Without graphing, determine the average rate of change for the function f(x)=x^4 on the intervals below:

Question

1. -1 ≤ x ≤ 1
2. -2 ≤ x ≤ 2
3. -10 ≤ x ≤ 10 

What can you notice about the answers found for the intervals above? Why did that happen? If possible, graph the function and suggest why this behaviour occurs.

^ I found out that all of the intervals for the given function equal zero. Moreover, I’m just a bit confused on how to answer the questions above asking why this happens and why the behaviour occurs when graphed. Thank you.

Answer 1

You are right that all the intervals' average rate of change is 0. This occurs because, if we graph x⁴, we notice it's symmetrical about x=0.

In number 1, the rate of change from x= -1 to x= 0 is -1 (the slope). From 0 to 1 of the same graph, the slope is now 1. When we average it out, we get 0. Since the graph is symmetrical about x=0, as long as our domain is symmetrical (-10 to 10, -5 to 5), our average rate of change will always be zero. Of course, this changes if we have a domain of, say, -5 < x < 2. You can prove to yourself that the negative rate of change will win in this scenario.

Think of it like: I spent 10 dollars yesterday, and gained 10 dollars today. My average rate of income is 0. I've completely broken even.