Study Notes for MATH109: Engineering Mathematics

MATH109: ENGINEERING MATHEMATICS

1. THE REAL NUMBER SYSTEM

1.1 Natural Numbers (ℕ)

• Definition: Natural numbers are those numbers used for counting and ordering. These are colloquially referred to as cardinal numbers (for counting) and ordinal numbers (for ordering).
• Example: Natural numbers are represented as 𝔽 = {1, 2, 3, 4, …, n}.

1.2 Whole Numbers (ℕ₀)

• Definition: Whole numbers include all natural numbers plus the number 0. Whole numbers do not include fractions or decimals.
• Example: Whole numbers are represented as ℕ₀ = {0, 1, 2, 3, 4, …, n}.

1.3 Integers (ℤ)

• Definition: Integers include all positive and negative whole numbers, as well as 0. They are constructed from natural numbers under subtraction.
• Example: Integers are represented as ℤ = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}.

1.4 Rational Numbers (ℚ)

• Definition: A rational number is any number that can be expressed as the fraction of two integers, where the numerator is p and the denominator is q (∈ ℤ, q ≠ 0).
• Example: Rational numbers are represented as ℚ = {p/q | p, q ∈ ℤ, q ≠ 0}

1.5 Irrational Numbers (ℝ \ ℚ)

• Definition: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. They cannot be represented as terminating or repeating decimals.
• Example: Irrational numbers include π, e, √2, etc. represented as I = {π, e, √2}.

1.6 Real Numbers (ℝ)

• Definition: Real numbers encompass all rational and irrational numbers. This includes all integers and fractions as well.
• Example: Real numbers can be denoted as ℝ = {x | x is rational or irrational}.

1.7 Quadratic Inequalities

• Definition: A quadratic inequality is an expression of second degree involving an inequality sign instead of an equal sign. Common forms are:

  • $ax^2 + bx + c \geq 0$
  • $ax^2 + bx + c \leq 0$
  • $ax^2 + bx + c > 0$
  • $ax^2 + bx + c < 0$

• How to Solve:

  1. Write the quadratic inequality in standard form: $ax^2 + bx + c$ where $a, b, c$ are coefficients and $a <br />\neq 0$.
  2. Determine the roots of the inequality.
  3. Express the solution in either inequality notation or interval notation.
  4. For inequalities in the factored form such as (x - a)(x - b) ≥ 0, denote intervals based on the roots obtained.

1.8 The Absolute Value

• Definition: The absolute value of a number describes the distance from zero on the number line without considerating direction. It is never negative.
• Case-wise Formula:
  $|x| = \begin{cases} x, & \text{if } x > 0 \ -x, & \text{if } x \leq 0 \end{cases}$
• Solving Equations with Absolute Values: Isolate the absolute value and set its content equal to both positive and negative equivalents.
• Example: Solve $|x| + 2 = 5$ leads to:
  1. $|x| = 3$
  2. Solutions: $x = 3$ or $x = -3$.

1.9 Intervals

• Definition: An interval is a set of real numbers containing all numbers between two endpoints.
• Types of Intervals:
  1. Closed Interval: $I = [a, b]$, denoting all $x$ such that $a \leq x \\leq b$.
  2. Open Interval: $I = (a, b)$, denoting all $x$ such that $a < x < b$.
  3. Half Closed Interval: $I = [a, b)$ or $I = (a, b]$.

1.10 Exercises

Exercise Set 1: Quadratic Inequalities

1. Solve $x^2 > x + 2$
2. Solve $(x + 3)^2 \geq 2(x^2 + 7)$
3. Solve $y^2 - 17y + 70 < 0$
4. Solve $x(x + 1) > 11 - 5x$
5. Solve $2x^2 \leq 5x - 2$
6. Solve $-8 < 4(x - x^2)$

Exercise Set 2: Absolute Value

1. Solve: $3x + 12 + 7 = 7$
2. Solve: $|3x - 7| + 7 = 9$
3. Solve: $|x + 5| = |2x - 1|$

2. FUNCTION

2.1 Function Definition

• Definition: A function is a relation that assigns each input from a set of real numbers (domain) to exactly one output in another set of numbers (range).
• Notation: For a function $f$ with domain X and range Y, we write $f: X \rightarrow Y$ where $f(x)$ denotes the output corresponding to the input $x$.

2.2 Linear Functions

• Definition: A linear function is a function of the form $f(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept.
• Example: For the function $f(x) = x + 3$, the output would be:
  • $f(2) = 5$
  • $f(-6) = -3$

2.3 Plotting the Graph of a Function

• Definition: The graphical representation consists of plotting all points $(x, f(x))$ on a coordinate system.
• Example: For $f(x) = 3x^2 - 4$, compute outputs for various inputs to plot.

2.4 Validity of a Function

• A function must provide a unique output for every input; having multiple outputs for the same input violates the definition of a function.
• Example: $f(x) = \sqrt{x}$ is valid for non-negative arguments.

2.5 Root of a Function

• Definition: The root of a function is a value where the function evaluates to zero.
• Finding Roots: To find roots compute $f(x) = 0$.
• Example: For $f(t) = 9t^3 - 18t^2 + 6t$, solve $9t^2(t - 2) + 6t = 0$.

2.6 Composite Function

• Definition: The composition of functions $(f \circ g)(x) = f(g(x))$ represents applying one function to the result of another.

2.7 Inverse Function

• Definition: The inverse function reverts the output back to the input. If $f(a) = b$ then $f^{-1}(b) = a$.
• Finding Inverses: Replace $f(x)$ with $y$ and reverse the roles of $x$ and $y$ to solve for $y$.

2.8 Exercises

Exercise Set 1: Function

1. Determine the argument and dependent variable of $f(x) = 2x^2 + 5x - 3$.
2. Plot the graph of the function using tables.
3. Estimate and calculate $f(1.5)$ exactly.

Exercise Set 2: Root of Functions

1. Find roots for several given polynomial functions.

Exercise Set 3: Composite Function

1. Find $(f \circ f)(x)$ for given function.

3. PRINCIPLE OF MATHEMATICAL INDUCTION

3.1 Introduction

• Mathematical induction is a proof technique used for proving assertions for all natural numbers.

3.2 The Principle of Mathematical Induction

• Steps:
  1. Base Case: Show the statement is true for the first natural number (usually n=1).
  2. Inductive Step: Assume true for $n=k$ and prove true for $n=k+1$.

3.3 Exercises

Exercise Set 1: Prove by Induction

1. Prove $1^2 + 2^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}$.
2. Prove the divisibility of $(7^n - 3^n)$ by 4.