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# Mean value theorem (old)

## Video transcript

I've gotten several requests to explain or teach the mean value theorem so let's do that in this video so this is the mean value theorem mean value theorem and I have mixed feelings about the mean value theorem it's kind of neat but when you what what you'll see is kind of it might not be obvious to prove but the intuition behind it's pretty obvious and the reason why I have mixed feelings about it is that even though as you'll hopefully see the theorem as you'll hopefully see that the the intuition is pretty obvious but if they stick it in the math books and people are just trying to learn calculus and get to what matters and then they put the mean value theorem in there and they have all of this function notation and they have all of these words and it just confuses people so hopefully this video will clarify that a little bit and and I'm curious to see what you think of it so let's see what's what does the mean value theorem say let me draw some axes I'll do a visual explanation first I mean value theorem I think this calls for magenta so that's my x axis this is my y axis and let's say I have some function f of X some function f of X so let me draw my f of X Oh that's as good as any and this is some function f of X and I'm going to put a few conditions on f of X f of X has to be continuous continuous continuous and differentiable differentiable and I know a lot of you probably differentiable get intimidated when you hear these words it sounds like you know what a mathematician would say and it sounds very abstract all continuous means is that the curve is connected to itself as you go along it and in here the conditions are over a closed interval this is another very Matthew term you'll see so they'll often say on a closed interval from A to B all that means is an interval let's say a is a low point let's say this is a we don't know what number that is I could be minus 5 or who knows and let's say this is the eeee all right here let me make B right here let's say that's B so when people talk about a closed interval defined on a closed interval that means that the function needs to be defined at every number between a and B and the function needs to be defined at a and at B if they said over an open interval between a and B that means that it's only defined at every value between a and B but not necessarily at a and B so it has to be continuous differentiable and let's say it's defined it's defined over the closed interval and this is just the notation for it a B so that means it has to be defined at all of the X values from A to B including a and B if it was an open interval you would write it like this you would write a and B that means an interval for all the numbers between a and B but not including those so let's ignore that for now so back to the mean value theorem so you know hopefully what continuous means let me draw here a function that's not continuous so a function that's not continuously would look like this it would go like this and then would start up here go like that right so this would be example of a function let's say this you know st. Maxie's let me draw it in a different color if that was our Y no that's not a good if that was our y-axis and that was our x-axis just to give you a reference what I drew so if the function you know is continuous continuous continuous and then it jumps that that disconnect that would make this function discontinuous or it would not be a continuous function so that so a function has to be continuous and now what is differentiable mean differentiable means that at every point over the interval that we care about you have to be able to find a derivative that means you can take the derivative of it it's differentiable and what else does that mean well that means that if you were to graph the derivative of this function that it is also continuous and I'll let you think about that for a second and actually in this video I'm going I'm going to show you an example of a function that is continuous but not differentiable and because of that the mean value theorem breaks down but anyway let's get back to the mean value there most of the functions we deal with satisfy all three of these things most of unless you know you're doing limit problems and they try to make these things break down anyway back to the function so this function meets all of these requirements so all it says is if I were to take the average slope between point a and point B so what is the slope the average slope between point a and point B well slope is just rise over run right so what is it let me see if I can draw the average slope so the run would be this distance that'd be the run right and this would be the rise so this is the point right here that's the point a F of a over here this is the point B F of B so what's the average slope between a and B what's rise over run so what's the Rise what's what's this distance how much have we gone up from F of a to f of B well the rise over the rise will just be F of B this height F of B minus F of a F of B minus F of a and what's the run what's this distance what's just B minus a B minus a and if I were to draw a line that has that average slope it would look something like this we could make it go through those two points but it really doesn't have to so let me do it enough let me do it in a blue blue so that's the average slope between those two points right so what does the mean value theorem tell us it says if we if f of X is defined over this closed interval from A to B and f of X is continuous and it's differentiable that you can take the derivative any point that there must be there must be some point C there has to be some point C some point C where F prime of C is equal to this thing so is equal to F prime of C I didn't I shouldn't have written it here so what is that telling us so all that's telling us is if it's we're continuous differentiable defined over the closed interval that there's some point C oh and C has to be between a and B there's some point between a and B and it could be at one of the points but there's some point C where the derivative at C or the slope at C or the instantaneous slope at C is exactly equal to the average slope over that interval so what does that mean so we could look at it visually is there any point along this curve where the slope looks very similar to this to this average slope we calculated well sure let's see it looks like maybe this point right here just the way I drew it this is very inexact but that point looks like the slope you know I could say that the slope is something like that right there so we don't we don't know what analytically this function is but visually you can see that this at this point C the derivative so I just picked that point so this could be our point C and how do we say that well because f prime of C is this slope and it's equal to the average slope so F prime of C is this thing and it's going to be equal to the average slope over the whole thing and this curve actually probably has another point where the slope is equal to the average slope let's see this one looks like right right around there just the way I drew it looks like the slope there could look something like could be parallel as well these lines should be parallel the tangent line should be parallel so hopefully that makes a little sense to you another way to think about it is is that your average let's let's actually let me draw a graph just to make sure that we hit the point home let's draw my position as a function of time so this is something this will this will make it applicable to the real world so that's my x-axis or the time axis that's my position axis and this is going back to just you know our original intuition of what even a derivative is so this is time and I call this position or distance or it doesn't matter position and if I was moving at a constant velocity my position as a function of time would just be a straight line right and the velocity is actually your slope but let's say I had a varying velocity and in reality if you're driving a car you are always at a very variable velocity so let's say I started a standstill at time T equals zero and then and then I accelerate then I decelerate a little bit decelerate a little bit I keep decelerating and then I come to a standstill so my position stays still then I accelerate again decelerate accelerate etc all right so this could be you know I have a variable velocity and this could be my position as a function of time so all this says let's say that after this is time zero position zero let's say after one hour let's say that that is one hour that's time equals one hour let's say I have gone 60 miles 60 miles so what can you say you could say that my average velocity so velocity average it equals just change in distance divided by change in time so it equals 60 miles per hour miles per hour so what the mean value theorem says is okay your average velocity so you could almost view it as the average slope between this point and this point was 60 if your average velocity was 60 miles per hour there was some there was some point in time maybe more but there was at least one point in time where you were going exactly 60 miles per hour that makes sense right if you average 60 miles per hour maybe you're going 40 miles per hour some of the point but at some point you went 80 and in between you have to be going 60 miles per hour so let me see if I can draw that graphically so this is this slope is my average velocity and the way I drew it there's probably two points let's see probably right around here I was probably going 60 miles per hour the slope is probably 60 there the instantaneous velocity probably there as well so before I leave let's do this analytically just to just to I guess work with numbers and the reason why I have mixed feelings about the mean value theorem is it's useful later on if probably if you become a math major you'll you'll maybe use it to prove some theorems or maybe you'll prove it itself but if you're just applying calculus for the most part you're not going to be using the mean value theorem too much but anyway if you got to know it you got to know it and it tells you something else about the world so it's interesting that way so let's say we have the function f of X is equal to x squared minus 4x and the interval that I care about here is between is the closed interval so I'm including 2 from 2 to 4 now the mean value theorem tells us that if this function is defined on this interval and it is right we could put any number the domain of this is actually all real numbers I could put any number here so obviously it's going to depart be defined over this interval but so it's defined over the interval it's this is continuous this is differentiable you could take the derivative and the derivative is continuous so the mean value theorem should apply here so let's see what value of C is equal to the average slope between 2 & 4 so what's what's the average slope between 2 & 4 well it's going to be F of for so the change in the function f of 4 minus f of 2 divided by the change in X so 4 minus 2 so this is equal to the average slope so f of 4 is 16 minus 16 right so that's 0 let me make sure that 4 times 4 16 minus 4 times 4 is 16 right minus F of 2 F of 2 is 2 squared is 4 right and then minus 4 times 2 so minus 8 let me show that right 4 minus 8 right all of that over 2 and so this equals minus 4 so this equals 4 over 2 so the average slope from X is equal to 2 to X is equal to 4 is 2 and now the mean value theorem tells us that there must be some point that's between these two maybe including one of those where the slope at that point is exactly equal to 2 let's figure out what point that is that's C let's take the derivative because the derivative at C is going to be equal to 2 so we just take the derivative so let's say f prime of X is equal to 2x minus 4 and we want to figure out at what x value does this equal to so we say 2x minus 4 is equal to 2 where does the slope equal to and you get 2x is equal to 6 X is equal to 3 so that X is equal to 3 the derivative is exactly equal to the average slope and let me see if I can let me get the graphing calculator here let me see if what I can do okay so here's the graph of x squared minus 4x let me see if I can make it a little bit bigger x squared minus 4x the interval that we care about is from here from here to here so the average slope the average slope over that interval was 2 so if we were to draw the slope it was like that the slope would look like that and the at the point three the slope is exactly two so let me actually draw it this isn't too hard to draw for myself let me see so if that's the x-axis that's I want that graph out of the way that's the y-axis so the graph goes to the point zero zero I want to draw this as neatly as possible nope that's not neat so the graph goes something like this dips up and then it goes like that and then I actually keeps going straight up like that it's a parabola this is the point for the point two is here and at 2 or negative 4 so the vertex is that the point 2 minus 4 so what we said is the average slope over so the closed interval that we care about between 2 & 4 it's from 2 here to 4 here that's the interval 2 to 4 the average slope is 2 it doesn't look like it only because I've kind of compressed the y-axis and we're saying at the point X is equal to 3 the slope is equal to exactly that so at X is equal to 3 the slope is equal to the same thing that's all to mean the V value theorem is I know it sounds complicated people talk about continued continuity and differentiability and F prime of C and all of this but all it says is the there's some point between these two points where the instantaneous slope or the slope exactly at that point is equal to the slope between these two points hope I didn't confuse you