Abitur Mathematics cheat sheet

Power rule: (x^n)' = n·x^(n−1) for all real n ≠ 0; combined with linearity (a·f + b·g)' = a·f' + b·g'.

Product rule (u·v)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule [f(g(x))]' = f'(g(x))·g'(x).

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

Tangent line at x₀: y = f(x₀) + f'(x₀)(x − x₀). Normal slope is −1/f'(x₀) where f'(x₀) ≠ 0.

Monotonicity: f' > 0 on an interval ⇒ f strictly increasing there; f' < 0 ⇒ strictly decreasing.

Extrema: necessary condition f'(x₀) = 0; sufficient sign change of f' (or f''(x₀) > 0 minimum, f''(x₀) < 0 maximum).

Curvature: f'' > 0 convex (linksgekrümmt), f'' < 0 concave; inflection point requires f'' = 0 with sign change of f''.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) with F' = f; area between curves = ∫ₐᵇ |f(x) − g(x)| dx.

Integration techniques: substitution ∫ f(g(x))·g'(x) dx = F(g(x)) + C; integration by parts ∫u v' = uv − ∫u' v.

Dot product a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos φ; a ⊥ b ⇔ a·b = 0.

Cross product a×b gives a normal vector; |a×b| equals the area of the parallelogram spanned by a and b.

Line: x = p + t·u. Plane in normal form: n·(x − p) = 0, equivalently n·x = d; coordinate form ax + by + cz = d.

Distance point–plane: d = |n·(q − p)| / |n| = |aq₁ + bq₂ + cq₃ − d| / √(a² + b² + c²).

Distance point–line: d = |(q − p) × u| / |u|. Skew lines: d = |(p₂ − p₁)·(u₁ × u₂)| / |u₁ × u₂|.

Relative position of line and plane: substitute the line into the plane equation; unique t ⇒ intersection, no solution ⇒ parallel, identity ⇒ line lies in plane.

Matrix–vector product: (A·x)ᵢ = Σⱼ aᵢⱼxⱼ; A·x = b is solvable iff b lies in the column space of A.

Determinant 2×2: det [[a,b],[c,d]] = ad − bc; det = 0 ⇔ rows/columns linearly dependent ⇔ matrix not invertible.

Gaussian elimination: row operations preserve the solution set; number of free parameters = number of variables − rank(A).

Laplace probability P(A) = |A|/|Ω|; complement P(Ā) = 1 − P(A); addition rule P(A∪B) = P(A) + P(B) − P(A∩B).

Independence: P(A∩B) = P(A)·P(B). Conditional probability P(A|B) = P(A∩B)/P(B), B ≠ 0.

Binomial: P(X = k) = C(n,k)·p^k·(1−p)^(n−k); E(X) = n·p; Var(X) = n·p·(1−p); σ = √(np(1−p)).

Cumulative binomial P(X ≤ k) is summed term by term; for large n use the normal approximation with μ = np, σ = √(np(1−p)) and continuity correction ±0.5.

Standardisation z = (x − μ)/σ; standard normal table gives Φ(z) = P(Z ≤ z), with Φ(−z) = 1 − Φ(z).

Normal quantiles: z₀.₉₀ ≈ 1.282, z₀.₉₅ ≈ 1.645, z₀.₉₇₅ ≈ 1.960, z₀.₉₉ ≈ 2.326 (one-sided upper tail).

One-sided binomial test: reject H₀: p ≤ p₀ at level α if P(X ≥ k | p₀) ≤ α (upper-tail test); mirror condition P(X ≤ k | p₀) ≤ α for the lower tail.

Two-sided test: reject if the two-sided p-value ≤ α, i.e. each tail probability ≤ α/2. The significance level α bounds the type-I error, not the type-II error.

Expected value of a discrete variable: E(X) = Σ xᵢ·P(X = xᵢ); variance Var(X) = E(X²) − (E(X))².

Model cycle: real situation → assumptions → mathematical model → solution → interpretation and validation against the real situation.

Exponential growth/decay: f(t) = a·e^(kt); k > 0 growth, k < 0 decay; doubling time T = ln 2 / k.

Logistic (bounded) growth: f(t) = S / (1 + c·e^(−kt)) with saturation S; inflection point at f = S/2.

Rates as derivatives: if a quantity is given by f(t), then f'(t) is the instantaneous rate of change and ∫ f'(t) dt over an interval gives the total change.

Optimisation: build the objective as a function of one variable using the constraint, then set the derivative to zero and verify with the second derivative or an interval check.

Recovering a quantity from a rate: total = ∫ₐᵇ r(t) dt; mean value of f on [a,b] is (1/(b−a))·∫ₐᵇ f(x) dx.

Linear regression: slope m = (Σ(xᵢ−x̄)(yᵢ−ȳ)) / Σ(xᵢ−x̄)²; correlation coefficient r ∈ [−1,1] measures linear fit strength.

Check model plausibility: units must be consistent, boundary values must match the data, and long-term behaviour must not contradict the situation.

Domain restrictions: ln x requires x > 0; √x requires x ≥ 0; denominators require ≠ 0; tan x undefined at odd multiples of π/2.

Limits at infinity: lim (x→±∞) of a rational function is determined by the highest powers; if degrees are equal the limit is the ratio of leading coefficients.

Asymptotes: vertical where the denominator vanishes without cancellation; oblique asymptote found by polynomial division.

Symmetry: f(−x) = f(x) is even (y-axis symmetry); f(−x) = −f(x) is odd (point symmetry about the origin).

Curve sketching steps: domain, intercepts with the axes, symmetry, limits and asymptotes, f' for monotonicity and extrema, f'' for curvature and inflection points.

Even/odd integrals: for odd f, ∫₋ₐᵃ f(x) dx = 0; for even f, ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx.

Volume of revolution about the x-axis: V = π·∫ₐᵇ (f(x))² dx.

Solve equations of the form e^(g(x)) = c by taking ln of both sides; solve a·e^(kx) + b = 0 by isolating the exponential term first.

Vector between two points: AB = b − a; magnitude |a| = √(a₁² + a₂² + a₃²); midpoint M = (a + b)/2.

Angle between vectors: cos φ = (a·b)/(|a||b|); angle between line and plane: sin φ = |u·n|/(|u||n|).

Plane through three points: form two direction vectors, take their cross product as the normal, then use n·(x − p) = 0.

Intersection of two planes: solve the two linear equations; the solution is a line with one free parameter.

Matrix multiplication is not commutative: A·B ≠ B·A in general; (A·B)⁻¹ = B⁻¹·A⁻¹.

Identity matrix I satisfies A·I = I·A = A; inverse exists iff det(A) ≠ 0; for 2×2, A⁻¹ = (1/det)·[[d,−b],[−c,a]].

A system A·x = b has a unique solution iff det(A) ≠ 0; otherwise zero or infinitely many solutions depending on consistency.

Orthogonal projection of a onto b: (a·b/|b|²)·b; the component of a perpendicular to b is a minus this projection.

Bernoulli experiment: exactly two outcomes; a binomial variable counts successes in n independent trials with constant p.

Binomial coefficient C(n,k) = n!/(k!(n−k)!); symmetry C(n,k) = C(n,n−k); Σₖ C(n,k) p^k (1−p)^(n−k) = 1.

At least one success: P(X ≥ 1) = 1 − P(X = 0) = 1 − (1−p)^n.

Tree diagrams: multiply along a path, add across paths; the law of total probability P(B) = Σ P(Aᵢ)·P(B|Aᵢ) over a partition.

Bayes' theorem: P(A|B) = P(B|A)·P(A) / P(B), useful for reversing conditional probabilities.

Normal distribution is symmetric about μ; P(μ − σ < X < μ + σ) ≈ 0.683, P(μ − 2σ < X < μ + 2σ) ≈ 0.954, P(μ − 3σ < X < μ + 3σ) ≈ 0.997.

Hypothesis test setup: H₀ and H₁ stated before data; test statistic computed; decision by comparing p-value with α or the statistic with a critical value.

Errors: type-I error = rejecting a true H₀ (probability ≤ α); type-II error = not rejecting a false H₀. Reducing α generally increases the type-II error for fixed n.

Proportionality: y = k·x; inverse proportionality y = k/x; identify the type from the ratio or product of data pairs.

Compound interest: K(t) = K₀·(1 + r)^t; continuous compounding K(t) = K₀·e^(rt); relate the two via r_cont = ln(1 + r).

Half-life: after time T, amount halves; T = ln 2 / |k| for f(t) = a·e^(kt).

Newton cooling / limited growth: rate of change proportional to the difference from an ambient value gives an exponential approach to that value.

Piecewise models: choose different formulas on subintervals; check continuity at the join points and differentiability if a smooth curve is required.

Sensitivity: a small change in a parameter changes f by approximately f'(p)·Δp; use this to judge whether a model is robust.

Compare models by residuals: the model with smaller sum of squared residuals fits better; never extrapolate far beyond the data range without justification.

When interpreting a definite integral in context, state the unit as (unit of f) × (unit of x) and describe it as an accumulated total.