## MAS110 Mathematics Core I

 Semester 1, 2019/20 20 Credits Lecturer: Dr Jayanta Manoharmayum uses MOLE 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%)

Five quizzes in the Problems Classes/Practicals. (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.

Type Author(s) Title Library Blackwells Amazon
C Adams Calculus: A Complete Course
C Jordan and Smith Mathematical Techniques
C Ross and Wright Discrete Mathematics, 5th edition.
C Smith and Minton Calculus
C Stewart Calculus
C Stewart, Redlin and Watson Precalculus
C Thomas (and Finney) Calculus

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable

 Mon 10:00 - 10:50 tutorial (group 11) (weeks 1,3,5,8,10,12) Jessop West 114 Mon 10:00 - 10:50 lab session (group 61) (weeks 2,4,6,9,11) RC JP LTB Mon 11:00 - 11:50 tutorial (group 21) (weeks 1,3,5,8,10,12) Jessop West 412 - Sem Rm 5 Mon 11:00 - 11:50 tutorial (group 22) (weeks 1,3,5,8,10,12) Hicks I16 Mon 12:00 - 12:50 tutorial (group 30) (weeks 1,3,5,8,10,12) Jessop West 114 Mon 12:00 - 12:50 tutorial (group 31) (weeks 1,3,5,8,10,12) Hicks I12 Mon 12:00 - 12:50 tutorial (group 32) (weeks 1,3,5,8,10,12) Jessop West 215 Mon 12:00 - 12:50 tutorial (group 36) (weeks 1,3,5,8,10,12) Jessop West 412 - Sem Rm 5 Mon 13:00 - 13:50 lecture Richard Roberts Auditorium Mon 16:00 - 16:50 tutorial (group 105) (weeks 1,3,5,8,10,12) Hicks Seminar Room F35 Mon 16:00 - 16:50 tutorial (group 106) (weeks 1,3,5,8,10,12) Hicks Lecture Theatre 11 Mon 16:00 - 16:50 tutorial (group 107) (weeks 1,3,5,8,10,12) Hicks Seminar Room F38 Mon 16:00 - 16:50 tutorial (group 108) (weeks 1,3,5,8,10,12) Hicks Seminar Room F41 Mon 16:00 - 16:50 tutorial (group 40) (weeks 1,3,5,8,10,12) Hicks I12 Mon 16:00 - 16:50 tutorial (group 42) (weeks 1,3,5,8,10,12) Jessop West 412 - Sem Rm 5 Mon 16:00 - 16:50 tutorial (group 63) (weeks 1,3,5,8,10,12) Hicks Lecture Theatre 10 Tue 10:00 - 10:50 tutorial (group 13) (weeks 1,3,5,8,10,12) Jessop West 412 - Sem Rm 5 Tue 10:00 - 10:50 lab session (group 16) (weeks 2,4,6,9,11) Hicks Seminar Room F38 Tue 10:00 - 10:50 tutorial (group 43) (weeks 1,3,5,8,10,12) Jessop West 401 Tue 10:00 - 10:50 tutorial (group 50) (weeks 1,3,5,8,10,12) Hicks Seminar Room F38 Wed 10:00 - 10:50 tutorial (group 101) (weeks 1,3,5,8,10,12) Jessop West Seminar Room HUB01 Wed 10:00 - 10:50 tutorial (group 99) (weeks 1,3,5,8,10,12) 301 Glossop Road Seminar Room A1