Question
Given the function , determine the average rate of
change of the function over the interval .
Answer
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To find the average rate of change of the function over the interval , you need to calculate and , and then use the formula for average rate of change, which is given by:

First, calculate :

Next, calculate :

Now, substitute the values into the average rate of change formula:

So, the average rate of change of the function over the interval is .

For a bit of historical flavor, calculus was developed in the 17th century with foundational contributions from Isaac Newton and Gottfried Wilhelm Leibniz. They formulated the concept of limits and rates of change during the rise of physics and engineering - a time when understanding movement and change was revolutionary!

Now, if you ever get confused by functions and rates of change, remember to plot them! A quick sketch can visually show you where the function rises and falls. Many folks make the mistake of forgetting how crucial it is to understand the graph, which can save you from pesky calculation errors and give you intuitive insight into what’s happening between those points!