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0112 formulas

Sets & inequalities

  • Union: A ∪ B = { x : x ∈ A or x ∈ B }
  • Intersection: A ∩ B = { x : x ∈ A and x ∈ B }
  • Complement (relative to universe U): Ac = { x ∈ U : x ∉ A }
  • Difference: A − B = { x : x ∈ A and x ∉ B }
  • Inclusion-exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
  • Subset count of an n-element set: 2n subsets; 2n − 1 proper subsets; 2n − 1 nonempty subsets
  • Empty set ∅ is a subset of every set
  • Quadratic inequality ax2+bx+c>0: solution keyed to root(s) and sign of a / discriminant Δ=b2−4ac (Δ>0 two roots, Δ=0 one, Δ<0 none)
  • Roots straddle k (one >k, one <k) for a>0: f(k) < 0
  • Rational inequality f(x)/g(x) ≤ 0: sign chart on numerator/denominator zeros (g(x) ≠ 0)
  • Absolute value: |x − a| < b ⇔ a − b < x < a + b (b>0)
  • Inequality multiplicative property: multiplying/dividing by a negative reverses the inequality

029 formulas

Functions (general, power/exp/log)

  • Function: y = f(x), domain D, range R; equal functions require same domain AND same rule
  • Domain conditions: denominator ≠ 0; even-root radicand ≥ 0; log argument > 0; base of non-integer power > 0; intersect all conditions
  • Even: f(−x) = f(x) (symmetric about y-axis); Odd: f(−x) = −f(x) (symmetric about origin); parity requires a domain symmetric about 0
  • Power function: y = xa (coefficient must be 1)
  • Exponential: y = ax (a>0, a≠1); natural base e
  • Logarithm: y = loga x, inverse of ax; ln = loge, lg = log10
  • Log rules: loga(MN)=loga M+loga N; loga(M/N)=loga M−loga N; loga(Mk)=k·loga M
  • Change of base: loga b = logc b / logc a; identities aloga N=N, loga(an)=n
  • Periodicity: f(x + T) = f(x)

037 formulas

Sequences and series

  • Arithmetic nth term: an = a1 + (n − 1)d
  • Arithmetic sum: Sn = n(a1 + an)/2 = n·a1 + n(n−1)d/2 (quadratic in n)
  • Geometric nth term: an = a1·rn−1
  • Geometric sum (r ≠ 1): Sn = a1(1 − rn)/(1 − r)
  • Geometric mean of a,b: G = ±√ab
  • Term from partial sum: an = Sn − Sn−1 for n ≥ 2, a1 = S1
  • Geometric-series partial-sum form Sn = A·rn + C requires A + C = 0

0411 formulas

Trigonometry

  • Degree-radian: radians = degrees × π/180
  • Pythagorean identity: sin2 x + cos2 x = 1; tan x = sin x / cos x
  • Periods: 2π for sin and cos, π for tan
  • Sum/difference: sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB; tan(A±B)=(tanA ± tanB)/(1 ∓ tanA tanB)
  • Double-angle: sin 2x = 2 sin x cos x; cos 2x = cos2 x − sin2 x = 2cos2 x − 1 = 1 − 2sin2 x
  • Half-angle and product-to-sum / sum-to-product formulas (standard set)
  • Triangle angle sum: A + B + C = π (so C = π − (A+B))
  • Sine Rule: a/sin A = b/sin B = c/sin C = 2R
  • Law of Cosines: c2 = a2 + b2 − 2ab cos C
  • Triangle area: S = (1/2)ab sin C
  • y = A sin(ωx + φ): amplitude A, period 2π/ω; horizontal stretch by factor k uses ω → ω/k

0511 formulas

Calculus (derivatives)

  • Definition: f'(x) = limh→0 [f(x+h) − f(x)]/h
  • Tangent-line slope at x0: k = f'(x0); tangent line y − f(x0) = f'(x0)(x − x0)
  • Power rule: d/dx xn = n xn−1; d/dx (1/x) = −1/x2; d/dx √x = 1/(2√x)
  • d/dx ex = ex; d/dx ax = ax ln a
  • d/dx ln x = 1/x; d/dx loga x = 1/(x ln a)
  • d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec2 x
  • Rules: (u±v)'=u'±v'; (uv)'=u'v+uv'; (u/v)'=(u'v−uv')/v2; (cu)'=cu'
  • Chain rule: [f(g(x))]' = f'(g(x))·g'(x)
  • Monotonicity: f'(x) > 0 increasing, f'(x) < 0 decreasing
  • Extrema: f'(x0)=0 and f' changes + to − ⇒ local max; − to + ⇒ local min; check endpoints for closed-interval max/min
  • Convexity (per CSCA convention): f''(x) < 0 on an interval defines a 'convex' function there

069 formulas

Analytic geometry (lines, circles, conics)

  • Slope from two points: k = (y2 − y1)/(x2 − x1); vertical line slope undefined; slope from inclination θ: k = tan θ
  • Line forms: point-slope y−y1=k(x−x1); slope-intercept y=kx+b; intercept x/a+y/b=1; general Ax+By+C=0
  • Parallel: k1=k2 (or A1B2=A2B1); Perpendicular: k1·k2=−1 (or A1A2+B1B2=0)
  • Point-to-line distance: d = |Ax0+By0+C| / √A2+B2
  • Circle standard: (x−a)2+(y−b)2=r2; general x2+y2+Dx+Ey+F=0, center (−D/2,−E/2), r=√D2+E2−4F/2 (need D2+E2−4F>0)
  • Ellipse: x2/a2 + y2/b2 = 1 (a>b>0); a2 = b2 + c2; |PF1|+|PF2| = 2a; e = c/a < 1
  • Hyperbola: x2/a2 − y2/b2 = 1; c2 = a2 + b2; asymptotes y = ±(b/a)x; ||PF1|−|PF2|| = 2a; e = c/a > 1
  • Parabola: y2 = 2px (focus (p/2,0), directrix x=−p/2); e = 1; distance to focus = distance to directrix
  • Common chord of two circles: subtract the two general equations to get the chord (radical) line

076 formulas

Vectors

  • Coordinates of AB: B − A (component-wise); magnitude |a| = √ax2 + ay2 (+ az2)
  • Dot product: a·b = |a||b| cos θ = ax bx + ay by (+ az bz)
  • Angle between vectors: cos θ = (a·b)/(|a||b|)
  • Perpendicular: a·b = 0; Parallel/collinear (2D): ax by − ay bx = 0 (components proportional)
  • Magnitude of a sum: |a+b|2 = |a|2 + 2(a·b) + |b|2
  • Projection vector of a onto b: ((a·b)/|b|2) b; scalar projection (a·b)/|b|

086 formulas

Complex numbers

  • z = a + bi, i2 = −1, a,b ∈ R; Re(z)=a, Im(z)=b
  • Modulus: |z| = √a2 + b2; Conjugate: z̄ = a − bi
  • Multiplication: (a1+b1 i)(a2+b2 i) = (a1a2 − b1b2) + (a1b2 + a2b1)i
  • Division: multiply numerator and denominator by the conjugate of the denominator
  • Polar form: z = r(cos θ + i sin θ); De Moivre: zn = rn(cos nθ + i sin nθ)
  • Rotation by ±π/2 about origin corresponds to multiplication by ±i

099 formulas

Solid geometry

  • 3D point P(x,y,z); 3D distance AB = √(x2−x1)2 + (y2−y1)2 + (z2−z1)2
  • Cuboid volume = lwh; surface = 2(lw+lh+wh); cube volume = s3, surface = 6s2
  • Cylinder: V = πr2 h, S = 2πr2 + 2πrh
  • Cone: V = (1/3)πr2 h, lateral S = πrl, total S = πrl + πr2, slant l = √r2+h2
  • Sphere: V = (4/3)πR3, S = 4πR2
  • Cone lateral unrolls to a sector: arc length 2πr = l·(central angle); base radius r = l·(angle)/(2π)
  • Tetrahedron from coordinates: V = (1/6)|scalar triple product|; pyramid V = (1/3)·base·height
  • Oblique (cavalier) projection area ratio: S' = (√2/4) S
  • Equilateral triangle of side s: area = (√3/4)s2

1015 formulas

Probability & statistics

  • Classical probability: P(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1; P(certain)=1, P(impossible)=0
  • Combinations C(n,k)=n!/(k!(n−k)!); permutations A(n,k)=n!/(n−k)!
  • Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)
  • Complement: P(Ac) = 1 − P(A); at-least-one: P(≥1) = 1 − P(none)
  • Conditional: P(A|B) = P(A∩B)/P(B); Multiplication: P(A∩B)=P(A)P(B|A); Independence: P(A∩B)=P(A)P(B)
  • Mean: x̄ = (Σ xi)/n; weighted/combined mean weights by group sizes
  • Variance: s2 = (1/n) Σ (xi − x̄)2; standard deviation s = √s2; range = xmax − xmin
  • Linear transform: E(aX+b)=aE(X)+b; D(aX+b)=a2 D(X)
  • Pooled variance of two groups: weight within-group variances by size AND add between-group (mean-shift) term
  • Normal PDF: f(x) = 1/(σ√2π) · e−(x−μ2/(2σ2)); curve symmetric about μ; total area = 1
  • Empirical rule: ≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
  • Standard normal: Z = (x−μ)/σ; Φ(x)+Φ(−x)=1
  • Estimated count in an interval = P(interval) × population size
  • Least-squares slope: b̂ = (Σ xi yi − n x̄ ȳ)/(Σ xi2 − n x̄2); intercept â = ȳ − b̂ x̄
  • Normal approximation to binomial (De Moivre-Laplace): X~B(n,p) ≈ N(np, np(1−p))