In mathematics the average rate of change of a function is the instantaneous rate of change of the function averaged over some interval. This interval can be a point a line a curve a plane or even space itself. The average rate of change over an interval is usually denoted by a Average Rate of Change symbol (∆) followed by the interval over which it is averaged. For example the average rate of change of a function f(x) over the interval from x = a to x = b is denoted ∆f/∆x and is defined as follows:
∆f/∆x = (f(b) – f(a))/(b – a)
The average rate of change of a function can be thought of as the slope of the function’s graph averaged over some interval. In other words it is the slope of a line that is tangent to the graph of the function at some point in the interval. The average rate of change of a function can be found by taking the derivative of the function.
The average rate of change of a function can be used to approximate the instantaneous rate of change of the function at some point in the interval. This is because the average rate of change is a good approximation of the instantaneous rate of change when the interval over which it is averaged is small. For example if we want to approximate the instantaneous rate of change of a function at x = c we can take the average rate of change of the function over the interval from x = c – Δx to x = c + Δx. This will give us an approximation of the instantaneous rate of change of the function at x = c.
The average rate of change of a function can also be used to find the equation of a tangent line to the graph of the function at some point in the interval. This is because the equation of a tangent line to a graph at a point is the equation of the line that has the same slope as the graph at that point. For example if we want to find the equation of the tangent line to the graph of a function at x = c we can take the average rate of change of the function over the interval from x = c – Δx to x = c + Δx and use this to find the equation of the tangent line.
The average rate of change of a function can also be used to find the equation of a normal line to the graph of the function at some point in the interval. This is because the equation of a normal line to a graph at a point is the equation of the line that is perpendicular to the graph at that point. For example if we want to find the equation of the normal line to the graph of a function at x = c we can take the average rate of change of the function over the interval from x = c – Δx to x = c + Δx and use this to find the equation of the normal line.
The average rate of change of a function can also be used to estimate the value of the function at some point in the interval. This is because the average rate of change is a good approximation of the instantaneous rate of change when the interval over which it is averaged is small. For example if we want to estimate the value of a function at x = c we can take the average rate of change of the function over the interval from x = c – Δx to x = c + Δx and use this to estimate the value of the function at x = c.
How do you find the average rate of change from a table?
You take the difference in y-values divided by the difference in x-values for any two points on the graph.
What is the average rate of change for the points (24) and (48)?
The average rate of change would be 2 since the y-values double when the x-values increase by 2.
What is the average rate of change for the points (12) and (26)?
The average rate of change would be 4 since the y-values increase by 4 when the x-values increase by 1.
What is the average rate of change for the points (21) and (44)?
The average rate of change would be 1.
5 since the y-values increase by 1.
5 when the x-values increase by 2.
What is the average rate of change for the points (34) and (68)?
The average rate of change would be 2 since the y-values double when the x-values increase by 3.
What is the average rate of change for the points (32) and (96)?
The average rate of change would be 1 since the y-values increase by 1 when the x-values increase by 3.
What is the average rate of change for the points (41) and (84)?
The average rate of change would be 1 since the y-values increase by 1 when the x-values increase by 4.
What is the average rate of change for the points (45) and (1220)?
The average rate of change would be 4 since the y-values increase by 4 when the x-values increase by 8.
What is the average rate of change for the points (54) and (108)?
The average rate of change would be 2 since the y-values double when the x-values increase by 5.
What is the average rate of change for the points (56) and (1518)?
The average rate of change would be 3 since the y-values increase by 3 when the x-values increase by 5.
What is the average rate of change for the points (65) and (1210)?
The average rate of change would be 2.
5 since the y-values increase by 2.
5 when the x-values increase by 6.
What is the average rate of change for the points (64) and (1812)?
The average rate of change would be 2 since the y-values double when the x-values increase by 6.
What is the average rate of change for the points (76) and (1412)?
The average rate of change would be 2 since the y-values double when the x-values increase by 7.
What is the average rate of change for the points (87) and (1614)?
The average rate of change would be 2 since the y-values double when the x-values increase by 8.
What is the average rate of change for the points (85) and (2420)?
The average rate of change would be 4 since the y-values increase by 4 when the x-values increase by 8.
