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CCSS.Math:

we are asked what is the average rate of change of the function f and this function is f up here is the definition of it over the interval from negative 2 to 3 and it's a closed interval because they put these brackets around it instead of parenthesis so that means it includes both of these boundaries pause this video and see if you can work through that all right now let's work through it together so there's a couple of ways that we can conceptualize average rate of change of a function one way to think about it is it's our change in the value of our function divided by our change in X or it's our change in the value of our function per X on average so you could view this change in the value of function divided by your change in X if you say that Y is equal to f of X you could also express it as change in Y over change in X on average how much does a function change per unit change in X on average and we could do this with a table or we could try to conceptualize it visually but let's just do this one with a table and then we'll try to connect the dots a little bit with a visual so if we have X here and then if we have y is equal to f of X right over here when X is equal to negative 2 what is y going to be equal to or what is f of negative 2 well let's see F of so Y is equal to F of negative 2 which is going to be equal to negative 8 that's negative 2 to the third power minus 4 times negative 2 so that's minus negative 8 so that's plus 8 that equals 0 and then when X is equal to 3 I'm going to the right end of that interval well now Y is equal to F of 3 which is equal to 27 3 to the third power minus 4 times 3 minus 12 which is equal to 15 so what is our change in Y over our change in x over this interval well our Y went from 0 to 15 so we have a increase of 15 in Y and what was our change in X well we went from negative 2 to positive 3 so we had a plus 5 change in X so our change in X is plus 5 and so our rate of change of Y with respect to X or the rate of change of our function with respect to x over the interval is going to be equal to 3 if you wanted to think about this visually I could try to sketch this so this is the x-axis the y-axis and our function does something like this so at x equals negative 2 f of X is 0 and then it goes up and then it comes back down and then it does something like this it does something like this and it does and what's going before this and so the interval that we care about we're going from negative 2 to 3 which is right about there so that's x equals negative 2 to x equals 3 and so what we want to do at the left end of the interval our function is equal to 0 so we're at this point right over here I'll do this in a new color we're at this point right over there and at the right end of our function f of 3 is 15 so we are up here someplace let me connect the curve a little bit we are we are going to be up there and so when we're thinking about the average rate of change over this interval we're really thinking about the slope of the line that connects these two points so the line that connects these two points looks something looks something like this and we're just calculating what is our change in Y which is going to be this our change in Y and we see that the value of our function increased by 15 divided by our change in X so this right over here is our change in X which we see we went from negative 2 to 3 that's going to be equal to 5 so that's all we're doing when we're thinking about the average rate of change