## MAS110 Mathematics Core I

Note: This is an old module occurrence.

You may wish to visit the module list for information on current teaching.

 Semester 1, 2015/16 20 Credits Lecturer: Dr Jayanta Manoharmayum Home page (also MOLE) Timetable Reading List 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. Limits
5. Differentiation
6. Integration
7. Logarithms, exponentials and series
8. Complex numbers
9. Differential equations

## Office hours

Just drop in to J22 Hicks, or book an appointment.

43 lectures, 6 tutorials

## Assessment

Entirely on a two-hour examination. All questions compulsory, number and length of questions variable, but 60 marks altogether.

## 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. Pascal's triangle. 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 (3 lectures)
Radians, circles and periodicity, geometrical definitions of trigonometric functions. Their relation with triangles and applications. Addition and double angle formulas. Inverse trigonometric functions. Addition formula for inverse tan and Pi.
4. Limits (4 lectures)
Idea of a limit, including at infinity. Left and right limits. 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 and applications: differentiation of rational powers, tangents to curves. 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 by substitution and integration by parts. Trigonometric substitution.
7. Logarithms, exponentials and series (4 lectures)
The natural logarithm as an integral, and the exponential function as its inverse. Differentiation of arbitrary powers. Maclaurin series, arithmetical definitions of sin, cos and exp via infinite series.
8. Complex numbers (5 lectures)
Square roots of negative numbers, complex numbers. Argand diagram, modulus, amplitude and triangle inequality. Geometrical realisations of addition and multiplication, de Moivre's Theorem, nth roots of unity. Euler's formula, exponential form, new insight into addition formulas etc.
9. Differential equations (9 lectures)
Exponential growth and decay, 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 Jordan and Smith Mathematical Techniques 510 Blackwells Amazon
C Kreyszig Advanced Engineering Mathematics 510.2462 Blackwells Amazon
C Ross and Wright Discrete Mathematics, 5th edition. 510 Blackwells Amazon
C Smith and Minton Calculus 515 Blackwells Amazon
C Stewart Calculus 515 Blackwells Amazon
C Thomas (and Finney) Calculus Blackwells Amazon

(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 20) Hicks F9b Mon 10:00 - 10:50 tutorial (group 21) Hicks I14 Mon 11:00 - 11:50 tutorial (group 23) Hicks G19 Mon 11:00 - 11:50 tutorial (group 30) Richard Roberts Room B79 Mon 12:00 - 12:50 tutorial (group 43) Hicks J15 Mon 12:00 - 12:50 tutorial (group 44) Hicks J18b Mon 16:00 - 16:50 tutorial (group 46) Jessop West Seminar Room HUB01 Mon 16:00 - 16:50 tutorial (group 51) Hicks I23 Tue 10:00 - 10:50 tutorial (group 52) Hicks I7 Tue 10:00 - 10:50 tutorial (group 53) Hicks I16