7 ptsQuestion 7The average value of the function f (c) = (2 _ 1)2 on the interval €1to x = 564/3256/3416/316/3...

Find the average value of the function on the given interval. $$f(x)=5 x^{2} ; \quad[1,4]$$

So we have that The average of ffx from 0 to 2 is five and the average of ffx from 2 to 6 is 11. That means that the area under half a box from zero to to is equal to two times the average on average in that region, which is five making a tent and then we can do the same thing from two 26 This time it has a width of four and the average of 11 making this 44. So that means the average value of ffx across the across the whole interval can be found by the total area divided by the total interval. The total area is 10 plus 44 making this 54 and the total interval is gonna be from zero all the way up to six. So six, which gives the final answer of nine

If we want to find that average value of the function to minus three x squared over or on Central three, two, one. Remember that average the value is going to be one who be minus a integrated from aid be of aftereffects DX. So average value on A to B and now we just go ahead and plug everything in that we have. So we have one over, so R B is going to be three and our is going to be one so three minus one and then we integrate from one to three and f of x is two minus three x squared pix, So three months ones, too. So we get one half here, integrated. So now let's go ahead and interpret this riposte. So this is power. So it's going to be too X minus a ring X now to the third power divided by three. And then we evaluate from one to three. So this three and that three camps float. So I have one here and now we can go out and plug in three and plug one. So these two times three But Chancellor two times three, minus three. Cute, which will be twenty seven twenty seven minus. Will we plug in one? So two times one would be too. And then, when I plug in one toe, excuse will be minus one. So just writing this two times three six. But in one purposes now, adding all this up, so two minus one is one. So we end up with six minus twenty seven minus one, which is negative, twenty two. Then divide that by two and we end up with negative of others. So the average value on a TV or the function to minus three X squared is negative.

The average value of X to the negative two over the interval from 1 to 4 is given by the following. And we can evaluate the integral of X and -2 by using the power rule. So we know that we just add one and divide by that some. So the integral Of X. and -2. That's equal to X to the negative two plus one. All over -2 Plus one. Because this is an indefinite integral. We would add C. When we get that this is minus X. to the negative one plus C. But we have a definite integral. So in this case we get that this is equal to one third times negative X To the -1 From 1- four. And this is equal to negative one third Times 1/4 minus and negative one third Times 1/1 which is equal to one third minus 1/12. And this is equal to 4/12 -1/12, Which is 3/12, which is 14 So we can say that the average value of this function is 1 4th And now we want to find a point in the interval from 1 to 4 that gives us this average value when we plug it into F of X. So we can find that point by setting F of X equal to this average value. So we have X to the -2 -2 is equal to 1/4 next weekend. Cross multiply. So we get four is equal to x squared and finally X is equal to plus or -2. But we're dealing with the interval from 1-4. So we get that X must be equal at two and that completes the problem.

In this given question we able to find the average value of the function and uh function given us is affects it was too four minus X squared and in the end develop minus toto today. Okay to find the average value you can say that formula. Yes. To calculate the average value one. Dubai Dubai B -A international effects city extended into a love affair too theory or values of N we r minus two and three. So one do I get by 3 -2. That is just too limits are -2-3 and function is four minus access square. Okay now we integrate here four minus makes a square. So we know the integration of bodies products and nexus calories excused by three. And in this integration we have to put the limits and it is divided by right okay. No they make here further calculations. So if you put the parliament That is three than it reserves you four in 2312 three Q battery that is 9. 3 square. Okay? Minus If I book minus two then minus eight and -1 exchanges Hindu plus And it divided by three. No, we know that 12 -9 is three. So three plus it- Age divided by three, terrible Assaidi's 11 and no, You're right. It does 11 -8 x three. That results in you 33 minus eight. 33- here. It is 25. And if you cancel it by five, okay if you cancel it by fine then finally I can say that I should get five x 3. This is the answer here. Thanks.