## MA135 & MA160-II ALGEBRA & CALCULUS (SEMESTER II) LECTURER: GRAHAM ELLIS

### SYLLABUS & LEARNING OUTCOMES

The syllabus is described by the 60 algebra and calculus Semester II problems listed in this document.  The learning outcomes are simply that, having completed the module, you should be able to answer these 60 problems and closely related problems. Assessment is via an end-of-semester exam.

### FORTNIGHTLY HOMEWORKS

The continuous assessment in Semester II consists of six fortnightly algebra/calculus homeworks of equal weight.

Please click here to access the MA135/MA160 homework sheets. The first homework will be due on 30th January.  Late submissions will not be graded.

Student feedback on the module is available here.

### WEEKLY TUTORIALS

The tutorial starts in the second week of semester. It will be at 3.00pm on Wednesdays in IT203.

### WHAT IS MATHEMATICS?

I'm not too sure of the answer. But whatever it is it is possibly something a bit larger than what was taught in your school mathematics classes. If you are interested in the question then you should browse this article by Fields Medallist William Thurston. He won the Fields Medal for his work in geometry. You could also take a look at the lovely little book A Mathematicians Apology by G.H. Hardy which is available online here.

### ALGEBRAMATERIAL

Algebra text:
Algebra & Geometry: An introduction to University Mathematics by Mark V.Lawson.
A pre-publication pdf version of this text is available on blackboard. This version is for private use only and the pdf version must not be made available on the internet.

Algebra outline:

This module introduces the student to propositional logic (8 lectures), systems of equations (8 lectures) and complex numbers (8 lectures).

Algebra lectures 2015-16:

The algebra lecture slides will be uploaded to the web after each lecture and links to the slides will be given below. A brief outline of each lecture will be added/modified below shortly after each lecture.

I'll place student feedback here.

### 1

Lecture 1:
Recalled basics on modular arithmetic and then explained how arithmetic modulo 2 is related to logic. Explained the logical connectibe "AND".

### 2

Lecture 2:
Introduced the logical connectives AND and NOT. Explained how to calculate a truth table for a complicated logical expression constructed using AND, OR and NOT.

### 3

Lecture 3 (Guest Lecturer):
Introduced the logical connective "implies".

### 4

Lecture 4 (Guest Lecturer):
Introduced the notion of a tautology and the notion of a contractiction.

### 5

Lecture 5:
Explained what is meant by a "logically valid argument" and investigated some examples.

### 6

Lecture 6:
Investigated the logical validity of several arguments.

### 7

Lecture 7:
Introduced complex numbers and explained how they are added, subtracted and multiplied.

### 8

Lecture 8:
Explained how to divide one complex number by another. Then introduced the modulus of a complex number and the argument of a complex number. STUDENT ATTENDANCE WAS RECORDED.

### 9

Lecture 9:
Proved that |wz|=|w||z| and Arg(wz)=Arg(w)+Arg(z). Then used this result to make some calculations.

### 10

Lecture 10 :
Explained De Moivre's Theorem and used it to make some calculations. Explained how to factorize the polynomials x4-1 and x5-1.

### 11

Lecture 11:
Introduced the notation eix=cos x + i sin x . Then factorized the polynomial x5-1 as a product of real linear and quadratic polynomials. The same method will work for factorizing the polynomials x6-1, x7-1, ... .

### 12

Lecture 12:
Solved a complex numbers exam question from the 2006-07 MA123 exam paper. (This year's MA133/MA135 module is following essentially the same syllabus as the former MA121 & MA123 modules. In the old days there were two 3-hour papers in summer, one devoted to calculus and one devoted to algebra, and no Christmas exams. These days we hold a 2-hour calculus & algebra exam at Christmas and a 2-hour calculus & algebra exam in summer.)

### 13

Lecture 13:
Solved a system of linear equations using Gaussian elimination.

### 14

Lecture 14:
Explained how we can regard a linear equation as a plane, and how we can regard the solutions to a system of linear equations as being those points in the intersection of a system of planes.

### 15

Lecture 15:
Explained that a system of linear equations may have no solution, or it may have just one solution, or it may have infinitely many solutions lying on a line, or it may have infinitely many solutions lying on a plane, or ... . Solved some systems of linear equations.

### 16

Lecture 16:
Solved some systems of linear equations.

### 17

Lecture 17:
Explained how matrix notation, and the inverse of a matrix, can be used to solve a system of n equations in n unkowns.

### 18

Lecture 18:
Explained the Gauss-Jordan (or row operation) method for finding the inverse of a square matrix.

### 19

Lecture 19:
Revised Logic using problems from the problem sheet.

### 20

Lecture 20:
Revised problems on complex numbers.

### 21

Lecture 21:
Revised more problems on complex numbers. Student attendance was recorded.

Lecture 22:

Lecture 23:

Lecture 24:

### CALCULUS MATERIAL

Calculus text:
We'll continue to use Stewart's calculus text.

Calculus outline:

This module introduces the student to integration (8 lectures), techniques of integration (8 lectures) and differential equations (8 lectures).

Calculus lectures 2015-16:

The calculus lecture slides will be uploaded to the web after each lecture and links to the slides will be given below. A brief outline of each lecture will be added/modified below shortly after each lecture.

### 1

Lecture 1:
Recalled basic formulae for areas; explained how limits are needed to derive a formula for the area of a circle; introduced the notion of a definite integral.

### 2

Lecture 2:
Calculated definite integrals for the functions y=mx+c, y=|x+c| and y=x2.

### 3

Lecture 3 (Guest Lecturer):
Introduced the floor function. Eplained how to calculate the integral of a sum and of a scalar multiple.

### 4

Lecture 4 (Guest Lecturer):
Introduced the notion of an anti-derivative. Stated the Fundamental Theorem of Calculus and used it to compute the integral of a product.

### 5

Lecture 5:
Recalled the Fundamental Theorem of Calculus, and then gave a proof of it.

### 6

Lecture 6:
Used integration to solve a range of word problems.

### 7

Lecture 7:
Solved an integration problem involving acceleration (= rate of change of velocity). Then started to talk about techniques of integration. Covered Technique 1: algebraic simplification.

### 8

Lecture 8:
More on techniques of integration. Covered Technique 2: spotting integrals of some standard functions. Started Technique 3: simple substitutions.

### 9

Lecture 9:
More on techniques of integration. Covered Technique 3: simple substitutions.

### 10

Lecture 10 :
More techniques of integration. Covered integration by parts (ie integration of a product). STUDENT ATTENDANCE WAS RECORDED.

### 11

Lecture 11:
More techniques of integration: Covered trigonometric substitutions.

### 12

Lecture 12:
Started to cover the final technique of integration: partial fractions.

### 13

Lecture 13:
Gave more details and examples on partial fractions.

### 14

Lecture 14:
Introduced the notion of a differential equation and its solution. Found all solutions of the differential equation dy/dx = ky . Then investigated a cooling cup of coffee.

### 15

Lecture 15:
Finished the investigation of a cooling cup of coffee. Then explained the Malthusian model of population growth and used it to model the World's population.

### 16

Lecture 16:
Introduced the Logistic Model of population growth. The Logistic Model is a separable differential equation - so we covered the technique for solving a separable differential equation.

### 17

Lecture 17:
Solved the Logistic Model and predicted the limiting world population.

### 18

Lecture 18:
Revised problems on integration from Section 10.7 in the problem sheet.

### 19

Lecture 19:
Revised more problems on integration.

### 20

Lecture 20:
Revised problems from Section 10.9 on differential equations.

### 21

Lecture 21:
Revised more problems on differential equations.

### 22

Lecture 22:
Revised problems on techniques of integration.

### 23

Lecture 23:
Revised more problems on techniques of integration. (Recorded attendance.)

### 24

Lecture 24:
Revised more problems on techniques of integration. (Recorded attendance.)