Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Limits of functions

A limit describes what happens to the values of a function as the input approaches a particular value. The limit of a function exists and equals L when the function values get arbitrarily close to L as x approaches a specific value from both sides.

Average gradient (average rate of change)

The average gradient measures how much a function changes over a specific interval. It represents the slope of the straight line connecting two points on a curve.

Formula: $\text{Average gradient} = \frac{f(x + h) - f(x)}{h}$

This formula calculates the change in y-values divided by the change in x-values between two points that are h units apart.

Gradient at a point (instantaneous rate of change)

The gradient at a point (or instantaneous rate of change) measures how fast a function is changing at one specific point. This is found by taking the limit of the average gradient as the interval becomes infinitesimally small.

Derivative notation

There are several ways to write derivatives, all meaning the same thing:

Differentiating from first principles

First principles differentiation uses the fundamental definition of a derivative as a limit. This method provides the foundation for all differentiation rules.

Rules for differentiation

General power rule

For any function of the form $x^n$ where n is a real number and $n \neq 0$:

Derivative of a constant

The derivative of any constant is zero because constants don't change:

Formula: $\frac{d}{dx}[k] = 0$

Constant multiple rule

When a constant multiplies a function, the derivative equals the constant times the derivative of the function:

Formula: $\frac{d}{dx}[k \cdot f(x)] = k \cdot \frac{d}{dx}[f(x)]$

Sum and difference rules

The derivative of a sum equals the sum of the derivatives:

Formula: $\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

The derivative of a difference equals the difference of the derivatives:

Formula: $\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]$

Second derivatives

The second derivative is the derivative of the first derivative. It measures how the rate of change itself is changing.

Applications to sketching graphs

Stationary points occur where the gradient of the curve equals zero. At these points, the tangent line is horizontal.

Types of stationary points:

• Local maximum: highest point in the immediate area
• Local minimum: lowest point in the immediate area
• Point of inflection: where the curve changes concavity but may not be a maximum or minimum

The gradient at stationary points is zero, making the tangent line horizontal at these locations.

Optimisation problems

Optimisation involves finding maximum or minimum values of functions, which is crucial for real-world problem-solving.

This method helps solve practical problems like maximising profit, minimising cost, or finding optimal dimensions.