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Mean value theorem example: polynomial

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.B (LO)
,
FUN‑1.B.1 (EK)
Sal finds the number that satisfies the Mean value theorem for f(x)=x²-6x+8 over the interval [2,5]. Created by Sal Khan.

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  • blobby green style avatar for user Ian Osea
    How did he get the vertex at 3?
    (7 votes)
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    • blobby green style avatar for user Creeksider
      It appears he was relying on the symmetry of a parabola, as he says it's halfway between the x-intercepts. There are other ways to get this however. Note that f'(x) = 2x - 6. Setting that equal to zero tells us the critical point is at x = 3, and in the case of a parabola, there is only one critical point, at the vertex.
      (29 votes)
  • piceratops seed style avatar for user brenda
    Would the average rate of change over an interval be the same as the value guaranteed by the Mean Value Theorem over the same interval?
    (2 votes)
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    • leaf green style avatar for user Steven Raft
      The Mean Value Theorem doesn't guarantee any particular value or set of values. Rather, it states that for any closed interval over which a function is continuous, there exists some x within that interval at which the slope of the tangent equals the slope of the secant line defined by the interval endpoints.
      (10 votes)
  • leafers seedling style avatar for user Chrissirhc5
    Why did he include the vertex on the graph about the mean theorem? plz explain this
    (1 vote)
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  • aqualine ultimate style avatar for user Hank Hueston
    So does c represent the instant slope/rate of change/derivative? Or is it an x value were the derivative and secant slope are equal?
    (2 votes)
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  • aqualine seed style avatar for user himmelcat13
    I'm just curious, if the equation wasn't able to be factored, would two be the only value then that has a y-value of zero? I just want to make sure for future understanding. Thank you!
    (2 votes)
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  • blobby green style avatar for user jishyhh
    How to calculate average f' over a period (a,b)? Is this equal to average rate of change over (a,b)? Thanks.
    (2 votes)
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  • aqualine sapling style avatar for user Derek Wang
    At , why does Sal convert from brackets to parenthesis? Don't those represent closed and open intervals?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      That's correct. Recall that the statement of the mean value theorem requires that the function be continuous on the closed interval [a, b], but differentiable only on the open interval (a, b). Sal switches to open intervals when he begins talking about the derivative of the function.
      (4 votes)
  • duskpin ultimate style avatar for user Maria Șandru
    My question is if this is the Rolle's theorem
    (2 votes)
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  • leaf green style avatar for user Michael Pannekoek
    how does he find that f'(x) = 2x-6
    (1 vote)
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  • piceratops sapling style avatar for user kat wil
    how did Sal get 3/3 when there is only one three present? @ in the video.
    (1 vote)
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    • leaf blue style avatar for user Stefen
      The average change is calculated using (y1-y1)/(x2-x1), or the slope, right?
      Here y is a function of f, that means we have (f(x2) - f(x1))/(x2 - x1)
      We are told that x1=2 and x2=5
      We are told that f(x) = x² - 6x + 8.
      That means f(x2) = f(5) = 5² - (6)(5) + 8 = 25 - 30 + 8 = 3
      And that f(x1) = f(2) = 2² - (6)(2) + 8 = 4 - 12 + 8 = 0.
      THEREFORE
      (y1-y1)/(x2-x1) = (f(x2) - f(x1))/(x2 - x1) = (3 - 0)/(5 - 2) = 3/3.
      (2 votes)

Video transcript

Let's say I have some function f of x that is defined as being equal to x squared minus 6x plus 8 for all x. And what I want to do is show that for this function we can definitely find a c in an interval where the derivative at the point c is equal to the average rate of change over that interval. So let's give ourselves an interval right over here. Let's say we care about the interval between 2 and 5. And this function is definitely continuous over this closed interval, and it's also differentiable over it. And it just has to be differentiable over the open interval, but this is differentiable really for all x. And so let's show that we can find a c that's inside the open interval, that's a member of the open interval, that's in the open interval such that the derivative at c is equal to the average rate of change over this interval. Or is equal to the slope of the secant line between the two endpoints of the interval. So it's equal to f of 5 minus f of 2 over 5 minus 2. And so I encourage you to pause the video now and try to find a c where this is actually true. Well to do that, let's just calculate what this has to be. Then let's just take the derivative and set them equal and we should be able to solve for our c. So let's see f of 5 minus f of 2, f of 5 is, let's see, f of 5 is equal to 25 minus 30 plus 8. So that's negative 5 plus 8 is equal to 3. f of 2 is equal to 2 squared minus 12. So it's 4 minus 12 plus 8. That's going to be a 0. So this is equal to 3/3, which is equal to 1. f prime of c needs to be equal to 1. And so what is the derivative of this? Well let's see, f prime of x is equal to 2x minus 6. And so we need to figure out at what x value, especially it has to be in this open interval, at what x value is it equal to 1? So this needs to be equal to 1. So let's add 6 to both sides. You get 2x is equal to 7. x is equal to 7/2, which is the same thing as 3 and 1/2. So it's definitely in this interval right over here. So we've just found our c is equal to 7/2. And let's just graph this to really make sure that this makes sense. So this right over here is our y-axis. And then this right over here is our x-axis. Looks like all the action is happening in the first and fourth quadrants. So that is our x-axis. And let's see, let's say this is 1, 2, 3, 4, and 5. So we already know that 2 is one of the zeroes here. So we know that our function if we wanted to graph it, it intersects the x-axis right over here. And you can factor this out as x minus 2 times x minus 4. So the other place where our function hits 0 is when x is equal to 4 right over here. Our vertex is going to be right in between at x is equal to 3. When x is equal to 3, let's see, 9 minus 18 is negative 9 plus 8, so negative 1. So you have the point 3, negative 1 on it. And so that's enough for us to graph. And we also know at 5, we're at 3. So 1, 2, 3. So at 5, we are right over here. So over the interval that we care about, our graph looks something like this. So that's the interval that we care about. And we're saying that we were looking for a c whose slope is the same as the slope of the secant line, same as the slope of the line between these two points. And if I were to just visually look at it, I'd say well yeah, it looks like right around there just based on my drawing, the slope of the tangent line looks like it's parallel. It looks like it has the same slope, looks like the tangent line is parallel to the secant line. And that looks like it's right at 3 and 1/2 or 7/2. So it makes sense. So this right over here is our c. c is equal to 7/2.