Mathematics Professional

Formally, lim_x→a f(x) = L means: for every ε > 0 there exists δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε.

A function is continuous at a if lim_x→a f(x) = f(a). Three conditions must hold:

1. f(a) is defined.
2. lim_x→a f(x) exists.
3. They are equal.

The Intermediate Value Theorem (IVT): if f is continuous on [a, b] and N is between f(a) and f(b), then there exists c ∈ (a, b) with f(c) = N.

Example — ε-δ proof that lim(x→2) 3x = 6:
  Given ε > 0, choose δ = ε/3.
  If |x − 2| < δ then |3x − 6| = 3|x − 2| < 3·(ε/3) = ε. □

The derivative f′(x) = lim_h→0 [f(x+h) − f(x)] / h measures the instantaneous rate of change.

Differentiation rules:

• Power rule: d/dx [xⁿ] = nxⁿ⁻¹
• Product rule: (fg)′ = f′g + fg′
• Quotient rule: (f/g)′ = (f′g − fg′) / g²
• Chain rule: [f(g(x))]′ = f′(g(x))·g′(x)

Common derivatives: d/dx [sin x] = cos x, d/dx [cos x] = −sin x, d/dx [eˣ] = eˣ, d/dx [ln x] = 1/x.

Example: differentiate f(x) = x³ sin x
  Product rule: f′(x) = 3x² sin x + x³ cos x

Example (chain rule): d/dx [sin(x²)] = cos(x²) · 2x = 2x cos(x²)

At a local extremum of a differentiable function, f′(x) = 0 (critical point). The First Derivative Test:

• If f′ changes from + to − at c → local maximum.
• If f′ changes from − to + at c → local minimum.

The Second Derivative Test: if f′(c) = 0 and f″(c) > 0 → local min; f″(c) < 0 → local max; f″(c) = 0 → inconclusive.

The global (absolute) extrema on a closed interval [a, b] occur at critical points or endpoints — evaluate f at all and compare.

Example: find extrema of f(x) = 2x³ − 9x² + 12x on [0, 3].
  f′(x) = 6x² − 18x + 12 = 6(x−1)(x−2)
  Critical points: x = 1, x = 2.
  f(0) = 0,  f(1) = 5,  f(2) = 4,  f(3) = 9.
  Global max = 9 at x = 3;  global min = 0 at x = 0.

The indefinite integral (antiderivative): ∫f(x)dx = F(x) + C, where F′(x) = f(x).

Basic antiderivatives:

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

The Fundamental Theorem of Calculus (FTC):

• Part 1: if F(x) = ∫_a^x f(t)dt, then F′(x) = f(x).
• Part 2: ∫_a^b f(x)dx = F(b) − F(a), where F′ = f.

Example: ∫₀² (3x² + 2) dx
  F(x) = x³ + 2x
  F(2) − F(0) = (8 + 4) − 0 = 12

u-substitution: let u = g(x), then du = g′(x)dx, transforming ∫f(g(x))g′(x)dx into ∫f(u)du.

Example: ∫2x cos(x²) dx
  Let u = x², du = 2x dx
  → ∫cos u du = sin u + C = sin(x²) + C

A series Σaₙ converges if the sequence of partial sums Sₙ has a finite limit. Key tests:

• Divergence test: if lim aₙ ≠ 0, the series diverges.
• Geometric series: Σ arⁿ converges to a/(1−r) iff |r| < 1.
• p-series: Σ 1/nᵖ converges iff p > 1.
• Ratio test: let L = lim|aₙ₊₁/aₙ|. L < 1 → converges; L > 1 → diverges; L = 1 → inconclusive.

A vector in ℝⁿ is an ordered n-tuple. In ℝ² and ℝ³: addition is componentwise, scalar multiplication scales each component. The dot product u·v = Σuᵢvᵢ = |u||v|cosθ, useful for finding angles.

A matrix A ∈ ℝ^(m×n) is a rectangular array of numbers. Matrix multiplication (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ; it requires the inner dimensions to match (m×n · n×p = m×p).

The identity matrix I satisfies AI = IA = A. If A is square and invertible, A⁻¹ exists and AA⁻¹ = I.

Example: (2×2 multiplication)
  [1 2] × [5 6]  =  [1·5+2·7  1·6+2·8]  =  [19 22]
  [3 4]   [7 8]     [3·5+4·7  3·6+4·8]     [43 50]

The determinant of a 2×2 matrix is det[[a,b],[c,d]] = ad − bc. For 3×3, expand along any row or column by cofactors. The determinant is zero iff the matrix is singular (non-invertible).

A scalar λ is an eigenvalue of A if there exists a non-zero vector v (the eigenvector) such that Av = λv. To find eigenvalues, solve the characteristic equation:

det(A − λI) = 0

Once eigenvalues are found, substitute each λ back into (A − λI)v = 0 to find the corresponding eigenvectors.

Example: A = [[3,1],[0,2]]
  det(A − λI) = (3−λ)(2−λ) = 0  →  λ₁ = 3, λ₂ = 2
  For λ₁ = 3: (A−3I)v = [[0,1],[0,−1]]v = 0  →  v = [1, 0]ᵀ
  For λ₂ = 2: (A−2I)v = [[1,1],[0,0]]v = 0   →  v = [1,−1]ᵀ

A first-order ODE relates a function y(x) and its derivative y′. Two standard types:

Separable equations: dy/dx = f(x)g(y). Rewrite as dy/g(y) = f(x)dx and integrate both sides.

Linear equations: dy/dx + P(x)y = Q(x). Multiply by the integrating factor μ(x) = e^(∫P(x)dx) to get d/dx[μy] = μQ, then integrate.

Example (separable): dy/dx = 2xy,  y(0) = 1
  dy/y = 2x dx  →  ln|y| = x² + C  →  y = Ae^(x²)
  y(0) = 1 → A = 1,  so y = e^(x²)

Example (exponential growth): dP/dt = kP → P(t) = P₀eᵏᵗ

Mathematical induction proves P(n) for all integers n ≥ n₀:

1. Base case: verify P(n₀).
2. Inductive step: assume P(k) (inductive hypothesis); prove P(k+1).

Prove: Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
  Base (n=1): 1 = 1·2·3/6 = 1 ✓
  Step: assume for k; add (k+1)²:
    k(k+1)(2k+1)/6 + (k+1)²
    = (k+1)[k(2k+1) + 6(k+1)]/6
    = (k+1)(2k²+7k+6)/6
    = (k+1)(k+2)(2k+3)/6 ✓

Set notation: A ∪ B, A ∩ B, A \ B, Aᶜ, |A| (cardinality). The empty set ∅ satisfies ∅ ⊆ every set. De Morgan: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ.

Propositional logic: compound statements built from ∧ (and), ∨ (or), ¬ (not), → (implies). A valid argument preserves truth from premises to conclusion.