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# What is calculus?

### The gradient of a function

• For the linear function, $$y = f(x) = mx+c$$, the gradient is represented by $$m$$, and is constant.
• The figure (below) shows how the gradient is calculated for a linear function.

For example, $$f(x) = 0.25(x+5)(x+2)(x+1)(x-1)(x-3)$$ is shown in the figure below. The average rate of change of the curve between two points $$x = -5$$ and $$x = -4$$, can be approximated by finding the gradient of the straight line connecting these two points.

The average rate of change, $$m$$ for the curve between the points $$(-5, 0)$$ and $$(-4, 52.5)$$ will be:
$m = \frac{\Delta f}{\Delta x} = \frac{f(x_{2}) - f(x_{1})}{x_{2}- x_{1}} = \frac{52.5 - 0}{-4-(-5)}=52.5$
Therefore, between $$-5$$ and $$-4$$ the curves changes at an average rate of $$52.5$$ units of $$y$$ for each unit of $$x$$.