Analysis: functions, derivatives and integrals
What this domain tests
Analysis is the heaviest block of the written Abitur paper. You must sketch and discuss function families (polynomial, exponential, rational, trigonometric), justify monotonicity and curvature with derivatives, and compute areas and volumes with integrals. Part (a) is usually technical calculus, part (b) an applied interpretation of the same function, part (c) a proof or a parameter discussion (Ortskurve, Scharparameter).
Rules and concepts
- Differentiation: product rule, quotient rule, and above all the chain rule, (f∘g)'(x) = f'(g(x))·g'(x); so (e^u)' = u'·e^u and (ln u)' = u'/u.
- Extrema: f'(x₀) = 0 is necessary. Sufficient: a sign change of f' at x₀, or f''(x₀) ≠ 0 with f'' > 0 giving a minimum.
- Wendepunkte: f''(x₀) = 0 plus a sign change of f''; f'''(x₀) ≠ 0 is sufficient.
- Hauptsatz: the integral of f from a to b equals F(b) − F(a) for any antiderivative F.
- Area between two graphs: integrate |f − g| over the interval bounded by the intersection abscissae; split where the order changes.
- Symmetry: f(−x) = f(x) gives axis symmetry, f(−x) = −f(x) point symmetry about the origin.
Worked example
f(x) = x²·e^(−x). Product and chain rule: f'(x) = 2x·e^(−x) − x²·e^(−x) = x(2 − x)e^(−x). Since e^(−x) > 0, the only critical points are x = 0 and x = 2. Differentiating again, f''(x) = (x² − 4x + 2)e^(−x). f''(0) = 2 > 0, so (0|0) is a local minimum; f''(2) = −2e^(−2) < 0, so (2|4e^(−2)) ≈ (2|0.54) is a local maximum. Setting f''(x) = 0 gives x = 2 ± √2, each with a sign change, hence two Wendepunkte. As x → +∞, f(x) → 0⁺ (x-axis as asymptote); as x → −∞, f(x) → +∞.
Traps that cost marks
- Dropping the inner derivative in e^(kx) or ln(kx) - the single most frequent error.
- Dividing by x without noting that x = 0 is a solution.
- Claiming a Wendepunkt from f''(x₀) = 0 alone; x⁴ at x = 0 is the counterexample.
- Integrating between the ends of a given interval instead of between the intersection points.
- Giving a bare number in a context task without unit or interpretation of the derivative as a rate of change.