Online Tutorial for Maths Revision

This article is intended to give prospective students an overview on what background a typical introductory maths course at a university requires. Readers should note, however, that there are differences between introductory maths courses at different universities, and even within universities. It also provides a study guide using online tutorials so that at the end, when successfully completing all tutorial sessions, students will be enabled with the skills to master a first years introductory maths course.

The online tutorials can be found here.

During your high school, you should have taken O-level maths or an equivalent course, and understood the material. The online part of this guide provides you with tutorials that enable you with the required knowhow for your introductory maths course.

You do not need a high school calculus or probability course; if you have taken one, you should not assume that you can skip classes, or not study, during the first part of the introductory maths course. University introductory maths course goes much deeper than most courses in high schools do, even if they appear to cover the same material. And remember to keep using your algebra and other mathematical skills, because this is the most important prerequisite.

The following description represents the minimum that you need to know before you begin introductory maths course in year one. In addition, you should also know something about linear algebra, geometry, statistics, and other areas of mathematics; you should have experience applying mathematics in other subjects; and you should be able to write clear explanations of what you know, and solve problems that require a certain amount of lateral thinking.

Below, you find a list of required skills. When you successfully complete all the tutorials step by step, you will have the required prerequisites for the introductory maths course and gained experience on the following fields:

Basic Arithmetic
You should be able to do basic arithmetic without a calculator, including operations on fractions, negative numbers, and decimals. You should be able to compute simple powers and roots. This material, which is from the elementary and junior high school curriculum, is fundamental for everything that follows.

Polynomials
You should know how to add, subtract, multiply, divide, and factor polynomials. You should know special forms such as the difference of powers. You should understand the connection between roots and factorizations, and be able to solve a quadratic equation using the quadratic formula. You should be able to work with a polynomial function of something nontrivial, like sin and cos functions.

Basic Algebra
You should know the basic rules for addition, subtraction, multiplication, division, and exponents, and be aware of the operations such as division by 0 and taking the square root of a negative number that cannot be done within the real number system. You should know how to solve a simple equation, simplify an algebraic expression, and evaluate an expression by plugging values into it.

Inequalities and absolute values
You should be able to solve simple inequalities and perform algebraic operations with them. In particular, you should know which operations reverse inequalities and which ones preserve them. You should understand interval notation, including open, closed, and half-open intervals, and intervals with limits at infinity. You should know how to compute an absolute value, and to do simple algebra using the absolute value function.

Functions
You should understand the concept of a function and its inverse function and know how to compute the composition of two or more functions. You should be able to determine the range and domain of a simple function. This will be important for understanding the Chain Rule, various methods of integration, and limits.

Algebra with functions
You should be able to simplify a fractional expression, convert a stacked fractional expression into a simple one, put fractional expressions over a common denominator, and perform a partial fraction expansion. These skills will be useful in finding various derivatives, simplifying derivatives and integrals, and in particular for the “partial fractions" technique of integration.

Rationalizing numerators and denominators
You should know how to eliminate square (and other) roots from the numerator or denominator of a fraction by multiplying both the numerator and denominator by an appropriate expression. This technique will be important in finding the derivatives of certain expressions involving roots.

Linear Graphs
You should be able to graph linear functions and inequalities, determine the slope and intercept of a line from its equation and vice versa, determine where two lines meet, use the negative-reciprocal rule for orthogonal lines, and find the distance between two points. Many of these ideas will be conceptually important in calculus, which deals a lot with slopes, tangent lines, secant lines, etc.

Graphs
You should be able to graph polynomials and rational functions, showing features such as zeroes, y intercept, horizontal, vertical, and slant asymptotes, and points of discontinuity. You should also be able to read significant features from a graph. You should be able to draw a graph by determining the main features and joining them together with smooth curves. It is not sufficient to plot five or six points and joining them with straight lines. In your introductory maths course, you will learn to extend these graphing skills by adding other features, such as maxima, minima, and points of inflection.

Exponents and roots
You should know the basic identities for exponents and roots, and be able to use them to solve equations and derive other identities. In particular, you should be able to convert a reciprocal to a negative power, or a root to a fractional power. These identities will be extremely important for differentiation and integration, because they let us use one rule to differentiate and integrate many apparently different expressions.

Logarithms
Logarithms are extremely important in many of the sciences, and it is important to be able to differentiate and integrate expressions using them. To do that, you have to be able to manipulate logarithms algebraically. You should know the definition of logarithms to various bases, their relation to powers and roots, and the change-of-base formula loga(b) x logb(c) = loga(c).

Problem solving skills
For solving applied problems, you should be able to pick out the important numerical quantities, known or unknown, from the problem description, and determine the relations between them. These yield a set of equations that must be solved to yield the desired quantity. You may also need to know certain quantities and relations that are not given in the problem. Do NOT try to learn the formula for each type of problem. Instead, learn basic relations and heuristics.