MTEL Mathematics (63) cheat sheet

Limit definition of the derivative: f'(x) = lim(h→0) [f(x+h) − f(x)]/h; a function is differentiable at a point only if it is continuous there, but continuity does not imply differentiability (e.g., |x| at x = 0).

Derivative rules: power d/dx xⁿ = n·xⁿ⁻¹; product (uv)' = u'v + uv'; quotient (u/v)' = (u'v − uv')/v²; chain d/dx f(g(x)) = f'(g(x))·g'(x).

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C.

Fundamental Theorem of Calculus: if F(x) = ∫ₐˣ f(t) dt with f continuous, then F'(x) = f(x); and ∫ₐᵇ f(x) dx = F(b) − F(a) for any antiderivative F.

L'Hôpital's rule applies to 0/0 or ∞/∞ indeterminate forms: lim f/g = lim f'/g', valid only when the resulting limit exists or is ±∞; rewrite products, powers, and differences to reach a quotient form first.

Convergence tests for series: geometric Σarⁿ converges iff |r| < 1 to a/(1−r); p-series Σ1/nᵖ converges iff p > 1; ratio test inconclusive when L = 1; alternating series test requires decreasing terms tending to 0.

Taylor/Maclaurin series: f(x) = Σ f⁽ⁿ⁾(a)(x−a)ⁿ/n!; key expansions sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …, eˣ = Σ xⁿ/n!, 1/(1−x) = Σ xⁿ for |x| < 1.

Critical points occur where f'(x) = 0 or is undefined; second derivative test: f''(c) > 0 gives a local minimum, f''(c) < 0 gives a local maximum, f''(c) = 0 is inconclusive.

Definite integral applications: area between curves ∫(top − bottom) dx; volume by disks V = π∫[R(x)]² dx; average value of f on [a,b] = (1/(b−a))∫ₐᵇ f(x) dx.

Vector operations: dot product u·v = |u||v|cos θ (scalar; zero iff perpendicular); cross product u×v is a vector perpendicular to both with magnitude |u||v|sin θ, and its sign follows the right-hand rule.

Matrix facts: (AB)ᵀ = BᵀAᵀ; a square matrix is invertible iff det ≠ 0; det(AB) = det(A)det(B); for 2×2 [[a,b],[c,d]], det = ad − bc and inverse = (1/det)[[d,−b],[−c,a]].

Eigenvalues solve det(A − λI) = 0; the characteristic polynomial has degree n, so an n×n matrix has n eigenvalues counted with multiplicity over the complex numbers.

Probability axioms: 0 ≤ P(E) ≤ 1; P(Eᶜ) = 1 − P(E); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); independence means P(A ∩ B) = P(A)P(B), and conditional probability P(A|B) = P(A ∩ B)/P(B).

Counting: permutations P(n,r) = n!/(n−r)!; combinations C(n,r) = n!/(r!(n−r)!); the binomial theorem gives (a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ, and the number of subsets of an n-set is 2ⁿ.

Expected value and variance: E(X) = Σ x·P(x); Var(X) = E(X²) − [E(X)]²; for independent variables E(X+Y) = E(X)+E(Y) and Var(X+Y) = Var(X)+Var(Y).

Binomial distribution: P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ with mean np and variance np(1−p); normal approximation is reasonable when np ≥ 5 and n(1−p) ≥ 5.

Confidence interval for a mean with known σ: x̄ ± z*(σ/√n); the margin of error shrinks with √n, so quadrupling sample size halves it; a 95% interval uses z* ≈ 1.96.

Hypothesis testing logic: the p-value is the probability of data at least as extreme as observed, assuming H₀ is true; it is not the probability that H₀ is true; a Type I error rejects a true H₀ (probability α), a Type II error fails to reject a false H₀ (probability β).

Euclidean algorithm: gcd(a,b) = gcd(b, a mod b); if d = gcd(a,b) then there exist integers x, y with ax + by = d (Bézout); lcm(a,b)·gcd(a,b) = a·b.

Modular arithmetic: a ≡ b (mod n) means n divides (a − b); congruence is preserved under addition and multiplication; inverses modulo n exist iff gcd(a,n) = 1.

The integers mod n form a field iff n is prime; in ℤₚ every nonzero element has a multiplicative inverse, and Fermat's little theorem gives aᵖ⁻¹ ≡ 1 (mod p) for a not divisible by p.

A group is a set with an associative operation, an identity, and inverses; a ring adds a second operation distributing over the first; an integral domain has no zero divisors; a field is a commutative ring in which every nonzero element is invertible.

Polynomial division algorithm: for f, g with g ≠ 0 there exist unique q, r with f = gq + r and deg r < deg g; over a field, r = 0 iff g divides f; a polynomial of degree n has at most n roots over a field.

Rational root theorem: any rational root p/q (lowest terms) of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.

Number theory: every integer n > 1 factors uniquely into primes; there are infinitely many primes (Euclid); the prime counting function π(x) grows like x/ln x (Prime Number Theorem).

Order of an element g in a finite group: the least k > 0 with gᵏ = e; k divides the group order (Lagrange); a cyclic group of order n has φ(n) generators, where φ is Euler's totient.

Equivalence relations require reflexivity, symmetry, and transitivity; equivalence classes partition the set; a function is injective (one-to-one), surjective (onto), or bijective, and a bijection has an inverse.

Triangle congruence criteria: SSS, SAS, ASA, AAS, and HL for right triangles; SSA is ambiguous and AAA guarantees similarity, not congruence.

Pythagorean theorem a² + b² = c² for right triangles; the converse identifies right triangles; special right triangles: 45-45-90 has sides in ratio 1:1:√2, 30-60-90 in ratio 1:√3:2.

Circle facts: inscribed angle equals half its intercepted arc; an angle inscribed in a semicircle is 90°; a tangent is perpendicular to the radius at the point of tangency; central angle equals its arc measure.

Area and volume formulas: triangle ½bh; circle πr²; trapezoid ½(b₁+b₂)h; prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³ and surface area 4πr².

Similar figures: corresponding angles equal and sides proportional; perimeters scale by k, areas by k², and volumes by k³ when lengths scale by factor k.

Coordinate geometry: distance √[(x₂−x₁)² + (y₂−y₁)²]; midpoint ((x₁+x₂)/2, (y₁+y₂)/2); slope (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular slopes multiply to −1.

Conic sections from focus/directrix: a parabola is the set of points equidistant from a focus and a directrix; an ellipse has constant sum of distances to two foci; a hyperbola has constant absolute difference.

Transformations: rotations and translations are isometries preserving distance and angle; reflections reverse orientation; dilations preserve shape but scale lengths by the scale factor.

Polygon angle sums: interior angles of an n-gon sum to (n−2)·180°; exterior angles of any convex polygon sum to 360°; a regular n-gon has each interior angle (n−2)·180°/n.

Solving quadratics: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac gives two real roots when positive, one when zero, and a conjugate pair of complex roots when negative.

Function transformations: y = a·f(b(x − h)) + k shifts right by h, up by k, scales vertically by a, and horizontally by 1/b; even functions satisfy f(−x) = f(x), odd satisfy f(−x) = −f(x).

Exponential vs. linear growth: linear changes by a constant difference, exponential by a constant ratio; exponential y = abˣ has growth factor b per unit increase in x, and continuous growth uses y = Pe^{rt}.

Logarithm laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x; log_b b = 1 and log_b 1 = 0; change of base log_b x = ln x / ln b.

Rational functions: vertical asymptotes where the denominator is zero but the numerator is not; a horizontal asymptote depends on degrees (equal degrees → ratio of leading coefficients; numerator degree higher → none, or a slant asymptote).

Arithmetic sequence aₙ = a₁ + (n−1)d with sum Sₙ = n(a₁ + aₙ)/2; geometric sequence aₙ = a₁rⁿ⁻¹ with sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Inverse functions: f has an inverse only if it is one-to-one; graphs of f and f⁻¹ reflect across y = x; the domain of f⁻¹ equals the range of f; composition f(f⁻¹(x)) = x.

Absolute value and inequalities: |x| < a means −a < x < a; |x| > a means x < −a or x > a; multiplying or dividing an inequality by a negative number reverses the inequality sign.

Complex numbers: i² = −1; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; the modulus |a + bi| = √(a² + b²); division uses the conjugate, dividing by (c + di) after multiplying by (c − di).