If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Mean value theorem

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.B (LO)
,
FUN‑1.B.1 (EK)

## Video transcript

let's see if we can give ourselves an intuitive understanding of the mean value theorem and as we'll see if you once you parse some of the mathematical lingo and notation it's actually a quite intuitive theorem and so let's just think about some function f so let's say I have some function f and we know a few things about this function we know that it is continuous continuous over continued us continued us over the closed interval between x equals a and x is equal to B and so when we put these brackets here that just means closed interval that means we're including so where I put a bracket here that means we're including the point a and if I put the bracket on the right hand side instead of a parenthesis that means that we are including the point B and continuous just means we don't have any gaps or jumps in the function over this closed interval now let's also assume that it's differentiable it's differentiable differentiable over the open interval between a and B so now we're saying well it's it's okay if it's not differentiable right at a if it's not differentiable right it being differentiable just means that there's a defined derivative that you can actually take the derivative at those points so it's differentiable over the open interval between a and B so those are the constraints we're going to put on ourselves for the mean value theorem and so let's just try to visualize this thing so this is my function that's the y-axis and this right over here is the x-axis and I'm going to see x-axis and let me draw my interval so that's a and then this is B right over here and so let's say our function looks something like this let's say it looks something draw an arbitrary function right over here let's say my function looks something like that so this point right over here the x value is a and the y-value is F of a F of a this point right over here the x value is B and the Y value of course is f of B F of B F of B so all the mean value theorem tells us is if we take the average rate of change over the interval that at some point the instantaneous rate of change at least at some point in this open interval the instantaneous change is going to be the same as the average change now what does that mean visually so let's calculate the average change the average change between point a and point B well that's going to be the slope of the secant line that's going to be the slope of the secant line so that's so this is the secant line so think about its slope all the mean value theorem tells us is that at some point in this interval the instant slope of the tangent line is going to be the same as the slope of the secant line and we can see just visually it looks like right over here the slope of the slope of the tangent line is it looks like the same as the slope of the secant line it also looks like the case right over here the slope of the tangent line is equal to the slope of the secant line and it makes intuitive sense at some point your instantaneous slope is going to be the same as the average slope now how would we write that mathematically well let's let's calculate let's calculate well let's calculate the average slope over this interval well the average slope of the over this interval is the average change the slope of the secant line is going to be our change in Y our change in Y right over here over our change in X over our change our change in X well what is our change in Y our change in Y is f of B minus F of a minus F of a and that's going to be over that is going to be over our change in x over B minus B minus a let me do that in that right color so let's just remind ourselves what's going on here so this right over here this is the graph of y is equal to f of X we're saying that the slope of the secant line or our average rate of change over the interval from A to B is our change in is our change in Y our change in Y that the Greek letter Delta is just shorthand for change in Y over our change in x over our change in X which of course is equal to this and the mean value theorem tells us that there exists so if if we know these two things about the function then there exists there exists some some x value in between a and B so in the open interval between a and B there exists some C there exists some C and we could say it's a member of the open interval between a and B between a and B or we could say some C such that a is less than such that a a is less than C which is less than B so some C in this interval so some see some C in in between it where the instantaneous rate of change at at that x value is the same as the average rate of change so there exists some C in this open interval where the average rate of change is equal to the instantaneous rate of change at that point that's all it's saying and as we saw on this diagram right over here this could be our C or this could be our C as well so nothing nothing really it looks in all-usa F is continuous over a B differentiable over over f is continuous over the closed interval differentiable over the open interval and you see all of this notation is like what does that telling us all it's saying is at some point in the interval the instantaneous rate of change is going to be the same as the average rate of change over the whole interval and the next video will try to give you a kind of a real-life example about when that makes sense