Grade 12 maths: curve sketching, optimisation and integration
Grade 12 uses differentiation to the full extent: you analyse functions systematically, solve optimisation problems from applications, and meet integration as the reverse of differentiation.
Reference: existing NRW curriculum mapping in Germany. Topics and sequence may vary by federal state, school type and classroom. This selection is not a complete curriculum.
Work through curve sketching in a fixed order: domain, derivatives, zeros, extrema, inflection points, behaviour at infinity. Optimisation always needs a constraint, otherwise there is no unique solution.
1. Analysing functions
The exponential function is its own derivative and grows faster than any power. In curve sketching the first derivative gives extrema and the second the curvature. Always list the zeros of the derivative and check each one.
2. Extremal problems
First write down the target function to be maximised or minimised. Use the constraint to express it in a single variable. Then examine the derivative and check the boundary values of the domain.
3. Integration
The indefinite integral is the set of all antiderivatives; the definite integral is the signed area between graph and axis. The fundamental theorem links the two: integrate using an antiderivative and substitute the limits.
Two questions to get started
Differentiate f(x) = e^x.
Answer and worked solution
f′(x) = e^x
The exponential function is its own derivative, so the expression stays unchanged.
Find the zeros of f(x) = x² − 4x.
Answer and worked solution
x = 0 and x = 4
Factor out x: x · (x − 4) = 0, so x = 0 or x = 4.
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