PAU Mathematics II (La Rioja) cheat sheet

AX = B gives X = A^(-1)B only if |A| != 0; for XA = B multiply on the right by A^(-1).

|kA| = k^n|A| for n x n matrices, and |A^(-1)| = 1/|A|.

rank(A) = rank(A|b) is necessary and sufficient for consistency.

Unique solution requires rank(A) = rank(A|b) = number of unknowns.

With rank r < n the solution has n - r free parameters.

A homogeneous system has non-trivial solutions iff rank(A) < n.

L'Hopital requires a 0/0 or infinity/infinity form; differentiate numerator and denominator separately.

sin x ~ x, 1 - cos x ~ x^2/2, ln(1+x) ~ x, e^x - 1 ~ x near 0.

Differentiability implies continuity, never the reverse.

Rolle needs continuity on [a,b], differentiability on (a,b) and f(a) = f(b).

Bolzano gives existence of a root; the sign of f' gives uniqueness.

Extrema come from f' = 0 or f' undefined, confirmed by a sign change of f'.

Definite integral equals F(b) - F(a); the constant of integration cancels.

Area always needs the intersection points and upper minus lower, split at crossings.

Volume about the x-axis: pi times the integral of f(x)^2; between curves, pi times the integral of f^2 - g^2.

Integration by parts: integral of u dv = uv - integral of v du.

Rational integrands are handled by partial fractions according to the type of factor.

Improper integrals are limits and converge only if the limit is finite.

[u,v,w] = 0 iff the three vectors are coplanar; |u x v| is a parallelogram area.

rank(u, v, PQ) = 3 means skew, 2 means intersecting, 1 means parallel or coincident.

Distance point-plane: |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2).

Distance point-line: |AP x u| / |u|; skew lines: |[u,v,AP]| / |u x v|.

Angle line-plane uses sin(t) = |u.n|/(|u||n|), never cos.

Always check the final plane or line with the points that generated it.