Calculus

The concept of a limit in mathematics describes the value that a function approaches as its input approaches a certain value. It's a foundational idea in calculus, allowing us to understand how functions behave near particular points, even if they're undefined at those points. The derivative of a function, on the other hand, represents the rate at which the function is changing. When interpreted geometrically, the derivative at a point gives the gradient (or slope) of the tangent to the curve of the function at that point. In real-world applications, the derivative often represents a rate of change, such as how an object's velocity changes over time.

Key Formulae:

$$ \text{Limit:} \quad \lim_{{x \to a}} f(x) = L $$

$$ \text{Derivative:} \quad f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} $$

Example:

Consider the function \( f(x) = x^2 \). Let's find the derivative of this function at the point \( x = 2 \).

Using the definition of the derivative:

$$ f'(2) = \lim_{{h \to 0}} \frac{(2+h)^2 - 2^2}{h} \\ f'(2) = \lim_{{h \to 0}} \frac{4h + h^2}{h} \\ f'(2) = \lim_{{h \to 0}} (4 + h) \\ f'(2) = 4 $$

This means that the gradient of the tangent to the curve \( y = x^2 \) at the point \( x = 2 \) is 4.

If you use a TI-Nspire GDC there are instructions useful for this topic.

This video on differentiation is from Revision Village and is aimed at students taking the IB Maths Standard level course.