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 \(x\).
To do
- Slope from graph activity (Khan Academy)
More info
- Introduction to differentiation quick reference (mathcentre)
- Slope and Rate of Change video (Khan Academy)
- Formal definition of the derivative as a limit video (Khan Academy)
- Calculating slope of tangent line using derivative definition video (Khan Academy)