Contact The Learning Centre

# What is calculus?

### What is Calculus?

Calculus is the study of change, with the basic focus being on:

1. Rate of change (Differential Calculus)
2. Accumulation (Integral Calculus)

### Notations for the derivative

• In functional notation, we can write a function $$y$$ in terms of $$x$$ as $$y=f(x)$$.
• We read $$y = f(x)$$ as $$y$$ is a function of $$x$$.
• There are a number of different notations for the derivative of a function, the two most common are: $$\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$$ or $$f’(x)$$.
• The average rate of change of a function is determined from the gradient of a secant between the points, $$(x, f(x))$$ and $$(x+h,f(x+h))$$ is:
$\mbox{Average rate of change} =\frac{\Delta f}{\Delta x}= \frac{f(x+h)-f(x)}{h}$
• Note the Greek symbol Delta, $$\Delta$$ represents change in, so $$\Delta x$$ reads as change in $$x$$.
• The instantaneous rate of change of a function is termed the derivative, which is the limit as $$h$$ (or step size) approaches $$0$$:
$f’(x) = \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$
• The derivative is also the gradient of the tangent line.