MATHS 102 Course Notes

The University of Auckland, Faculty of Science, Department of Mathematics

Course Overview

  • This course book covers the essential topics of MATHS 102 at The University of Auckland.

  • Topics include algebra, functions, trigonometry, and calculus, among others.

Chapter 1: Basic Algebra 1
1.1 Order of Operations

  • Definition: Order of operations refers to rules that dictate the sequence in which calculations should be performed in mathematical expressions.

  • Mnemonics: BEDMAS (Brackets, Exponents, Division/Multiplication, Addition/Subtraction).

  • Rules:

    • Perform calculations inside brackets first.

    • Handle exponents next.

    • Carry out multiplication and division from left to right.

    • Finally, perform addition and subtraction from left to right.

1.2 Expanding Brackets

  • Distributive Law: a(b + c) = ab + ac

  • Examples:

    • Expand x(x - 3) = x^2 - 3x

    • Use the distributive law consistently to simplify expressions with brackets.

1.3 Factors

  • Definitions:

    • Product: The result of multiplying terms.

    • Factor: A term that divides another term without leaving a remainder.

  • Common Factors are factors shared by two or more terms.

  • Highest Common Factor is the largest of these shared factors.

1.4 Fractions

  • Understand algebraic fractions, which contain unknowns.

  • Key Concepts:

    • Numerator: The top part of a fraction.

    • Denominator: The bottom part of a fraction.

    • Multiplication and division rules for fractions include:

    • Multiplying: Multiply the numerators and the denominators.

    • Dividing: Multiply by the reciprocal of the divisor.

Chapter 2: Basic Algebra 2
2.1 Surds

  • Definition: A surd is the root of a number that cannot be simplified to remove the radical.

  • Learn to operate with surds, including simplifying, adding, and working with surds in denominators.

2.2 Equations

  • Understand the basics of manipulation techniques for solving equations:

    • Isolating the variable on one side of the equation.

    • Balancing the equation by applying the same operation to both sides.

2.3 Quadratic Equations

  • Definition: Polynomial equations of the form ax^2 + bx + c = 0.

  • Methods include**: factoring, completing the square, and using the quadratic formula.

  • Roots: The solutions to the quadratic equation.

Chapter 3: Basic Algebra 3
3.1 Straight Lines

  • Gradient: A measure of how steep a line is, calculated as: m = (y2 - y1) / (x2 - x1).

  • Equation of a Line: y - y1 = m(x - x1).

Chapter 4: Functions
4.1 Function Notation

  • Definition: A rule that defines a relationship between input and output values.

  • Domain and Range: Understanding the set of inputs for which the function is defined (domain) and the possible outputs (range).