In this module, we will examine average rate of change and instantaneous rate of change, making connections to the slope of secants and tangents.

 We are interested in examining two types of rates of change.

The first is average rate of change which is measured over an interval.

The second is instantaneous rate of change which is measured at a particular instant.

A secant is a line which passes through a curve in at least two distinct points.

More than one secant can be drawn through a point on a curve.

A secant through a single point is not unique.

The slope of the secant between two points represents the average rate of change and can be found as follows:

The notation \(m_{\text{secant}}\) is read “\(m\) subscript secant” and is used to represent the slope of the secant.

For something like \(m_{PQ}\), we read “\(m\) subscript \(PQ\).”

This notation denotes the slope of a secant through \(PQ\).

A tangent is a line which touches a curve at a point.

The point is called a point of tangency. At the point of tangency, the tangent (line) does not cross the curve but it may or may not cross the curve at some other point.

On the diagram, the line is tangent to the curve at point \(P\) and crosses the curve at point \(Q\).

The line is not tangent to the curve at point \(Q\).

A tangent most resembles the curve near that point.

The slope of the tangent to a specific point on the curve represents the instantaneous rate of change of the curve at that point.

This idea will be pursued later in the module.

Consider the linear function \(y = 2x -2\).

A graph of \(y = 2x - 2\) is shown with three specific points \(P\,(0,-2)\), \(Q\,(3,4)\), and \(R\,(5,8)\) also plotted on the graph.

Again, the notation \(m_{\text{PQ}}\) is read “\(m\) subscript \(PQ\)” and is used to represent the slope of the secant through \(PQ\).

The slope of every line segment contained on the line \(y = 2x - 2\) is the same as the slope of the line.

In fact, the average rate of change for any two points on a linear function is constant.

The average rate of change for a linear function is the same as the slope of the line segment which, in turn, is the same as the slope of the linear function.

It follows that the instantaneous rate of change at any point on the linear function is also the same as the average rate of change between any two points on the line.

However, finding the average rate of change and the instantaneous rate of change of a curve presents a different challenge: the slope is not constant at every point.