Explain the statement that a continuous function on an interva...

Question

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

Accepted Answer

Welcome back everyone. Suppose H of X is continuous on the interval from 0 to 5. What does the mean value theorem for integrals guarantee? A says it guarantees that H of X is always equal to its average value on the interval from 0 to 5. B that H 0 equals H 5. C that the average value of the function is always 0, and D that there exists at least 1. C on the interval from 0 to 5, such that H of C equals 1/5 of the integral between 0 to 5 of H of X with respect to X. Now to figure out which one of these statements or conditions it guarantees, it would be a good idea for us to note what the mean value theorem for integrals actually says. And I recall that the theorem basically states. OK, that if If a function H of X. Is continuous. On a closed interval, in this case on the closed interval from 0 to 5. OK. Then There exists at least 1 point, at least 1 point. The point that we cause see. In the given interval. In this case, again from 0 to 5. Such that HFC Properly H of C. Is going to be equal to 1 divided by the difference between the endpoints in the interval, that is 5 minus 0, multiplied by the integral, OK, between those limits. Of age of X with respect to X. No, 5 minus 0 is 5, so this, in this case or in this problem H of C would be equal to 1/5 of the integral between 0 to 5 of H of X with respect to X. This is what the mean value theorem for integrals can guarantee if H of X is continuous on our interval. This means that there is some value C in the interval where the function attains its average value over the interval. Therefore, D is the correct answer. We can sure we are, we can be sure that we are right, that answer D is right if we think about the other options. Now, in our first option, to say that H of X is always equal to its average value on the interval is not necessarily true because the function is not necessarily constant. It could be, but we don't know if it's constant, right? Next, we can't say that H of 0 is not equal to H of 5. The function doesn't have to satisfy that. And we can't say the average value of the function is always zero, because that's not always true. So the rest of things we can't guarantee. I mean, they could be possible, but we can't guarantee them. But we know for sure that answered sorry option D has to be true. Thanks a lot for watching everyone. I hope this video helped.

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

Key Concepts

Continuous Function

Average Value of a Function

Mean Value Theorem

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