Lesson Category: Grade 12 Lessons

Full explanation on this topic and how to answer questions in an exam.

Notes for Inventory valuation.

1. Revise earlier work on the necessary and sufficient conditions for polygons to be similar.

2. Prove (accepting results established in earlier grades):
   • That a line drawn parallel to one side of a triangle divides the other two
   sides proportionally (and the Midpoint Theorem as a special case of the converse of this theorem);
   • That equiangular triangles are similar;
   • That triangles with sides in proportion are similar; and the Pythagorean
   Theorem by similar triangles

1. Revise the following including grade 10 concepts:
   • The equation of a line through two given points;
   • The equation of a line through one point and parallel or perpendicular to
   a given line; and
   • The inclination (θ) of a line, where 𝑚 = tan 𝜃 is the gradient of the line (0° ≤ 𝜃 ≤ 180°)

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2. Apply the equation

that defines a circle with radius r and centre (a ; b)

3.Determine the equation of a tangent to a given circle.

Differential Calculus Including Polynomials

1. Factorise third-degree polynomials. Apply the Remainder and Factor
   Theorems to polynomials of degree at most 3 (no proofs required).
2. An intuitive understanding of the limit concept, in the context of
   approximating the rate of change or gradient of a function at a point

Patterns: Revise number patterns leading to those where there is a constant second difference between
consecutive terms and the general term is therefore quadratic.

Differential Calculus Including Polynomials

3. Use limits to define the derivative of a function f at any 𝑥 :

Generalise to find the derivative of f at any point x in the domain of f , i.e., define the derivative function f ‘(x) of the function f (x) . Understand intuitively that f ‘(a) is the gradient of the tangent to the
graph of f at the point with x -coordinate a.

4. Using the definition (first principle), determine the derivative, f ‘(x) where a, b
   and c are constants: