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Solve the problem:Suppose that f is continuous and that f(z) dz =t andE(z) dz = 8 FindAf(x) dx...

Question

Solve the problem:Suppose that f is continuous and that f(z) dz =t andE(z) dz = 8 FindAf(x) dx

Solve the problem: Suppose that f is continuous and that f(z) dz =t and E(z) dz = 8 Find Af(x) dx



Answers

If $f$ is continuous and $\int_{0}^{8} f(x) d x=32,$ find $\int_{0}^{4} f(2 x) d x$

So here we're asked to evaluate this integral. We have the integral of f prime of t d t from really e and were given a to known values of our original question three is equal to 4.5 of eight is equal to and they want us to evaluate this integral over here. So what's important to notice with this is that we're taking the integral of f prime of tea, so actually taking the integral of the derivative of our original function. So when we take the in, a girl of a derivative actually cancels out that produce. So Thean a girl of this is actually just going to be f of T. We're going to evaluate it from our to bounce there. So for hate three, so we have half of eight is equal to 20. We're going to subtract of three, which is for that's gonna give us 60. That's what our interval is able to

All right. Were given that F is a continuous function and were given the value of an integral of F X. And were asked to show that ffx takes a particular value over a particular interval. The given information in the question, and what the question is asking for immediately suggests that we need to use the main value fear. Um specifically the mean value theorem for integral, excuse me, Which is a firm that states that the function F of X takes its average value at some point ah in the interval where the average value of the function F of X on the interval A to B is given by the formula on screen. Right now, that formula gives you the average value Ff. The mean value theorem for integral states that the function F of X takes the value F at some point in the interval A to B. So if it turns out that four is the average value of F of X on the interval from 1 to 3, then that will have solved this problem. So Let's go ahead and compute the average value of F on the interval from 1-3. So by the formula, we want to compute the average value which equals one over three minus one Times the integral from 1 to 3 of F of x D X. And this is the other reason why the phrasing of the problem heavily suggests that this is the correct way to solve this problem because we are not told what F of X is at all. So the integral from 1 to 3 of F of XDX is not something we can evaluate on our own. Fortunately, the problem tells us exactly what it is. A problem tells us that this integral equals eight. So we can compute this anyway, this expression equals one half times eight, which equals four. This is exactly the value. We were hoping that this is exactly what we were hoping the average value would be, and the result of the problem showing that F takes the value for that is exactly what the mean value theorem implies. So we're done.

We know we're gonna be using the mean value theorem, which essentially means that because of his continuous there exists a numbers see between the bounds of 123 such that the average value won over B minus A from A to B of our function after vax d X equals F f c. So we know this means that half of sea is 1/2 times ate half of eight is four. Therefore, there does exist to see such the F of CS four.

We're taking a look at this, we know that what we've got Specific definite intervals that we have here from 0-3 and then again from 0-4. So this What that tells us that from 0 to 3 from C. D. C. Plus from 3 to 4. I'll have to see D. C. That's going to be equal to seven. All right. And since we know that from 0-3 is equal to three here than this, Most equal four here. So the answer to our first part, Let's be equal to four for the second part. Doesn't matter if we change the variability, how it's T. Here T here still functioned of F which is being continuous here. So the important thing that we have to notice here is that the boundaries have flipped and when we flip the boundaries in order to get it to what we had before of three or four. Uh the Tgt or to do this, we would just have to indicate the outside here Because we know from the first part that that's equal to four from 3 to 4 and dedication of it would equal to thank you for. That's right sir.


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