PAU Applied Math ACS II (Aragón) flashcards

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

Rank of a matrix

The order of the largest nonzero minor. It equals the number of nonzero rows after Gaussian elimination, and determines whether a linear system has zero, one, or infinitely many solutions.

Rouché–Capelli theorem

A system AX = B is compatible exactly when rank(A) equals rank(A|B). If both equal the number of unknowns the solution is unique; if smaller, there are infinitely many solutions.

Cramer's rule

For a square system with det(A) not zero, each unknown equals the determinant of A with its column replaced by B, divided by det(A). Gives a unique solution directly.

Inverse of a 2×2 matrix

For A = [[a, b], [c, d]], A⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]], provided ad − bc is nonzero. Swapping diagonal entries and negating off-diagonal entries is the rule.

Determinant product rule

For square matrices of the same order, det(AB) = det(A)·det(B). Also det(Aᵀ) = det(A) and det(kA) = kⁿ det(A) for order n. A matrix is singular when its determinant is zero.

Feasible region in linear programming

The set of points satisfying all constraints simultaneously, including non-negativity. It is a convex polygon or polyhedron, and any optimal linear objective is attained at one of its vertices.

Vertex method for linear programs

Find all vertices by solving pairs of binding constraint equations, evaluate the objective at each vertex, and select the largest or smallest value. This suffices because the optimum lies at a vertex.

Binding constraint

A constraint satisfied with equality at a given point of the feasible region. Optimal vertices of a two-variable linear program are intersections of at least two binding constraints, including possibly the axes.

Unbounded feasible region

A region not enclosed on all sides. The objective may still attain a maximum or minimum at a vertex, or it may grow without bound along an open direction, in which case no finite optimum exists.

Objective function level lines

Lines c₁x₁ + c₂x₂ = k, parallel for different k. Moving them in the direction of increasing k until the last contact with the feasible region locates the optimal vertex graphically.

L'Hôpital's rule

For indeterminate forms 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g' when this latter limit exists. Differentiate numerator and denominator separately, never as a quotient.

Limit sin x over x

As x tends to zero, sin x divided by x tends to 1. This basic trigonometric limit underlies the derivatives of sine and cosine and resolves many 0/0 indeterminations involving trigonometric functions.

Limit defining e

As x tends to infinity, (1 + 1/x)ˣ tends to e, and as x tends to zero, (1 + x)^(1/x) tends to e. Used to resolve indeterminations of type 1 raised to infinity.

Continuity at a point

Three conditions must hold: f(a) is defined, the two-sided limit of f at a exists, and that limit equals f(a). Differentiability at a point implies continuity there, but the converse fails.

Chain rule

The derivative of f(g(x)) is f'(g(x))·g'(x). For example, the derivative of sin(x²) is 2x·cos(x²). Essential for composite functions, implicit differentiation, and related rates.

Implicit differentiation

When y is defined implicitly by F(x, y) = 0, differentiate both sides with respect to x, treating y as a function of x, then solve the resulting equation for y'.

Fundamental theorem of calculus

If f is continuous on [a, b] and F is any antiderivative, then the definite integral of f from a to b equals F(b) − F(a). It links areas under curves with antiderivatives.

Integration by substitution

Set u = g(x), du = g'(x)dx, transforming ∫f(g(x))g'(x)dx into ∫f(u)du. For definite integrals, change the limits to g(a) and g(b) instead of back-substituting.

Integration by parts

∫u dv = uv − ∫v du. Choose u as the factor that simplifies when differentiated, often following the order logarithm, inverse trigonometric, algebraic, trigonometric, exponential.

Area between two curves

Compute the integral of the absolute difference |f(x) − g(x)| between the intersection abscissas. Split the interval wherever the curves cross so the upper function is identified on each piece.

Improper integral convergence

The integral of 1/xᵖ from 1 to infinity converges exactly when p > 1. The integral of 1/xᵖ from 0 to 1 converges exactly when p < 1.

Volume by disks

Rotating y = f(x) about the x-axis between a and b gives volume π∫[f(x)]²dx. Each cross-section perpendicular to the axis is a disk of radius f(x).

Conditional probability

P(A|B) = P(A ∩ B)/P(B), defined when P(B) > 0. It restricts the sample space to B and is the basis for independence, total probability, and Bayes' theorem.

Independent events

A and B are independent when P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A). Independence differs from mutual exclusivity: exclusive events with positive probability are dependent.

Bayes' theorem

P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) divided by the sum of P(Aⱼ)P(B|Aⱼ) over a partition. It reverses conditional probabilities, updating prior beliefs after observing evidence B.

Binomial distribution

For n independent trials with success probability p, P(X = k) = C(n, k)pᵏ(1 − p)^(n−k). Mean is np, variance np(1 − p). Models counts of successes in fixed numbers of trials.

Poisson distribution

P(X = k) = e^(−λ)λᵏ/k!, where λ is the mean number of events in a fixed interval. Mean and variance both equal λ. Models rare, independent events counted over time or space.

Standardizing a normal variable

If X is normal with mean μ and standard deviation σ, then z = (x − μ)/σ follows the standard normal. Tables give probabilities for z, which are then translated back to X.

Standard error of the mean

The standard deviation of the sample mean X̄ equals σ/√n. It decreases as sample size grows, and it appears in confidence intervals and test statistics for the population mean.

Confidence interval for a mean

With known σ, x̄ ± z·σ/√n, where z = 1.96 for 95% confidence. With unknown σ and small n, replace z by the t value with n − 1 degrees of freedom and σ by s.

Sample size for margin of error

Given a target margin E and known σ, the required sample size is n = (zσ/E)², always rounded up. Doubling precision requires roughly four times the sample size.

Type I and type II errors

Type I error rejects a true null hypothesis, with probability α, the significance level. Type II error fails to reject a false null hypothesis, with probability β; power equals 1 − β.