Essential Math Formulas for Algebra, Geometry and Calculus

Matrices and Determinants

• Inverse Matrix: |A|⁻¹ = 1/|A|; A⁻¹ = (Adj A)ᵀ / |A|; A⁻¹ · A = I.
• Multiplication: Row by column (fila por columna).
• Determinants: |A₃,₃| = 4, det(2A) = 2³ · 4.
• Resolution (Cramer's Rule): x = |B c₂ c₃| / |A|, y = |c₁ B c₃| / |A|...

Geometry and Spatial Relationships

Areas and Volumes

• Area of a Parallelogram: |a × b|
• Area of a Triangle: |a × b| / 2
• Volume of a Parallelepiped: [u, v, w]
• Volume of a Tetrahedron: [u, v, w] / 6

Relative Positions

• Position of 3 Planes:
  • rg(A) = rg(A*) = 3: System Consistent Determined (1 point).
  • rg(A) = rg(A*) = 2: Line (recta).
  • rg(A) = rg(A*) = 1: Coincident.
  • rg(A) = 1, rg(A*) = 2: Parallel.
  • rg(A) = 2, rg(A*) = 3: Two cases: if two rows are proportional, two planes are parallel and one cuts; otherwise, they cut two by two.
• Position of Line-Plane: Substitute x, y, z into the plane equation. mt = n:
  • m ≠ 0: Secant.
  • m = 0, n = 0: Coincident.
  • m = 0, n ≠ 0: Parallel.

Angles and Distances

• Angle between 2 Vectors: cos α = (a · b) / (|a| · |b|)
• Angle between 2 Lines: cos α = (vᵣ · vₛ) / (|vᵣ| · |vₛ|)
• Angle between 2 Planes: cos α = (v_π · v_π') / (|v_π| · |v_π'|)
• Angle between Line and Plane: sen α = (uᵣ · u_π) / (|uᵣ| · |u_π|)
• Distance Point to Plane: d(P, π) = |Ax + By + Cz + D| / √(A² + B² + C²)
• Distance Point to Line: d = |PAᵣ × uᵣ| / |uᵣ|
• Distance between 2 Lines:
  • Parallel: |AᵣBₛ × uₛ| / |uₛ|
  • Crossing: [uᵣ, uₛ, AᵣBₛ] / |uᵣ × uₛ|

Mathematical Analysis

Function Study

• Study: Domain, intersection with axes.
• Asymptotes:
  • Vertical (AV): lim x → no-domain.
  • Horizontal (AH): lim x → ±∞.
  • Oblique (AO): y = mx + n; m = lim x → ±∞ f(x)/x; n = lim x → ±∞ [f(x) - mx].
• Monotony: First derivative.
• Inflection Point (PI): Second derivative.

Theorems and Indeterminate Forms

• Rolle's Theorem: If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), there exists at least one c ∈ (a, b) such that f'(c) = 0.
• Lagrange's Theorem: If f(x) is continuous and differentiable on (a, b), there exists at least one c ∈ (a, b) such that f'(c) = [f(b) - f(a)] / (b - a).
• Indeterminate Forms:
  • k/0: Lateral limits.
  • 0/0: L'Hôpital's Rule.
  • 0 · ∞: Transform into 0/0 and simplify.
  • 1^∞: e ^ lim x → ±∞ q(x) · (p(x) - 1).

Integration Techniques

• Integration by Parts: ∫ u dv = uv - ∫ v du. Choose u using ALPES: Arcs, Logarithms, Polynomials, Exponentials, Sines/Cosines.
• Rational Functions:
  • If degree P(x) ≥ Q(x): Divide = C(x) + R(x)/Q(x).
  • If degree P(x) < Q(x): Factorize denominator. For non-real roots: 1st fraction is log, 2nd write Q(x) = (x - a)² + b² and solve for arctg.

Probability and Statistics

Probability Rules

• p(Ā ∪ B⁻) = p(Ā) + p(B⁻) - p(Ā ∩ B⁻)
• p(A ∪ B) = p(A) + p(B) - p(A ∩ B)
• p(A ∪ B)⁻ = p(A⁻ ∩ B⁻)
• Incompatible Events: p(A ∩ B) = 0, and p(A ∪ B) = p(A) + p(B).
• Conditional Probability: p(A/B) = p(A ∩ B) / p(B); p(A ∩ B) = p(A/B) · p(B).
• p(B⁻ ∩ A) = p(A) - p(A ∩ B)
• Independent Events: p(A ∩ B) = p(A) · p(B).
• Total Probability: Use tree diagrams (árboles).
• Bayes' Theorem: p(A/B) = p(A ∩ B) / p(B).

Distributions

• Normal Distribution: N(μ, σ), Z = (X - μ) / σ.
• p(Z > a) = 1 - p(Z < a)
• p(Z < -a) = 1 - p(Z < a)
• p(Z > -a) = p(Z < a)
• p(a < Z < b) = p(Z < b) - p(Z < a)

Fundamental Integration Rules

• ∫ f(x)ⁿ · f'(x) dx = f(x)ⁿ⁺¹ / (n + 1)
• ∫ f'(x) / f(x) dx = ln|f(x)|
• ∫ f'(x) · eᶠ⁽ˣ⁾ dx = eᶠ⁽ˣ⁾
• ∫ f'(x) · aᶠ⁽ˣ⁾ dx = aᶠ⁽ˣ⁾ / ln a
• ∫ f'(x) · sen(f(x)) dx = -cos(f(x))
• ∫ f'(x) · cos(f(x)) dx = sen(f(x))
• ∫ f'(x) / cos²(f(x)) dx = tg(f(x))
• ∫ f'(x) · (1 + tg²(x)) dx = tg(f(x))
• ∫ f'(x) / √(1 - f(x)²) dx = arcsen(f(x))
• ∫ f'(x) / (1 + f(x)²) dx = arctg(f(x))