You can approximate the instantaneous rate of change at a point using average rate of change. To do this, select two points that are very close to each other and near , so that the slope of the line between them is nearly the slope of the tangent at .
Approximation is performed because it is difficult to position the tangent accurately. It can often be too slack or too steep, and then the numbers become wrong. When the tangent is too slack, you get a lower value than the answer, and when the tangent is too steep you get a value too high for the answer you are looking for.
There is another, more accurate way to calculate the instantaneous growth compared to trying to work with the tangent line. By using the formula for average rate of change and choosing very close to the point where you will find the growth, , you’ll get an approximate value for the instantaneous rate of change. Here is an example.
Example 1
Look at . You want to find the instantaneous rate of change when by approximation
You use the average rate of change formula:
You’ll now choose (or another value that is very close to ). You then have to find and , and those are
You enter the values into the formula and get:
The exact instantaneous rate of change is . You can see by how close the answer here is, that approximation is a pretty good approach.
You can get an even better result if you choose an that is even closer to . For example, if you select , then the calculation works out like this:
You then enter the values into the formula and get:
So as you can see, the closer the -values are to each other, the better the approximation will be. As we saw before, the exact value for instantaneous growth for is , so this method works well if you use two values that are very close to each other.