8th Ed., 2017). Discrete Mathematics, Pearson.
2. Rosen, K.H. & Krithivasan, K. (7th Ed., 2011). Discrete Mathematics and Its Applications,
McGraw-Hill.
3. Kreyzig, E. (8th Ed., 2006). Advanced Engineering Mathematics, Wiley Student Edition.
4. Susanna, S. E. (2010). Discrete Mathematics with Applications, Cengage Learning.
5. Larson, R. & Falvo, D. (6th Ed., 2009). Elementary Linear Algebra, Houghton Mifflin Harcout
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