Maths 2 Lecture Notes

Introduction

• Review of concepts from Maths 1.
• Focus on functions, geometry, and calculus.

Topics Reviewed from Maths 1

• Straight Lines
• Quadratic Equations
• Polynomials
• Functions
  • Specifically, functions from real numbers to real numbers (R to R).

Understanding Functions

• Definition: A function is a relation from a set of inputs (domain) to a set of outputs (codomain), where each input corresponds to exactly one output.
• Notation: If f maps from x to y, x is the domain and y is the codomain.
• Range: The subset of codomain which the function takes values from (values of f(x)).

Function of One Variable

• Denoted as f from a subset D of R to R.
• Example of a Linear Function:
  • Generally of the form f(x) = mx + c.
  • Example: f(x) = 5x + 2
    • f(0) = 2
    • f(20) = 102
• Graph of a Linear Function: A straight line, with:
  • m as the slope (related to the angle with x-axis).
  • c as the y-intercept.

Quadratic Functions

• Typically of the form f(x) = a(x - b)² + c, where:
  • Graph: Parabola (U-shaped curve).
  • a > 0: Opens upwards; a < 0: Opens downwards.
  • Minimum value occurs at x = b.
• Expanded Form:
  • f(x) = ax² + bx + c (polynomial degree 2).

Polynomial Functions

• General form: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_0.
• Example: f(x) = x³ - 4x.
• Zeros of Polynomial: Points where f(x) = 0 (e.g., x = -2, 0, 2 for f(x) = x³ - 4x).

Exponential and Logarithmic Functions

• Exponential Function: f(x) = a^x (where a > 0).
  • Defined for all real numbers.
• Logarithmic Function: Inverse of the exponential function.
• Important Note: Both functions are defined only for positive bases (a > 0).
• Example: f(x) = log₂(x).

Monotonic Functions

• Monotonic Increasing: If x1 ≤ x2 then f(x1) ≤ f(x2).
• Monotonic Decreasing: If x1 ≤ x2 then f(x1) ≥ f(x2).
• Graphical Examples:
  • Increasing function: y = x (linear).
  • Decreasing function: f(x) declines as x increases.

Comparison of Functions

• Growth Rates:
  • Exponential Functions grow faster than any polynomial functions.
  • Logarithm Functions grow slower than all polynomial functions.

Tangent Lines

• Definition: A line that touches a curve at a single point without crossing it.
• Example: Tangent to y = x² at point (1, 1).
• Future Study: Use calculus to determine the equation of tangent lines.

Conclusion

• Review and understanding of concepts from Maths 1 are crucial for progressing in Maths 2.
• Students encouraged to revisit Maths 1 materials if concepts feel unclear.