Lesson Category: Grade 12 Lessons

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

Notes for Inventory valuation.

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:

Number patterns, including arithmetic and geometric sequences and series

2. Sigma notation
3. Derivation and application of the formulae for the sum of arithmetic:
   3.1 𝑆𝑛 = 𝑛/2 [2𝑎 + (𝑛 − 1)𝑑];
   𝑆𝑛 = 𝑛/2 (𝑎 + 𝑙)

3. Derivation and application of the formulae for the sum of geometric series:

3.2 𝑆𝑛 = 𝑎(𝑟 𝑛−1) 𝑟−1 ; (𝑟 ≠ 1); and 3.3 𝑆𝑛 = 𝑎 1−𝑟 ; (−1 < 𝑟 < 1), (𝑟 ≠ 1)

Functions: Formal Definition , Inverse, exponential and logarithmic

1. Definition of a function.

2. General concept of the inverse of a
function and how the domain of the
function may need to be restricted (in
order to obtain a one-to-one function)
to ensure that the inverse is a function.

3. Determine and sketch graphs of the
inverses of the functions defined by
𝑦 = 𝑎𝑥 + 𝑞;

Focus on the following characteristics: domain and range intercepts with the axes,
turning points, minima, maxima, asymptotes (horizontal and vertical), shape and
symmetry, average gradient (average rate of change), intervals on which the function increases /decreases.