Summary Freshman Year Math Study Notes

Math Study Notes


1. Algebra

• Definition: Study of symbols and rules for manipulating them.
• Key Formulas:
• Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
• Difference of squares: a² - b² = (a - b)(a + b)
• Expansion: (a + b)² = a² + 2ab + b²
• Example: Solve x² - 5x + 6 = 0 → (x-2)(x-3) = 0 → x = 2, 3




2. Geometry

• Definition: Study of shapes, sizes, and properties of space.
• Key Formulas:
• Area of triangle: A = 1/2 × base × height
• Area of circle: A = πr²
• Circumference: C = 2πr
• Pythagoras: a² + b² = c² (right triangle)
• Example: Right triangle with legs 3 & 4 → Hypotenuse c = 5




3. Trigonometry

• Definition: Study of relationships between angles and sides in triangles.
• Key Ratios:
• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent
• Important Identity: sin² θ + cos² θ = 1
• Example: 30° angle in a 1-2-√3 triangle, sin 30° = 1/2




4. Calculus

• Definition: Study of change (derivatives) and accumulation (integrals).
• Key Formulas:
• Derivative: f'(x) = lim(h→0) [(f(x+h)-f(x))/h]
• Common derivatives: (xⁿ)' = nxⁿ⁻¹, (sin x)' = cos x
• Integral: ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C
• Example: d/dx(x³) = 3x²




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