Basque PAU Mathematics II flashcards

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

Determinant of a 2×2 matrix

For A = [[a, b], [c, d]], det(A) = ad − bc. The matrix is invertible exactly when this value is non-zero.

Rouché–Frobenius compatibility criterion

A linear system is compatible if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. If both equal the number of unknowns, the solution is unique.

Inverse of a 2×2 matrix

A⁻¹ = (1/det A)·[[d, −b], [−c, a]]. Swap the diagonal entries, change the sign of the off-diagonal ones and divide by the determinant.

Homogeneous system trivial solution

Every system Ax = 0 admits the zero solution. It has infinitely many solutions only when det(A) = 0, giving non-trivial solutions.

Dot product formula

u·v = u1v1 + u2v2 + u3v3 = |u||v|cos θ. The result is a scalar, and it equals zero when the vectors are perpendicular.

Cross product geometric meaning

u×v is a vector perpendicular to both whose modulus |u||v|sin θ equals the area of the parallelogram they span. Its direction follows the right-hand rule.

Scalar triple product

[u,v,w] = u·(v×w) equals the signed volume of the parallelepiped formed by the three vectors. It is zero when they are coplanar.

Distance point to plane

For P(x0,y0,z0) and plane Ax + By + Cz + D = 0, the distance is |Ax0 + By0 + Cz0 + D| divided by √(A² + B² + C²).

Angle between line and plane

If d is the line direction and n the plane normal, sin α = |d·n|/(|d||n|). The complementary angle is used because n is perpendicular to the plane.

L'Hôpital rule conditions

It applies only to indeterminate quotients 0/0 or ∞/∞. Differentiate numerator and denominator separately and repeat if the new quotient is still indeterminate.

Differentiability implies continuity

If f is differentiable at a point then it is continuous there. The converse fails: the absolute value function is continuous but not differentiable at zero.

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, a horizontal tangent.

Mean value theorem

Under continuity on [a,b] and differentiability on (a,b), there is a point c where f′(c) equals the average rate of change (f(b) − f(a))/(b − a).

Oblique asymptote calculation

Compute m = lim f(x)/x and n = lim (f(x) − mx) as x → ±∞. If both limits are finite, y = mx + n is an asymptote.

Fundamental theorem of calculus

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

Area between two curves

If f(x) ≥ g(x) on [a,b], the enclosed area is ∫(f − g) dx from a to b. Find intersection points first and split where the order changes.

Volume of revolution about x-axis

Rotating y = f(x) between a and b around the x-axis gives V = π∫_a^b [f(x)]² dx, the sum of disc areas.

Integration by parts

∫u dv = uv − ∫v du. Choose u by LIATE order so that the remaining integral is simpler than the original one.

Improper integral convergence

An integral over an infinite interval is defined as a limit of proper integrals. It converges when that limit is a finite real number.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. It restricts the sample space to B and rescales probabilities accordingly.

Independence of two events

A and B are independent when P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A). Dependence means the occurrence of one changes the other's probability.

Bayes' theorem

P(Ai|B) = P(Ai)P(B|Ai) divided by Σ P(Aj)P(B|Aj). It reverses the conditioning using prior probabilities over a partition.

Binomial distribution parameters

For B(n,p), P(X = k) = C(n,k)p^k(1−p)^{n−k}. The mean is np and the variance is np(1−p), with independent trials.

Standardizing a normal variable

Transform X ~ N(μ,σ) into z = (x − μ)/σ, which follows N(0,1). Then read probabilities from the standard normal table.

Normal approximation continuity correction

When approximating a discrete binomial by a normal, replace P(X = k) by P(k − 0.5 ≤ X ≤ k + 0.5) to correct for discreteness.

Confidence interval for the mean

With known σ, the interval is x̄ ± z_{α/2}·σ/√n. For 95% confidence z = 1.96, and the interval narrows as n grows.

Cramer's rule

For a square system with det(A) ≠ 0, each unknown equals det(Ai)/det(A), where Ai replaces column i with the constant vector.

Rank via non-zero minors

The rank is the largest order of a non-zero minor. It also equals the number of non-zero rows after Gaussian elimination.

Line–line relative position

Two lines are parallel if directions are proportional. Otherwise solve the system: a solution means they intersect, none means they are skew.

Plane through three points

For non-collinear A, B, C, compute n = (B − A) × (C − A), then write n·(r − A) = 0 as the plane equation.

Tangent line equation

At x = a the tangent is y − f(a) = f′(a)(x − a). The normal line has slope −1/f′(a) when f′(a) ≠ 0.

Critical point classification

Where f′(x) = 0, a sign change from + to − gives a maximum, − to + a minimum. If f″ ≠ 0, its sign decides directly.