Grade 11 maths: vectors, limits and differentiation
Grade 11 marks the start of upper secondary maths. Vectors describe positions and directions in space, and analysis leads through limits to the central tool of the final years: the derivative.
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.
A vector is not a line segment but a displacement; it has length and direction. For limits, what matters is behaviour near a point, not the value at it. The derivative is a limit of slopes, not a new arithmetic rule.
1. Vectors in space
Add vectors component by component and multiply them by a scalar. The magnitude of a vector is its length. Two vectors are parallel when one is a multiple of the other.
2. Limits and continuity
A limit describes the value a function approaches. A function is continuous at a point when the left-hand and right-hand limits agree and equal the function value. Discontinuities show up where the function is undefined.
3. The derivative
The derivative gives the instantaneous rate of change, that is the slope of the tangent. It arises as the limit of the difference quotient. The power rule differentiates polynomials term by term, keeping the factor.
Two questions to get started
Add a = (2; 3; 1) and b = (4; −1; 5).
Answer and worked solution
(6; 2; 6)
Add component by component: 2 + 4 = 6, 3 + (−1) = 2 and 1 + 5 = 6.
Find the limit of f(x) = 3x + 1 as x tends to 2.
Answer and worked solution
7
For a polynomial you may substitute directly: 3 · 2 + 1 = 7.
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