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Grade 12 — Formula Reference Sheet

Grade 12 — CAPS (NSC) · 15 formulas

Grade 12 — CAPS (NSC)

Grade 12

Differential Calculus, Integral Calculus, Advanced Finance, Counting Principles, Statistics.

15 formulas · opens your print dialog

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Calculus

Derivative from First Principles

f^{'} (x) = lim [h \to 0] \frac{f ( x + h ) - f ( x )}{h}

The formal definition of the derivative — rate of change at a point.

When to use

When specifically asked to use "first principles" in a calculus question.

Example

f(x)=x²: f'(x) = lim(2x+h) = 2x as h→0

Calculus

Power Rule (Differentiation)

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Differentiates a power function — bring down the exponent, reduce it by 1.

When to use

Differentiating any polynomial or term of the form axⁿ.

Example

f(x) = 5x³ → f'(x) = 15x²

Calculus

Product Rule

\frac{d}{d x} (f \cdot g) = f^{'} g + f g^{'}

Differentiates the product of two functions.

When to use

When you need to differentiate two expressions multiplied together.

Example

h(x) = x²·sin x → h'(x) = 2x·sin x + x²·cos x

Calculus

Chain Rule

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Differentiates a composite function (function inside a function).

When to use

When you see a function applied to another function, e.g. (3x+1)⁵.

Example

h(x) = (2x+1)³ → h'(x) = 3(2x+1)²×2 = 6(2x+1)²

Calculus

Anti-differentiation (Power Rule)

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)

Reverses differentiation — increases the power by 1 and divides.

When to use

Finding the integral of any power of x.

Example

∫3x² dx = x³ + C

Calculus

Definite Integral

\int_{a}^{b} f (x) d x = F (b) - F (a)

Finds the signed area under a curve between x = a and x = b.

When to use

Area under a curve, displacement from velocity, net change problems.

Example

∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3

Sequences & Series

Infinite Geometric Series

S \infty = \frac{a}{1 - r} for ∣ r ∣ < 1

Sum of an infinite geometric series — only converges when |r| < 1.

When to use

Any infinite geometric series problem where the series converges.

Example

1 + 1/2 + 1/4 + … → S∞ = 1/(1 − 0.5) = 2

Sequences & Series

Sigma Notation

k = 1 \sum n k = \frac{n ( n + 1 )}{2}; k = 1 \sum n k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

Standard closed-form formulas for common sigma summations.

When to use

Evaluating sigma notation expressions without adding each term individually.

Example

Σₖ₌₁¹⁰⁰ k = 100×101/2 = 5050

Financial Mathematics

Future Value Annuity

F V = \frac{x [( 1 + i ) ^{n} - 1 ]}{i}

Total value of regular payments (e.g. monthly savings) at the end of the period.

When to use

Retirement fund, savings plan — equal regular payments, finding the final lump sum.

Example

R1000/month at 8% p.a. (0.667%/month) for 60 months: FV ≈ R73 476

Financial Mathematics

Present Value Annuity

P V = \frac{x [ 1 - ( 1 + i ) ^{- n} ]}{i}

Lump sum today that is equivalent to a series of future equal payments.

When to use

Home loan, car loan — finding the loan amount given monthly payments.

Example

R8000/month at 9% p.a. for 20 years: PV ≈ R900 000 loan

Counting Principles

Factorial

n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1; 0! = 1

The product of all positive integers up to n.

When to use

Counting arrangements, permutations, combinations.

Example

5! = 5×4×3×2×1 = 120

Counting Principles

Permutations

^{n} P_{r} = \frac{n !}{( n - r )!}

Number of ordered arrangements of r objects chosen from n distinct objects.

When to use

Counting arrangements where order matters.

Example

Arranging 3 from 5 books: ⁵P₃ = 60

Counting Principles

Combinations

^{n} C_{r} = \frac{n !}{r ! ( n - r )!}

Number of ways to choose r objects from n where order does not matter.

When to use

Choosing team members, lotto numbers — when order is irrelevant.

Example

Choosing 3 from 5: ⁵C₃ = 10

Statistics

Pearson Correlation Coefficient

r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}

Measures the strength and direction of a linear relationship between two variables.

When to use

Scatter plot analysis, bivariate data, regression problems.

Example

r close to +1: strong positive correlation; r close to −1: strong negative

Statistics

Least Squares Regression Line

\overset{y}{^} = a + b x where b = \frac{n \sum x y - \sum x \sum y}{n \sum x ^{2} - ( \sum x ) ^{2}} and a = \overset{y}{ˉ} - b \overset{x}{ˉ}

The line of best fit that minimises the sum of squared residuals.

When to use

Predicting a y-value from an x-value using bivariate data.

Example

Use ŷ = 2.3 + 1.4x to predict sales (ŷ) from advertising spend (x)

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Grade 11

First Year