Saxony Abitur Mathematics cheat sheet

Power rule: for f(x) = x^n with n ≠ 0, f'(x) = n·x^(n−1); for n = 0 the derivative is 0. Combined with the sum rule and constant-factor rule this covers all polynomials.

Product rule: (u·v)' = u'·v + u·v'. Quotient rule: (u/v)' = (u'·v − u·v')/v², valid where v(x) ≠ 0. Chain rule: [u(v(x))]' = u'(v(x))·v'(x).

Derivatives to memorise: (e^x)' = e^x; (ln x)' = 1/x for x > 0; (sin x)' = cos x; (cos x)' = −sin x; (√x)' = 1/(2√x) for x > 0.

Monotonicity decision rule: on an interval, f' > 0 implies f strictly increasing, f' < 0 implies strictly decreasing. A sign change of f' at x₀ with f'(x₀) = 0 gives an extremum; f' changing from + to − gives a local maximum.

Second-derivative test: if f'(x₀) = 0 and f''(x₀) < 0, then x₀ is a local maximum; if f''(x₀) > 0, a local minimum; if f''(x₀) = 0 the test is inconclusive and a sign-change check of f' is required.

Curvature: f'' > 0 on an interval means f is convex (linksgekrümmt, opening upward); f'' < 0 means concave (rechtsgekrümmt). A point where f'' changes sign and f is continuous is a Wendepunkt (inflection point).

Tangent line at x₀: t(x) = f'(x₀)·(x − x₀) + f(x₀). Normal line slope is −1/f'(x₀) where f'(x₀) ≠ 0.

Main theorem of calculus: if f is continuous on [a; b] and F is any antiderivative, then ∫ₐᵇ f(x) dx = F(b) − F(a). Antiderivatives: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫e^x dx = e^x + C.

Area between graphs: for f(x) ≥ g(x) on [a; b], A = ∫ₐᵇ (f(x) − g(x)) dx. If the graphs cross inside [a; b], split the interval at each intersection and integrate |f − g| on each part.

Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos φ. Orthogonality test: a·b = 0 (with a, b ≠ 0). Angle: cos φ = (a·b)/(|a||b|).

Cross product: a×b = (a₂b₃ − a₃b₂; a₃b₁ − a₁b₃; a₁b₂ − a₂b₁). It is orthogonal to both a and b, and |a×b| = |a||b|sin φ equals the area of the parallelogram spanned by a and b.

Line forms: g: x = a + t·u (point–direction form) and the coordinate equations x = a₁ + t·u₁, y = a₂ + t·u₂, z = a₃ + t·u₃. Two lines are parallel iff their direction vectors are scalar multiples.

Plane forms: normal form n·(x − a) = 0, coordinate form n₁x + n₂y + n₃z = d with d = n·a. A plane is fixed by three non-collinear points; the normal is the cross product of two spanning vectors.

Distance from point P to plane E: n·x = d is |n·p − d|/|n|, where p is the position vector of P. Distance from point to line: use the perpendicular foot or |u×(p − a)|/|u|.

Positional relations: a line is parallel to a plane iff u·n = 0; it lies in the plane iff additionally the point satisfies the plane equation; otherwise there is exactly one intersection point, found by substituting the line into the plane equation.

Matrix–vector product: (A·x)ᵢ = Σⱼ aᵢⱼxⱼ. A linear system A·x = b is solved by Gaussian elimination (row reduction to staircase form); the solution set is empty, a single point, a line, or a plane depending on rank.

Inverse of a 2×2 matrix: for A = [[a, b], [c, d]] with det A = ad − bc ≠ 0, A⁻¹ = (1/det A)·[[d, −b], [−c, a]]. A matrix is invertible iff det A ≠ 0.

Cramer-style check: a square system A·x = b has a unique solution iff det A ≠ 0. If det A = 0, the system has either no solution or infinitely many; decide by comparing ranks of A and the augmented matrix.

Laplace model: for a finite sample space with equally likely outcomes, P(A) = |A|/|Ω|. Complement rule: P(Ā) = 1 − P(A). Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Independence: A and B are independent iff P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A) when P(B) > 0. Conditional probability: P(A|B) = P(A ∩ B)/P(B).

Binomial distribution: X ~ B(n; p) counts successes in n independent trials with success probability p; P(X = k) = C(n, k)·p^k·(1 − p)^(n−k). Mean μ = n·p, variance σ² = n·p·(1 − p), standard deviation σ = √(n·p·(1 − p)).

Cumulative binomial probabilities are read from tables or computed as P(X ≤ k) = Σⱼ₌₀ᵏ C(n, j)pʲ(1 − p)^(n−j). Complement identities: P(X ≥ k) = 1 − P(X ≤ k − 1) and P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1).

Expected value of a discrete variable: E(X) = Σ xᵢ·P(X = xᵢ). Variance: Var(X) = E(X²) − (E(X))². For constants: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).

Normal approximation rule of thumb: for B(n; p) with σ = √(n·p·(1 − p)) > 3, approximate by a normal distribution with μ = n·p and σ as above, using continuity correction (e.g. P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ)).

Standard normal: Z = (X − μ)/σ has mean 0 and standard deviation 1. Φ(−z) = 1 − Φ(z). For a symmetric interval around μ: P(μ − zσ ≤ X ≤ μ + zσ) = 2Φ(z) − 1.

Hypothesis testing decision rule: choose a significance level α, compute the probability of the observed or more extreme result under H₀, and reject H₀ if that probability is ≤ α. A one-sided test uses P(X ≥ k) or P(X ≤ k); a two-sided test uses both tails.

Confidence interval for p in a binomial setting: p̂ ± z·√(p̂(1 − p̂)/n), where z is chosen so that the central probability equals the confidence level (z ≈ 1.96 at 95%).

Necessary vs sufficient: 'A implies B' means A is sufficient for B and B is necessary for A. The contrapositive (¬B implies ¬A) is logically equivalent to A implies B; the converse (B implies A) is not.

Proof by contradiction: assume the negation of the claim and derive a contradiction. Proof by complete induction: show the base case and the step 'if true for n then true for n + 1'.

Quantifiers: ∀ means 'for all', ∃ means 'there exists'. Negating a statement swaps the quantifier and negates the predicate: ¬(∀x: P(x)) is equivalent to ∃x: ¬P(x).

Modelling cycle: real situation → assumptions and simplification → mathematical model → computation → interpretation → validation against the situation, with iteration if the result is implausible.

When choosing a model, name the assumptions explicitly (e.g. linear growth, constant rate, independence) and state the domain of validity; an extrapolation beyond the fitted data is a claim requiring justification.

Rounding and significance: give results with the precision the context allows, state units, and note that a rounded intermediate result can change the final digit; report exact values where the problem asks for them.

Reading derivative and integral in context: f'(x₀) is the instantaneous rate of change at x₀ (units: unit of f per unit of x); ∫ₐᵇ f(x) dx is the accumulated total (units: unit of f times unit of x).

Counterexample rule: a single counterexample disproves a universal claim ('for all n, …'), but examples never prove it; a claim of existence needs only one explicit example.

Justification standard: a correct answer without reasoning earns partial credit only; cite the theorem, criterion, or computation that links the given data to the conclusion, and state the conclusion in the terms of the original problem.