## MAS110 Mathematics Core I

 Semester 1, 2021/22 20 Credits Lecturer: Dr Jayanta Manoharmayum Timetable Reading List Outcomes Assessment Full Syllabus

The star players in this module are the trigonometric and exponential functions, and their inverses. These are the first functions we meet that go beyond polynomial and rational functions, which are generated by simple arithmetical operations. We first meet trigonometric functions in right-angled triangles, and exponential functions in considering how few powers of 10 separate atoms from galaxies. In this module we shall see how these two types of functions, coming from such different sources, are intimately linked (but only when we allow complex numbers, involving the square root of minus one). We shall be especially concerned with their special role in calculus, exemplified by their appearance together in the solutions of differential equations in the last part of the module. In preparation, we shall seek a thorough understanding of differentiation and integration, and we begin with some foundational material on sets, functions and counting, on which much of mathematics can be built.

There are no prerequisites for this module.

The following modules have this module as a prerequisite:

 MAS111 Mathematics Core II MAS112 Vectors and Mechanics MAS220 Algebra MAS222 Differential Equations

## Outline syllabus

1. Sets, functions and counting
2. Summation and Induction
3. Trigonometry
4. Complex numbers
5. Limits
6. Differentiation
7. Sequences and series
8. Integration
9. Logarithms, exponentials and series
10. Differential equations

## Learning outcomes

By the end of the unit, a candidate will be able to 1. work with sets, and use them in a formal logic framework; 2. use permutations and combinations to count sizes of sets; 3. understand exponentials, logarithms, trigonometric functions, hyperbolic functions and identities relating them; 4. perform algebraic and geometric calculations involving complex numbers; 5. understand the geometric and numerical meaning of differentiation; 6. calculate or compute derivatives of some standard functions, and apply rules for calculating derivatives. 7. calculate implicit derivatives and higher-order derivatives; 8. calculate integrals of some standard functions and use techniques for finding integrals (e.g., by parts and by substitution); 9. compute Taylor series and understand their relation to higher derivatives; 10. solve simple first and second order differential equations; 11. sum standard finite and infinite series.

43 lectures, 6 tutorials, 5 practicals

## Assessment

One formal 2 hour exam. All questions compulsory; format varies. (80%)

Engagement in Tutorials. (10%)
Online quizzes for the structured independent learning content. (10%)

## Full syllabus

1. Sets, functions and counting (7 lectures)
Sets, subsets, finite and infinite sets. Natural numbers, integers, rational and real numbers. Set operations: unions, intersections, difference, cartesian products of sets. Functions between arbitrary sets. Surjections, injections, bijections and inverse functions. Real-valued functions of real numbers, their domains and images, R2 and R3 in geometry. The fundamental role of sets in mathematics, Russell's paradox. Counting elements of finite sets. Counting permutations and combinations. Binomial Theorem.
2. Summation and Induction (2 lectures)
Proof by induction. Summation of geometric and arithmetic series, and of the first n squares.
3. Trigonometry (2 lectures)
Radians, circles and periodicity, geometrical definitions of trigonometric functions and their relation with triangles. Addition and double angle formulas. Inverse trigonometric functions. Addition formula for inverse tan and Pi.
4. Limits and continuity (4 lectures)
Idea of a limit and continuity. Left and right limits, limit at infinity. Sandwich rule, standard limit formulas.
5. Differentiation (4 lectures)
Tangent lines, the derivative as a limit, justifications of the sum, product, quotient and chain rules. Implicit differentiation. Differentiation of trigonometric and inverse trigonometric functions. The derivative as a rate of change. L'Hospital's rule.
6. Integration (5 lectures)
Areas under graphs, Fundamental Theorem of Calculus. Reversing the Chain Rule and the Product Rule to get integration by substitution and integration by parts. Trigonometric substitutions.
7. Sequences and series (5 lectures)
Idea of convergence of sequences. Sandwich rule, bounded monotone sequences, standard limit formulas. Infinite series of positive terms. Review of basic examples including geometric and harmonic series. Absolute convergence.
8. Series expansions (5 lectures)
Differentiation of arbitrary powers. Maclaurin series, arithmetical definitions of sin, cos and exp via infinite series.
9. Complex numbers (Structured independent learning)
Square roots of negative numbers, complex numbers. Argand diagram, modulus, argument and triangle inequality. Geometrical realisations of addition and multiplication, de Moivre's Theorem, nth roots of unity. The complex exponential, Euler's formula, exponential form, new insight into addition formulas.
10. Differential equations (Structured independent learning)
Separation of variables, integrating factors, homogeneous equations. Second order homogeneous equations with constant coefficients, auxiliary polynomial. General solutions and initial conditions. Non-homogeneous equations, particular integrals.