OSAT Advanced Mathematics (211) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Counterexample

A single specific case that satisfies a statement's hypotheses but violates its conclusion, proving that a universal claim is false.

Deductive reasoning

Reasoning from accepted premises or axioms to a conclusion that follows necessarily; if the premises are true and the logic valid, the conclusion must be true.

Inductive reasoning

Reasoning from observed patterns or specific examples to a general conjecture; the conclusion is probable rather than guaranteed and requires proof before acceptance.

Dimensional analysis

Checking or converting units by treating them as algebraic quantities, multiplying by conversion ratios equal to one so unwanted units cancel and the result carries correct units.

Rational number

A number expressible as p/q where p and q are integers and q is nonzero; its decimal expansion terminates or eventually repeats.

Fundamental Theorem of Arithmetic

Every integer greater than 1 factors uniquely into primes up to the order of the factors; this guarantees a well-defined prime factorization.

Complex conjugate

For z = a + bi, the conjugate is a − bi; multiplying z by its conjugate gives the real number a² + b², equal to |z|².

GCD-LCM product rule

For positive integers a and b, gcd(a,b) · lcm(a,b) = a · b, so either value can be found once the other is known.

Quadratic discriminant

For ax² + bx + c = 0, the value b² − 4ac; positive gives two real roots, zero one repeated real root, negative two complex conjugate roots.

Remainder theorem

Dividing polynomial p(x) by x − c leaves remainder p(c); consequently p(c) = 0 exactly when x − c is a factor.

Rational root theorem

Any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.

Inverse function condition

A function has an inverse function exactly when it is one-to-one; restricting the domain, as with sine to [−π/2, π/2], can create an inverse.

Similarity area ratio

If two similar figures have linear scale factor k, their areas compare as k² and their volumes as k³; perimeter compares as k.

Inscribed angle theorem

An angle inscribed in a circle measures half the central angle that subtends the same arc, so angles subtending a diameter are right angles.

Sphere volume

The volume of a sphere of radius r is (4/3)πr³, and its surface area is 4πr².

Polygon interior angle sum

A convex polygon with n sides has interior angles summing to (n − 2)·180°; a regular polygon therefore has each interior angle equal to that sum divided by n.

Law of Cosines

For a triangle with sides a, b, c and angle C opposite side c, c² = a² + b² − 2ab·cos C; it handles SAS and SSS cases.

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ; dividing by cos²θ or sin²θ yields the tangent and cotangent versions.

Radian-degree conversion

Multiply degrees by π/180 to obtain radians, or radians by 180/π to obtain degrees; a full circle is 360° or 2π radians.

Ambiguous case SSA

Given two sides and a non-included angle, zero, one, or two triangles may exist; check whether the altitude and angle type permit each solution.

Fundamental Theorem of Calculus

If f is continuous on [a,b] and F is any antiderivative, the definite integral of f from a to b equals F(b) − F(a).

L'Hôpital's rule

For limits producing 0/0 or ∞/∞, the limit of f/g equals the limit of f′/g′, provided that latter limit exists or is infinite.

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) satisfies f′(c) = (f(b) − f(a))/(b − a).

Second derivative test

At a critical point where f′ = 0, f″ > 0 indicates a local minimum and f″ < 0 a local maximum; f″ = 0 is inconclusive.

Bayes' theorem

P(A|B) = P(B|A)·P(A) / P(B), updating a prior probability P(A) using the likelihood P(B|A) and the total probability of B.

Central Limit Theorem

For independent, identically distributed variables with finite variance, the sampling distribution of the mean approaches normal as sample size grows, regardless of population shape.

Binomial mean and variance

A binomial count with n trials and success probability p has mean np and variance np(1 − p); its standard deviation is the square root of that variance.

Type I error

Rejecting a true null hypothesis; its probability equals the significance level α, while a Type II error is failing to reject a false null.

Logistic growth model

Population growth that starts nearly exponential but slows as the population approaches a carrying capacity K, modeled by dP/dt = rP(1 − P/K).

Least squares regression

The line chosen to minimize the sum of squared vertical residuals between observed values and predicted values; it gives the best fit in that squared-error sense.

Extrapolation

Predicting outside the range of observed explanatory-variable values; such predictions rely on the model holding beyond the data and are therefore risky.

Coefficient of determination

R², the proportion of variability in the response variable explained by the regression model; it ranges from 0 to 1.