What is covered in Grade 12 Mathematics Calculus — Applications?

Cubic graphs: stationary points, concavity, point of inflection. Optimisation problems. Rates of change.

Is Grade 12 Calculus — Applications tested in Paper 1 or Paper 2?

Grade 12 Mathematics Calculus — Applications is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Calculus — Applications taught in Grade 12 Mathematics?

Calculus — Applications is taught in Term 3 of Grade 12 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 12 Mathematics Calculus — Applications on the CAPS curriculum?

Yes. Calculus — Applications is part of the official CAPS Mathematics curriculum for Grade 12, Term 3 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 12 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 12 Mathematics

Grade 12 · Term 3Mathematics

Calculus — Applications

We apply differentiation to sketch cubic graphs (finding stationary points, inflection points), solve optimisation problems, and interpret derivatives in real-world contexts (rates of change).

Week 1

7.1 Cubic Graph Sketching & Optimisation

Find stationary points (where $f'(x)=0$) and classify using second derivative
Sketch cubic functions showing all key features
Solve optimisation problems using calculus
Interpret derivatives in context (rate of change)

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Real-World Connection

Engineers use calculus to find the MINIMUM cost or MAXIMUM strength of a design — the optimal solution. A box maker needs to find the dimensions that maximise volume from a fixed sheet of cardboard: this is a classic calculus optimisation problem.

Property / Rule

Stationary Points

At a stationary point, $f'(x)=0$ (the tangent is horizontal). To classify: if $f''(x)>0$ , it is a LOCAL MINIMUM; if $f''(x)<0$ , it is a LOCAL MAXIMUM; if $f''(x)=0$ , further analysis is needed (possibly an inflection point).

f'(x)=0 \text{ at stationary pts};\quad f''(x)<0\Rightarrow\text{max};\quad f''(x)>0\Rightarrow\text{min}

Second Derivative

f''(x) = \frac{d^2y}{dx^2}

Differentiate $f'(x)$ again

Property / Rule

Inflection Point

A point where the concavity changes. For a cubic, there is always one inflection point where $f''(x)=0$ .

f''(x)=0 \text{ and sign of }f'' \text{ changes}\Rightarrow\text{inflection point}

Property / Rule

Cubic Sketching Checklist

1) Find $y$ -intercept (set $x=0$ ). 2) Find $x$ -intercepts (solve $f(x)=0$ ). 3) Find stationary points ( $f'(x)=0$ ). 4) Classify with $f''$ . 5) Determine end behaviour (leading coefficient).

a>0:\text{ rises right};\quad a<0:\text{ falls right}

Worked Examples

Worked Example

Sketch a cubic function

Problem

f(x)=2x^3-3x^2-12x+4

Worked Example

Optimisation — maximum volume

Problem

x

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge3 marks

Find the stationary points of

f(x)=x^3-6x

L1 · Knowledge2 marks

Classify the stationary point of

f(x)=x^2-4x+7

x=2

L2 · Routine Procedures3 marks

Find the inflection point of

f(x)=x^3-3x^2+1

L2 · Routine Procedures5 marks

For

f(x)=2x^3+3x^2-36x

, find stationary points and classify them.

L3 · Complex Procedures5 marks

A farmer has 200 m of fencing to enclose a rectangular plot against a wall (no fence needed on wall side). Find the dimensions for maximum area.

L3 · Complex Procedures4 marks

A cubic

f(x)=ax^3+bx^2

has stationary points at

x=0

and

x=4

. Find

a:b

L4 · Problem Solving5 marks

The volume of a spherical balloon is increasing at

200

cm³/s. Find the rate of change of radius when

r=10

cm.

L4 · Problem Solving4 marks

Show that the minimum sum of a positive number and its reciprocal is 2.

Probability