Worked example: average rate of change from equation

y equals 1/8 x to the third minus x squared. Over which interval does y of x have an average rate of change of 1/2? So let's go interval by interval and calculate the average rate of change. So first let's think about this interval right over here. x is between negative 2 and 2. So negative 2 is less than x, which is less than 2. So let's just think about what is the value of our function when x is equal to negative 2? So y of negative 2 is equal to 1/8 times negative 2 to the third power minus negative 2 squared, which is equal to-- let's see, negative 2 the third power is negative 8. Negative 8 divided by 8 is negative 1. Negative 2 squared is positive 4, but then you're going to have to subtract that, so it's minus 4. So this is equal to negative 5. And y of 2 is equal to 1/8 times 2 to the third power minus 2 squared. And that's going to be equal to 1/8 times 8 is 1 minus 4, which is equal to negative 3. So if you want to find your average rate of change, you want to figure out how much does the value of your function change, and divide that by how much your x has changed. So we could make a table here. x, y. When x is negative 2, y is negative 5. When x is positive 2, y is negative 3. So how much did your y change? Well, your y increased by 2, and your x increased by 4. And you could get these numbers-- you could just look at it that y increased from that point to that point, x increased from that point to that point. Or you could say hey, negative 3 minus negative 5 is positive 2. That's the difference between negative 3 and negative 5. If you said 2 minus negative 4, well that gives you, once again, the distance, or the difference. It would give you positive 4. But if you look at it here, it's clear. When y increased by 2-- Or we could say when x increased by 4, y increased by 2. So our average rate of change over this interval is going to be average rate of change of y with respect to x is going to be equal to, well, when x changed by 4, by positive 4, y changed by positive 2. So it's equal to 1/2. So it does look like the average rate of change over this interval right over here is 1/2. So we got lucky in this situation, our first choice-- and this is a multiple choice, so it's not a multiselect here. So our first one met our criteria. So we know that that's the answer. But let me do one more of these other ones to show you why that is not the answer. So let's find the average rate of change between that point and that point. So let's do another-- let me do it in another color. So I'll do this one in purple. So 0 is less than x, which is less than 4. And I'll just do the table right over here. x and y. So when x is 0, what is y? Well, it's going to be 1/8 times 0 minus 0, and y is just going to be 0. When x is 4, what is y? Well y is going to be-- let's see, it's going to be 1/8-- trying to do this in my head. 1/8 times 4 to the third. 4 to the third is 64. 1/8 of 64 is 8. It's going to be 8 minus 4 squared, which is 16. 8 minus 16 is negative 8. So in this one, x is increasing by 4, x has increased by 4, what happened to y? y has decreased by 8. So the average rate of change of y with respect to x here is y changed-- so I could write my change in y-- this Greek letter delta just literally is shorthand for change in. My change in y is negative 8 when my change in x is 4. So the average rate of change here is negative 2. It's negative because as x increased, your y decreased. And on average, for every 1 that x increased, y decreased by 2. That's where you get your negative 2 here. So this clearly, the average rate of change is not 1/2. So that confirms that that is not the answer. And we know that these other two, and I encourage you to try them out, will also give you an average rate of change of something other than positive 1/2.