Estonian Upper-Secondary Broad Mathematics Exam cheat sheet

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

State domain restrictions before simplifying; cancelled factors still exclude values.

Composition f(g(x)) is evaluated inside-out and is generally not commutative.

Inverses require one-to-one functions; domain and range swap.

Horizontal transformations act opposite to the sign or factor shown.

Log laws convert sums to products, enabling solution of log equations.

Always check candidate solutions against original domain restrictions.

Consistency requires rank A = rank [A|b]; uniqueness also needs full column rank.

det A = 0 means no unique solution, not necessarily no solution.

Trace = sum of eigenvalues; determinant = product of eigenvalues.

a×b is perpendicular to both vectors; its magnitude is the parallelogram area.

Distance from point to plane: |n·P − d| / |n|.

Median resists outliers; mean does not.

P(A ∪ B) subtracts the intersection to avoid double counting.

Conditional probability divides the joint probability by the condition's probability.

Combinations count selections; permutations count arrangements.

z = (x − μ)/σ places a value on the standard normal scale.

Random assignment, not random sampling alone, supports causal conclusions.

Similarity scales lengths by k, areas by k², volumes by k³.

Inscribed angle equals half the intercepted central arc.

Sine law requires a known side–opposite-angle pair; watch the ambiguous case.

Cosine law handles SAS and SSS configurations.

Arc length and sector area formulas require radians.

Pythagorean identity sin²θ + cos²θ = 1 underpins most simplifications.

AP sum: n/2 × (first + last) or n/2(2a + (n − 1)d).

GP sum to infinity exists only for |r| < 1 and equals a/(1 − r).

The general binomial term C(n, r)a^(n−r)b^r gives any coefficient directly.

Count the number of terms carefully when the index starts at zero.

A negative common ratio alternates signs, so check the parity of the power.