Differentiate Between the Rate of Change at a Point and the Average Rate of Change Over an Interval
The rate of change of a function at a point is the limit of the average rate of change over a small interval containing the point.
The average rate of change over an interval is the total change of the function over the interval divided by the length of the interval.
For a function that is continuous at a point the two rates of change are equal.
When finding the rate of change over an interval it is important to first identify whether you are looking for the average rate of change or the rate of change at a point. These concepts are related but they are not the same.
The rate of change of a function at a point is the limit of the average rate of change over a small interval containing the point. In other words the rate of change at a point is what you would get if you took the average rate of change over an interval that got smaller and smaller until it just contained the one point.
The average rate of change over an interval is the total change of the function over the interval divided by the length of the interval. This is just the average value of the function over the interval.
For a function that is continuous at a point the two rates of change are equal. This makes sense because if the function is continuous then it doesn’t matter how small or large the interval is the average rate of change should be the same.
It is important to note that when finding the rate of change over an interval you are looking for the average rate of change. This is different from the rate of change at a point which is the limiting case of the average rate of change. Keep these concepts straight in your mind when working with rates of change.
How do you find the rate of change over an interval?
There are a few steps in finding the rate of change over an interval.
The first step is to find the difference in the y-values of the two points.
Next take the difference of the x-values of the two points.
Finally divide the difference in the y-values by the difference in the x-values to find the rate of change.
Why is it important to find the rate of change over an interval?
The rate of change is important because it can give you information about how a function is behaving.
For example if the rate of change is positive then the function is increasing.
If the rate of change is negative then the function is decreasing.
Where can you use the rate of change?
The rate of change can be used in a variety of situations.
For example it can be used to find how fast something is moving.
It can also be used to find how fast a function is increasing or decreasing.
What does a rate of change of 0 mean?
A rate of change of 0 means that the function is not changing.
This means that the y-value is the same at both points.
What does a rate of change of infinity mean?
A rate of change of infinity means that the function is increasing or decreasing at an incredibly fast rate.
What does a rate of change of -infinity mean?
A rate of change of -infinity means that the function is decreasing at an incredibly fast rate.
What is the rate of change of a constant function?
The rate of change of a constant function is 0.
What is the rate of change of a linear function?
The rate of change of a linear function is constant.
This means that the rate of change is the same at every point.
What is the rate of change of a quadratic function?
The rate of change of a quadratic function is not constant.
This means that the rate of change can be different at different points.
What is the rate of change of an exponential function?
The rate of change of an exponential function is also not constant.
What is the rate of change of a logarithmic function?
The rate of change of a logarithmic function is 0.
Is the rate of change always positive?
No the rate of change can be positive negative or 0.
What does a positive rate of change mean?
A positive rate of change means that the function is increasing.
What does a negative rate of change mean?
A negative rate of change means that the function is decreasing.
How can you find the rate of change of a function if you only have two points?
To find the rate of change of a function with two points you must take the difference of the y-values and the x-values of the two points.
Then divide the difference of the y-values by the difference of the x-values.
