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## Differential Calculus (2017 edition)

### Unit 11: Lesson 7

Mean value theorem

# 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.

## Want to join the conversation?

• How did he get the vertex at 3?
• 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.
• 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?
• 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.
• Why did he include the vertex on the graph about the mean theorem? plz explain this
(1 vote)
• Sal did it to help himself graph the equation better.
• 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?
• c just represents the point where the instantaneous slope is equal to the average slope of the function in the closed interval.
• 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!
• If the equation is impossible/difficult to factor,
we would use the quadratic formula
to get the roots/zeros for the equation (read when y=0.. the x-intercept)
• How to calculate average f' over a period (a,b)? Is this equal to average rate of change over (a,b)? Thanks.
• At , why does Sal convert from brackets to parenthesis? Don't those represent closed and open intervals?
(1 vote)
• 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.
• My question is if this is the Rolle's theorem
• how does he find that f'(x) = 2x-6
(1 vote)
• `f(x) = x² - 6x +8`
`f'(x) = 2x - 6`

If you've watched the previous videos on taking derivatives, you should know the power rule which states:
if `f(x) = x^n`, then `f'(x) = n*x^(n-1)`.
You should also know the rule if `f(x) = g(x) + h(x)` then `f'(x) = g'(x) + h'(x)`
• how did Sal get 3/3 when there is only one three present? @ in the video.
(1 vote)
• 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.