PAU Applied Math ACS II (La Rioja) flashcards

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

Matrix product order rule

If A has order m×n and B has order n×p, then AB is defined and has order m×p. The inner dimensions must match; the product is generally not commutative.

Determinant of 2×2 matrix

For A = [[a,b],[c,d]], det(A) = ad − bc. A is invertible exactly when this number is nonzero; otherwise the matrix is singular.

Inverse of 2×2 matrix

For A = [[a,b],[c,d]] with ad−bc ≠ 0, A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]]. Swapping diagonal entries and negating off-diagonal entries gives the adjugate.

Rouché–Fröbenius consistency criterion

A linear system AX = B is consistent if and only if rank(A) equals rank of the augmented matrix (A|B). If this common rank is less than the number of unknowns, solutions are infinite.

Cramer's rule

When det(A) ≠ 0, each unknown is xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with column i replaced by the constant vector B.

Rank of a matrix

The rank is the number of nonzero rows in any row-echelon form, equivalently the order of the largest square submatrix with nonzero determinant. Row operations do not change it.

Nontrivial homogeneous solutions

A homogeneous system AX = 0 always has the trivial solution X = 0. It has infinitely many solutions exactly when rank(A) is smaller than the number of unknowns.

Feasible region of linear program

The set of points satisfying all constraints and non-negativity conditions. In two variables it is a convex polygon, possibly unbounded or empty.

Vertex optimality theorem

If a linear objective has a finite optimum over a feasible region with vertices, at least one optimal solution occurs at a vertex. Evaluate the objective at each vertex and compare.

Bounded versus unbounded region

A bounded feasible region always gives both maximum and minimum. An unbounded region may lack one of them if the objective increases without limit along an open direction.

Level line slope rule

For z = ax + by, level lines have slope −a/b. The optimal vertex is where this slope lies between the slopes of the two edges meeting at that vertex.

Redundant constraint

A constraint whose removal leaves the feasible region unchanged. It never determines an optimal vertex and can be ignored when solving the program.

Integer programming rounding caution

When variables must be integers, solve the continuous problem first, then test nearby integer points inside the region. The continuous optimum value is only a bound, not necessarily attainable.

Two-sided limit existence

lim(x→a) f(x) exists exactly when the left-hand and right-hand limits both exist and are equal. The value f(a), or its absence, does not affect the limit.

Notable limit sin x over x

As x tends to zero, sin(x)/x tends to 1. This underlies the derivatives of sine and cosine and is used to resolve 0/0 indeterminate forms.

Definition of continuity at a point

f is continuous at a when lim(x→a) f(x) = f(a). This requires f(a) to be defined, the limit to exist, and both values to coincide.

Derivative as a limit

f′(a) = lim(h→0)[f(a+h) − f(a)]/h, the slope of the tangent line. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b) and f(a) = f(b), then some c in (a,b) satisfies f′(c) = 0. The mean value theorem relaxes the equal-endpoint condition.

Second derivative test

At a critical point c with f′(c) = 0: f″(c) > 0 gives a local minimum, f″(c) < 0 gives a local maximum, and f″(c) = 0 is inconclusive.

Inflection point condition

A point where concavity changes, so f″ changes sign there. f″(c) = 0 alone is insufficient; the sign change must be verified on both sides.

Barro's rule for definite integrals

If F is any primitive of f, then the integral from a to b of f equals F(b) − F(a). Any primitive works because constants cancel in the difference.

Area between two curves

On an interval where f is above g, the enclosed area is the integral of f − g. Where the curves cross, split the interval so the upper curve is identified on each piece.

Integration by parts formula

The integral of u dv equals uv minus the integral of v du. Choose u by the LIATE order: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential.

Improper integral convergence test

The integral from 1 to infinity of 1/xᵖ converges exactly when p > 1. Convergence means the limit of the integral up to b is finite as b tends to infinity.

Volume of revolution about OX

Rotating y = f(x) between a and b about the x-axis gives volume equal to π times the integral of f(x) squared. This is the disc method.

Conditional probability formula

For P(B) > 0, P(A|B) = P(A∩B)/P(B). Events are independent exactly when P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).

Law of total probability

For a partition A₁,…,Aₙ, P(B) = Σ P(Aᵢ)P(B|Aᵢ). Bayes' theorem then reverses the conditioning using this denominator.

Mean of a discrete variable

The expectation is Σ xᵢp(xᵢ), a weighted average of values by their probabilities. Variance equals Σxᵢ²p(xᵢ) minus the square of the mean.

Binomial distribution parameters

For B(n,p), P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. The mean is np and the variance is np(1−p); trials are independent with constant success probability.

Standardizing a normal variable

If X is N(μ,σ), then Z = (X−μ)/σ is standard normal. Tables give P(Z ≤ z); convert back with X = μ + σZ.

Standard error of the mean

For samples of size n from a population with standard deviation σ, the sample mean has standard deviation σ/√n. This quantity is called the standard error.

Confidence interval for the mean

With known σ, the interval is x̄ ± z_(α/2)·σ/√n. At 95% confidence the critical value is 1.96; at 99% it is 2.576.

Sample size for a given error

To keep maximum error at E, take n = [z_(α/2)·σ/E]², rounding up to the next integer. A larger sample narrows the interval.