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### Course: Algebra (all content)>Unit 7

Lesson 12: Average rate of change

# Worked example: average rate of change from equation

Finding the interval where a function has an average rate of change of ½ given its equation. Created by Sal Khan.

## Want to join the conversation?

• Would the interval include -2 and 2?
• That's a good catch, Brian--the interval "-2<x<2" does not include the numbers -2 & 2 (it's what's called an open interval). However, the average value of the function is the same regardless.

Also, it's good to keep this in mind because this topic is especially important later on when you learn Calculus and the all-important Mean Value Theorem. Anyway, hope this helps!
• The video narrator just puts in −2 and 2 into the function to calculate the change.

But why? Isn't it an open interval? Aren't −2 and 2 outside the interval?
• Technically yes. However, for the purposes of computing the average, you can treat the otherwise open intervals as closed.
• How can we optimize our time in answering these type of questions (objective) without having to try out all the other options? (say, A, B, C, or D)
• There's no real way to do it besides trying out all the options. If you want to try and be faster you could look over the options and see if any of them look more likely and try that one first.
If you have a calculator that can use variables you may want to first work out f(x) for all the different end points of the intervals, keeping the function on your calculator, and then after this go back and calculate the slopes.
• Is there a quicker way to do this? Writing all 4 equations out is a real pain in the butt.
• Is there a way not to guess the numbers to solve for the possible value range for x if the average rate of change is an integer?
• i wanted to ask the same, can someone answer this pls? :)
• Is this for equations for rate of change?
• Isn't Sal committed a fallacy by including 2 and -2 into the average rate of change. Isn't it great if he includes the interval?
• Will it work to build the following equation?
1/8*x^3 = 1/2x (x= +-2 therefore between the points -2 and 2 the slope will be 1/2)
I was thinking, since the average rate of change equals to the slope of the function, if I find where the function`s slope is equal to 1/2x I could match it to the respective interval. The thing is that it`s not a linear functions with all these exponents, and I`m not sure if adressing to the 1/8x^3 as the slope in this formation is correct?
If this way is incorrect, is there any way of figuring the interval a function has a specific average slope(1/2x in this example) without being given specific interval?
I hope my question is clear.