Mathematics
Complete Notes — Easy English

Linear Equations

Form: ax + b = 0
Solution: x = -b/a

Quadratic Equations

Form: ax² + bx + c = 0
Using quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

Types of Solutions

Inequalities

• Linear inequalities: ax + b > c
• Quadratic inequalities: ax² + bx + c > 0
• Solution represented on number line

Function Definition

Relation where each input has exactly one output. Notation: f(x) = y

Types of Functions

• Linear: f(x) = mx + c
• Quadratic: f(x) = ax² + bx + c
• Polynomial: f(x) = aₙxⁿ + ... + a₁x + a₀
• Exponential: f(x) = aˣ
• Logarithmic: f(x) = logₐ(x)

Function Operations

• Addition: (f + g)(x) = f(x) + g(x)
• Multiplication: (f·g)(x) = f(x)·g(x)
• Composition: (f∘g)(x) = f(g(x))

Inverse Functions

If f(a) = b, then f⁻¹(b) = a

Trigonometric Ratios

Special Angles

• sin 30° = 1/2, cos 30° = √3/2
• sin 45° = √2/2, cos 45° = √2/2
• sin 60° = √3/2, cos 60° = 1/2

Trigonometric Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• sin(A ± B) = sinA cosB ± cosA sinB
• cos(A ± B) = cosA cosB ∓ sinA sinB

Law of Sines and Cosines

Sine: a/sinA = b/sinB = c/sinC
Cosine: c² = a² + b² - 2ab cosC

Vector Representation

v = (x, y, z) or v = xî + yĵ + zk̂

Vector Operations

• Addition: a + b = (a₁+b₁, a₂+b₂, a₃+b₃)
• Scalar Multiplication: k·a = (ka₁, ka₂, ka₃)
• Dot Product: a·b = |a||b|cos θ
• Cross Product: a × b (perpendicular to both)

Vector Properties

• Magnitude: |v| = √(x² + y² + z²)
• Unit Vector: û = v/|v|
• Parallel vectors: a × b = 0

Applications

• Displacement and velocity
• Force resolution
• Work calculation

Limit Concept

lim(x→a) f(x) = L means f(x) approaches L as x approaches a

Limit Laws

• Sum: lim(f + g) = lim f + lim g
• Product: lim(f·g) = (lim f)·(lim g)
• Quotient: lim(f/g) = (lim f)/(lim g), if lim g ≠ 0

Continuity Definition

Function f is continuous at x = a if:

• f(a) is defined
• lim(x→a) f(x) exists
• lim(x→a) f(x) = f(a)

Discontinuities

• Removable: Can be fixed
• Jump: Left and right limits differ
• Infinite: Function approaches infinity

Derivative Definition

f'(x) = lim(h→0) [f(x+h) - f(x)] / h

Differentiation Rules

• Power Rule: d/dx(xⁿ) = nxⁿ⁻¹
• Product Rule: d/dx(fg) = f'g + fg'
• Quotient Rule: d/dx(f/g) = (f'g - fg')/g²
• Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)

Derivatives of Special Functions

• d/dx(eˣ) = eˣ
• d/dx(ln x) = 1/x
• d/dx(sin x) = cos x
• d/dx(cos x) = -sin x

Applications

• Finding maximum and minimum values
• Determining increasing/decreasing intervals
• Analyzing motion (velocity, acceleration)

Indefinite Integral

∫f(x)dx = F(x) + C, where F'(x) = f(x)

Integration Rules

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• ∫eˣ dx = eˣ + C
• ∫(1/x) dx = ln|x| + C
• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C

Definite Integral

∫ₐᵇ f(x)dx = F(b) - F(a)

Integration Techniques

• Substitution: u-substitution
• Parts: ∫u dv = uv - ∫v du

Applications

• Finding area under curves
• Volume calculations

Sequences

• Arithmetic: aₙ = a₁ + (n-1)d
• Geometric: aₙ = a₁ · rⁿ⁻¹

Series Sums

• Arithmetic Series: Sₙ = n/2(a₁ + aₙ)
• Geometric Series: Sₙ = a₁(1-rⁿ)/(1-r)

Infinite Series

• Convergent series: Sum approaches a limit
• Divergent series: Sum does not converge

Taylor Series

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...

Applications

• Physics problems involving motion
• Engineering calculations
• Finance and economics