Question
Cnain (Fug,Ite rale pf reacion , opropluule units Is giyen by R'() Ina Cruo hom t= 2t0{-6 Round tu decrni pLaccs , MceshirWthmFirua 284 527 0 78 9 DO
cnain (Fug,Ite rale pf reacion , opropluule units Is giyen by R'() Ina Cruo hom t= 2t0{-6 Round tu decrni pLaccs , Mceshir Wthm Firua 284 527 0 78 9 DO
Answers
Use Table 7.9 . $$\begin{array}{c|c|c|c|c|c} \hline t & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline g(t) & 1.87 & 2.64 & 3.34 & 3.98 & 4.55 \\ \hline t & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline g(t) & 5.07 & 5.54 & 5.96 & 6.35 & 6.69 \\ \hline \end{array}$$ Estimate $\int_{0}^{0.6} g(t) d t$ using the midpoint rule with $n=3.$
All right. If I was given this equation here, Um and has to find the right handsome when and is equal to three. First thing I would do is realize that that's just fancy fancy talk for a right three that we've already been doing. Right, Okay. So um and equals three. So if we want to go 02.9 and three steps, Then our Delta T. is going to be 0.3. Right? So we're going to start at .9. So 0.9 F function of 0.9 For the times that by 0.3 we're going to add that. So we're going to go down three now 20.6. And then since we're only doing three of them When I go down to three now, let's use our table our handy dandy table um from the book to get our values for this. So .9 is going to be 6.69. That's multiplied with 0.3 and then 5.54, 0.3, And then three is 3.98 times 0.3. All right, and put that all into our nice calculator. And what you're going to get is 4.9. Mhm.
Okay for this problem, We are asked Teoh graph a scatter plot of some time and temperature data. So for time, we're gonna put that on the X axis. Notice I made a what's called the first quadrant grid. So it doesn't always have to be, like 00 like a big plus sign. So here into time, I'm going to start off with zero on this. Be one oclock in two o'clock three o'clock, oclock. I'm black. Six clock, seven o'clock. Okay, so, um, you label every other one. So I guess this is time starting off at noon and then on my y axis is kind of unusual. You kind of want to start off at zero whenever you can. So when my numbers up in the sixties and seventies, But I kind of counted out ahead of time. So I start off with temperature that we believe this to be a zero. So we're gonna make this, uh, miss 10. This is 20 30 40 50 60 70. And students don't like it when you go off the grid a little bit here, but it's OK. Probably change that arrowhead there, But what's more important is that we get a sense of scale of for how warm it is. So I want to start off from zero here from time. I'm starting out from my guess. 12 o'clock is really what I'm doing here. So let's go. One oclock uh, 71 degrees. So she was up here, Came in at three o'clock. It's 74 degrees, No warm right in the five o'clock in 68 degrees at seven o'clock. It is 63 degrees. So makes sense, because at the end of the day gets cooler. Okay, that's a scatter plot. So what I'm not gonna do? I'm not going to connect the dots. I'm not gonna put a line in there like this, cause that's not exactly how things work. So it's got a plot. You just have the dots like that. Okay? And you don't have to just coincidentally, didn't want I label that were the 1135 But you don't have to label the ones that go with your points. But this does give you a good example of what be scatter plot Looks like one of the thing I kind of left off was labels, so we definitely do want to have labels. So this is temperature. Oh, doesn't say degrees since Celsius or Fahrenheit, but it should. And then this is the time I did label. All right, if it helps you out.
All right. I was given this equation here with a table from 01 to two two and asked to find the mid of five. Okay. My number's for this table and you can look at the table um in your book or Wherever You Get It. It is a table 7.7. It's going from 1.0 To 2.1. Yeah. Right? With intervals intervals uh so like Delta T. It's equal to .1. Right? Okay. So and we want a mid of five. So since each one since it's 1-2 And it's given five and our intervals Or it will be 00.12 and that looks really bad. Fix it up one point oh To 1.2 and so on all the way to to. Right? That's how we're going to get five out of this. 1.01.2468. And then and then finally to so if I want the mid of those I want to I want the in between values. So that's gonna be 1.1. Okay? 1.1 1.3 1.5. Okay. And then just like all the other ones, we have to multiply this by our difference, our difference between Um 0.1 And your .2 is going to be 0.2. Of course. So we're just gonna have to do do this five times. So we're going to add F1.3, 0.2 At F 1.5 and 0.2 At F 1 .7 and 0.2. And maybe And then at F 1.9 It's 0.2. Okay. And from our table we can get all these values. So uh 1.1 is negative 2.9 times 0.2 plus negative three .7 times. 0.2. Sorry, I lost my table. There we go. Then, uh negative 3.2 times 0.2, And then 1.7 is negative 1.7 times 0.2. And finally, 1.9 is 0.5 times 0.2. You put all these numbers into your calculator, You end up with negative 2.2 and that is going to be your answer for uh mid five. So mid five is equal to uh -2.2.