Use tha following table estimatef(x)dx Assume that f (x)decreasing function2025f(x)433628ebumato the valuethe Intsgral wethe Ieft-hand Jum approximation with D* =T...

Estimating a Definite Integral Use the table of values to find lower and upper estimates of
$$\int_{0}^{10} f(x) d x$$
Assume that $f$ is a decreasing function.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {0} & {2} & {4} & {6} & {8} & {10} \\ \hline f(x) & {32} & {24} & {12} & {-4} & {-20} & {-36} \\ \hline\end{array}$$

This question asked us to use the following table to estimate the given integral what we know that we can first off to the left hand. Some we know that we have Delta acts is 14 minutes 10 which is four or n is 26 minus 10/4, which is simply for therefore for l. We have half of 10 times four, which is 100 plus half of 14 times forces. You can see I've put the foreign the outside, plus up of 18 times for was of 22 times Force. We could just do distribution of multiple assault by four to get 1240 and then for the right hand, some same thing. All these volatile times four. So I'm just gonna do distribution after 14 plus half of 18 plus half of 22 then we have one more than 50. You're off of 20 six. This gives us 952 and on last step. Don't forget to find the average of these values

So we have that. The linear approximation have x y minus F of 25 is equal to one times X minus two minus one times Y minus five, which will give us that the only approximation of 2.2 4.9 minus six is equal to X minus two. My guess why plus five. Or that the only approximation at 2.24 point nine is equal to X minus Y plus nine. So now if we just plug in our values for X and Y, we get that the linear approximation is equal to 2.2, but it's 4.9 plus nine, which is equal to 6.3.

As you look at the table to estimate, I'm just copying down the values. Uh it's up to you because they don't tell you which room on some to use left or right. Um But if you were to draw this graph out 15, 11 and nine into your X. Values and then these are your Y. Values. So if you were to graph this, uh starting at 00 up to 32 then three down to 22 I'm just gonna use left riemann sums. So the quick way of thinking in this area of a rectangle would be the length of its width, which is three because it goes from 0-3 and I'm going to make the height 32 because that's how high it went up. Um so as I look at the next rectangle because it goes from 322-6 15. Uh But if I continue to use a left Um from 3-6 still has a width of three. And then this height, I mean you use 22. I'm trying to color code is so you can see, so as I do the next one, uh so he goes to 9 11, So I'm still using the left, so I'm using this value has a width of three. Still 6-9 is a width of three, height is 15, and then the last one, I don't know, I'm switching to blue now of all the colors, Even though this last point is 12 9 I'm using left. So I mean it was the height of 11, but so is a base of three. So to do this problem, you just do you know base times height plus base times height plus based on type ran out of room, but plus base times height. And I'm just going to a calculator and my answer may be different than the answer keys, but it's an approximation, so it doesn't really matter. I got 240, but you might get a different approximation if you did the right sums, if you did the right sums b. These four values instead of 32 at 22, 15, 11 and nine. Um and there's other approximations to, but this is good.

Were given a function, and we're told that this is a differential function. What efforts to five equals six partially to the vet Perspective X, at 511 and the partial very good with F with respect to y five equals the negative one and rest to estimate the value f of 2.2 4.9 using a linear approximation. First of all, your approximation at this point given by the value at this point, which was after two fun. What's the mhm Thanks. Times X minus two plus the cooperated with that y at this point and why minus five substitution success six plus one. Execs minus two minus mhm. I'm is y minus five, which ultimately simplifies minus. Why what's therefore we had to value 0.2 four point. This is approx 2.2, minus 4.9 plus nine, which is 6.3