OSAT Advanced Mathematics (211) cheat sheet

Pólya's stages: understand the problem, devise a plan, carry out the plan, look back and verify the result.

Deductive reasoning derives a conclusion that follows necessarily from premises; inductive reasoning generalizes probable patterns from specific cases.

One counterexample disproves a universal statement; examples alone can never prove a universal statement.

Direct proof assumes the hypotheses and derives the conclusion; contrapositive proof proves ¬Q → ¬P; contradiction assumes the negation of the conclusion.

Dimensional analysis treats units as algebraic quantities; multiplying by conversion ratios equal to 1 cancels units and exposes inconsistent formulas.

Necessary condition must hold but may not suffice; sufficient condition guarantees the result. 'P only if Q' translates as P → Q.

Estimation check: round each quantity to one significant figure, compute mentally, then compare the order of magnitude with the exact answer.

Working backward from the goal, introducing auxiliary variables, and solving a simpler related problem are standard non-routine strategies.

Precision is the repeatability of a measurement; accuracy is closeness to the true value; systematic error shifts all readings in one direction.

N ⊂ Z ⊂ Q ⊂ R ⊂ C; irrational numbers have nonterminating, nonrepeating decimal expansions.

A rational number is p/q with integers p, q and q ≠ 0; its decimal expansion terminates or eventually repeats.

Fundamental Theorem of Arithmetic: every integer greater than 1 has a unique prime factorization up to the order of factors.

gcd(a,b)·lcm(a,b) = ab for positive integers a and b; prime factorizations handle larger values efficiently.

Divisibility: by 3 or 9 if the digit sum is; by 4 if the last two digits are; by 8 if the last three are; by 11 if the alternating digit sum is.

Complex arithmetic: (a+bi)(c+di) = (ac − bd) + (ad + bc)i, with i² = −1; modulus |a+bi| = √(a²+b²).

Absolute value satisfies |xy| = |x||y| and the triangle inequality |x + y| ≤ |x| + |y|.

Percent change = (new − old)/old; successive percent changes multiply factors (1 + r₁)(1 + r₂) rather than adding the rates.

Rationalize denominators by multiplying by the conjugate: 1/(√a + √b) = (√a − √b)/(a − b).

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real, one repeated real, or two complex roots.

Vertex of y = ax² + bx + c lies at x = −b/(2a); vertex form is y = a(x − h)² + k.

Remainder Theorem: dividing p(x) by x − c leaves remainder p(c); Factor Theorem: p(c) = 0 exactly when x − c is a factor.

Rational Root Theorem: candidate roots ±p/q have p dividing the constant term and q dividing the leading coefficient.

Transformations: y = a·f(b(x − h)) + k scales vertically by a, horizontally by 1/b, shifts right h and up k.

Domain rules: even-index radicals require a nonnegative radicand, denominators must be nonzero, logarithm arguments must be positive.

A function has an inverse function exactly when it is one-to-one; the graph of f⁻¹ reflects f across the line y = x.

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

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

Pythagorean theorem a² + b² = c²; its converse identifies right triangles; common triples include 3-4-5 and 5-12-13.

Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x.

Area: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector ½r²θ when θ is in radians.

Volume: prism or cylinder Bh; pyramid or cone ⅓Bh; sphere (4/3)πr³; sphere surface area 4πr².

Similar figures with linear scale factor k have perimeters in ratio k, areas in ratio k², and volumes in ratio k³.

Polygon interior angles sum to (n − 2)·180°; exterior angles of any convex polygon sum to 360°.

Inscribed angle equals half the central angle on the same arc; an angle inscribed in a semicircle is a right angle; a tangent is perpendicular to the radius at the point of tangency.

Intersecting chords: AP·PB = CP·PD; secant-tangent: tangent² = external segment × whole secant.

Coordinate geometry: distance √((x₂−x₁)² + (y₂−y₁)²); midpoint is the average of coordinates; parallel slopes are equal, perpendicular slopes multiply to −1.

Unit circle: the point at angle θ is (cos θ, sin θ); sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.

180° = π radians; with θ in radians, arc length s = rθ and sector area A = ½r²θ.

Law of Sines: a/sin A = b/sin B = c/sin C = 2R; use for AAS, ASA, or SSA data.

Law of Cosines: c² = a² + b² − 2ab cos C; use for SAS or SSS data.

SSA ambiguous case: with h = b sin A, an acute A and h < a < b gives two triangles; a = h or a ≥ b gives one; a < h gives none.

Sum identities: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Double-angle identities: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.

For y = A sin(B(x − C)) + D: amplitude |A|, period 2π/|B|, phase shift C, vertical shift D.

Triangle area = ½ab sin C; inverse ranges are arcsin [−π/2, π/2], arccos [0, π], arctan (−π/2, π/2).

Derivative definition: f′(x) = lim_{h→0} [f(x+h) − f(x)]/h; differentiability implies continuity, but continuity does not imply differentiability.

Rules: (xⁿ)′ = n xⁿ⁻¹; (fg)′ = f′g + fg′; (f/g)′ = (f′g − fg′)/g²; [f(g(x))]′ = f′(g(x))g′(x).

Common derivatives: (sin x)′ = cos x; (cos x)′ = −sin x; (eˣ)′ = eˣ; (ln x)′ = 1/x.

Extreme Value Theorem: a continuous function on a closed interval [a,b] attains a maximum and minimum; check critical points and endpoints.

First derivative test: sign change from + to − gives a local maximum, − to + gives a local minimum; f″ > 0 means concave up.

Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), some c satisfies f′(c) = (f(b) − f(a))/(b − a).

L'Hôpital's rule: for indeterminate forms 0/0 or ∞/∞, lim f/g = lim f′/g′ when the latter limit exists or is infinite.

Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a) for any antiderivative F; d/dx ∫ₐˣ f(t) dt = f(x).

Integration by parts: ∫u dv = uv − ∫v du; substitution reverses the chain rule.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(Aᶜ) = 1 − P(A); independent events satisfy P(A ∩ B) = P(A)P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B); Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B).

Permutations nPr = n!/(n−r)! count ordered selections; combinations nCr = n!/(r!(n−r)!) count unordered selections.

Binomial distribution: P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p).

Normal empirical rule: about 68% of values lie within 1σ, 95% within 2σ, 99.7% within 3σ of the mean; z = (x − μ)/σ.

Central Limit Theorem: sample means approach a normal distribution with standard error σ/√n as sample size increases.

Large-sample confidence interval for a mean: x̄ ± z*·σ/√n; margin of error decreases as n grows.

Type I error rejects a true H₀ with probability α; Type II error fails to reject a false H₀ with probability β; power = 1 − β.

Correlation r measures only linear association and ranges from −1 to 1; correlation does not imply causation.

Modeling cycle: formulate the problem, solve the model, interpret results, validate against data, and refine assumptions.

Constant first differences suggest a linear model; constant second differences suggest a quadratic; constant ratios suggest an exponential model.

Linear y = mx + b has rate of change m; exponential y = a·bˣ has growth factor b, with b > 1 growth and 0 < b < 1 decay.

Logistic growth dP/dt = rP(1 − P/K) has carrying capacity K and an inflection point at P = K/2.

Least squares regression minimizes Σ(yᵢ − ŷᵢ)²; R² is the fraction of variance in the response explained by the model.

Interpolation inside the observed data range is safer than extrapolation beyond it, where the model's assumptions may fail.

Dimensional consistency validates a model; each parameter should carry meaningful units that combine correctly.

Linear programming: for a linear objective over linear constraints, an optimum occurs at a vertex of the feasible region.

Sensitivity analysis shows how changes in parameters or assumptions alter model outputs and conclusions.