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# Justification with the mean value theorem: table

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.B (LO)
,
FUN‑1.B.1 (EK)
Example justifying use of mean value theorem (where function is defined with a table).

## Want to join the conversation?

• Why do we use the domain as values to calculate the slope of the secant line when < & > are non-inclusive?
• Think about what the MVT actually says:
`for a function f that is - differential over (a, b)- continuous over [a, b]There exists a c in open interval (a, b) where f'(c) = avg rate of change over the closedinterval [a, b]`
So in the questions, our c is
a < c < b
or in other words
c is a member of (a, b)
But because of what the MVT says, we still use a and b to find the average rate of change (or the secant line).
• In the first example, does the MVT not apply because there's no value of f'(x) that equals the average rate of change over the interval 4 < c < 6? Or is it because there's no value of c such that f'(c)= 5 AND the average rate of change?
Also I felt like Sal was in a rush.
• There may be a value of x that causes f'(x) to equal the rate of change over the interval [4,6], but there is no value of c such that f'(c) = 5 and the rate of change. This is because you are adding the condition that f'(c) has to equal 5 while the rate of change over that interval stays the same.
(1 vote)
• Why we use open interval in differentiabilty case??
(1 vote)
• Because if the function isn't defined outside the interval, the derivative won't exist at the endpoints. But this doesn't affect the main point of the theorem.