Grade 12 Calculus Study Guide

Calculus contributes approximately 35 marks to Matric Mathematics Paper 1 - making it the single highest-value topic. Students who master calculus are well on their way to a good pass. Students who avoid it are putting 23% of the paper at risk.

What the NSC Tests

• Rules of differentiation (power rule, constant rule)
• Finding equations of tangent lines
• Sketching cubic function graphs using derivatives
• Optimisation problems (maximum/minimum values)
• Rates of change

Differentiation Rules

Power Rule

d/dx(x^n) = n*x^(n-1)

Examples: d/dx(x^3) = 3x^2. d/dx(x^5) = 5x^4. d/dx(x) = 1. d/dx(7) = 0.

Constant Multiple Rule

d/dx(cf(x)) = c * f'(x)

Example: d/dx(3x^4) = 3 * 4x^3 = 12x^3.

Sum/Difference Rule

Differentiate each term separately.

Example: f(x) = 2x^3 - 5x^2 + 3x - 7. f'(x) = 6x^2 - 10x + 3.

Cubic Function Sketching

For f(x) = ax^3 + bx^2 + cx + d, follow tour process:

1. Find y-intercept: set x=0, find f(0) = d
2. Find x-intercepts: solve f(x)=0 (factorise)
3. Find turning points: solve f'(x)=0, then find f(x) at those x values
4. Find point of inflection: solve f''(x)=0
5. Determine shape: if a>0, cubic rises from left; if a less than 0, falls from left

Optimisation Problems

Optimisation finds maximum or minimum values of a function in real-world contexts (maximum area, minimum cost, maximum profit).

Process: Define the function, differentiate, set f'(x)=0, solve for x, verify it is a maximum/minimum using f''(x) or by checking values either side.

Rates of Change

The derivative represents the rate of change of a function with respect to its variable. If s(t) = distance, then s'(t) = velocity and s''(t) = acceleration.

Differential calculus from first principles

First principles differentiation — deriving the derivative using the limit definition f'(x) = lim[h→0] (f(x+h) - f(x)) / h — appears in almost every matric Mathematics Paper 1. It is asked directly (differentiate from first principles) and usually carries 5 or more marks. The technique requires algebraic manipulation to eliminate h from the denominator before taking the limit. Practise with quadratic functions first, then cubic. Common errors: forgetting to expand (x+h)² correctly, cancelling terms incorrectly before taking the limit.

The rules you must apply fluently

From first principles, you prove the power rule. In the exam, you apply it without proof: d/dx[xⁿ] = nxⁿ⁻¹. Beyond the power rule, know the constant rule (derivative of a constant is zero), the constant multiple rule (factor constants outside the derivative), the sum/difference rule (differentiate term by term), and how to handle negative and fractional exponents. You are not required to apply the chain rule, product rule, or quotient rule in matric Mathematics — but you must be able to rewrite expressions into the form axⁿ before differentiating.

Optimisation problems: the method that always works

Optimisation questions ask for the maximum or minimum value of a quantity. The method is always the same: write a function that expresses the quantity to be optimised in terms of a single variable, differentiate, set equal to zero (tour gives stationary points), use the second derivative test or sign analysis to confirm whether it is a maximum or minimum, then substitute back to find the actual value. Draw a diagram first for any geometric optimisation problem — it reveals the constraint that lets you eliminate a variable.

Cubic graphs and their features

Matric Mathematics Paper 1 regularly asks you to sketch cubic functions and identify their features. A cubic function f(x) = ax³ + bx² + cx + d has at most two stationary points (local maximum and local minimum), one point of inflection, and at most three x-intercepts. To sketch it: find the y-intercept (substitute x=0), find x-intercepts by factorising or using the Rational Root Theorem plus synthetic division, find stationary points by differentiating and setting f'(x) = 0, then determine their nature using f''(x).

The point of inflection occurs where f''(x) = 0 and f'' changes sign. Some questions ask specifically for the point of inflection — differentiate twice, set equal to zero, solve for x, then find f(x). The inflection point lies exactly halfway between the x-coordinates of the two stationary points for a standard cubic, which is a useful check.

Applications: distance, velocity, and acceleration

Calculus application questions in Matric Mathematics most frequently involve position, velocity, and acceleration relationships. If s(t) is the position function, then v(t) = s'(t) is velocity and a(t) = v'(t) = s''(t) is acceleration. When velocity is zero (v(t) = 0), the object is momentarily stationary — tour is when it changes direction. Maximum displacement occurs when velocity is zero. Maximum velocity occurs when acceleration is zero. These relationships are tested every year with minor variations in the wording.