5

33.87 30.45 32.51 31.53 36.57 35.64 36.14 35.18 36.02 39.62 44.32 42.61 42.24 42.45 49.76 45.93 48.78 47.67 45.9449.18 49.45 49.94 54.25 52.08 51.31 54.21 52.82 53....

Question

33.87 30.45 32.51 31.53 36.57 35.64 36.14 35.18 36.02 39.62 44.32 42.61 42.24 42.45 49.76 45.93 48.78 47.67 45.9449.18 49.45 49.94 54.25 52.08 51.31 54.21 52.82 53.15 54.06 50.19 53.76 51.35 54.51 54.45 55.23 57.75 56.62 55.8458.45 57.98 57.58 56.76 57.28 57.66 59.52 64.83 61.65 60.21 62.04 63.91 60.32 64.67 60.64 62.54 63.43 63.19 64.6660.27 64.55 60.32 69.01 65.38 65.84 65.16 68.16 65.08 67.88 68.27 67.13 65.88 70.29 71.19 70.79 72.06 74.68 81.61

33.87 30.45 32.51 31.53 36.57 35.64 36.14 35.18 36.02 39.62 44.32 42.61 42.24 42.45 49.76 45.93 48.78 47.67 45.94 49.18 49.45 49.94 54.25 52.08 51.31 54.21 52.82 53.15 54.06 50.19 53.76 51.35 54.51 54.45 55.23 57.75 56.62 55.84 58.45 57.98 57.58 56.76 57.28 57.66 59.52 64.83 61.65 60.21 62.04 63.91 60.32 64.67 60.64 62.54 63.43 63.19 64.66 60.27 64.55 60.32 69.01 65.38 65.84 65.16 68.16 65.08 67.88 68.27 67.13 65.88 70.29 71.19 70.79 72.06 74.68 81.61



Answers

(a) $\mathrm{CH}_{3} \mathrm{NHCH}_{3}$ (b) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{NHCH}_{2} \mathrm{CH}_{3}$ (c) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{NHCH}_{3}$ (d) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$

Yeah. Hello. So in our given function um the Y value of negative three corresponds to two 2, 2 different x values um namely here negative three corresponds the X values of one and and four. So therefore it is not a 1-1 function Right? It's not 1-1. And since the function is not 1-1, it does not have mm have an inverse.

If we want to evaluate the equation above, we can use the binomial theorem written ingredient. When the equation is in this form, we can see from each individual term that X in the binomial theorem corresponds toe one for a in the binomial theorem corresponds 23 over four. An end in the binomial theorem corresponds to five. We know that the binomial theorem is used to expand equations of the form X plus a the end. So we know that this entire equation here can be rewritten as 1/4 plus 3/4 to the fifth. This equals one to the fifth, which just equals one.

All right. So for the following question, we have a very long equation that we need to find a numerical value outs. I'm gonna radio now. It's may take a little bit. It's it is very long. So obviously as we can see, this is already a binomial expansion, and hopefully we all know the formula for that. But if lot luckily, this question so long that, um, you can probably find it by telling done writing this whole thing out, you know, we're gonna persevere. Write it all out. It's just easier when you write it out. It's a better way of learning. Should always right your questions out whenever doing it. The sword is a doozy. Still going a couple more lines and we're gonna be good. Some of you can probably already kind of guess the rest of them or not. Guess story. No, the rest of them a pattern or formula wherever we like to call it should be a four. Uh, and this is the last line. So all this we need to figure out what it equals. So we know that from the binomial theorem, whenever we're given an X plus a why to be X to the Y, and it could be equal to the sun of and J is equal starts at zero of an over J and then we have X to the n minus j. And then why Teoh the J So when we actually expand that out in just purely, um, mathematical form, it looks like this. So and over zero x to the end, plus and over one x to the n minus one. Why plus and over to extra the war and Linus to why? To the power of to and then plus, And it goes on essentially until you reach end over end. So in this case, we're gonna look for our aunts, RJ's and our, um, our xer wise and our ends essentially. So from the first term, you can see that it's gonna be end over zero and X to the end. So in this case, I was gonna point some things out. I think that five is our and it's our end, Um, and then from the first formula, and then this right here, the 1/4 is gonna be our X. And then, um, for the next one, we have that we once again have our end of five and then one is the next step in the formula. We once again have this as X still, and and now we have this which is actually gonna be our Why. So now I'm just gonna write out everything about we have. We have X is equal to 1/4. Why is equal to 3/4 and we have that end is equal to five. So the formula that would have started this binomial theorem would look like this 1/4, plus 3/4 to the power of five, and that's actually going to be a formula. So now, since there's no variables in it, we could just solve for the number in this case. So 1/4 plus 3/4 is actually is going to be one. It's gonna be one to the power five, and obviously that equals one. So therefore the equation slash binomial expansion. I'm just going right. B E for short is equal to one. And we found that by looking at the formula that were given and kind of reverse engineering to get it back to the formula that we know so we could solve for a numerical value

In this video, we're gonna go through the answer to question number 19 from chapter 9.3 to rush to find the inverse matrix off F S R E O X, which is a matrix as a function of time given here. First, let's recall that inverse off a product major sees a B is equal to the inverse off B plans by the invested a sharing all of the investors exists. So let's think about how we can write this in a slightly different way. So we kind of want toe, not have to worry about all the u to the t You need to mine it easy to tease. So let's just write the coefficients first 14 and then you see that all the first row almost quite by eating Timmy on the second row E to the minus t you know, 30 points to t so we can turns up by e to the t zeroes ever in the second row zero e to the minus t zero and 3rd 1 00 each of the two teams. Okay, let's call this one a on. Let's call, this one will be, Then we can use this formula to find the total invest. Okay, so first up, let's find inverse off, eh? Let's do it in the usual reduction way. So what we got 111 one minus one. See? You want one? Combine that with the identity. 100010 There. Is there a woman? Okay, we're reducing. Let's subtract the first row from the bottom room. That gives us 00 three minus 101 less. Attract the first road from the second road zero minus 21 Uh, then screw reminds 110 leave in the first row is it is one warning zeros era. Okay, so try it times in the bottom row by 1/3. We got 001 minus 1/3 zero 1/3. Get me. Okay, then this new bomb row, we can subtract that from the 1st 2nd most. So from the first room gonna be 10 because I want one. That one minus one is zero. It's gonna be one minus a bird. Sorry. One minus minus. A bird, which is one plus a bird, which is 4/3 zero minus 00 zero minus 1/3 as much bird. Then subtract the new bottom row from the middle road is your, uh, minus two zero minus one minus minus 30 miles. Off course, a bird which is minus two birds one minus zero is just 10 minus. The third is my herd. Okay, so bottom row stays the same. 001 Mines third, zero third. Let's multiply the middle Robot minds heart to get 010 Ah, my hard times minus 2/3 is 1/3 then one times minus half is mine minus half minus. 1/3 is 16 Then let's do the top road minus this new middle road. Then we're gonna get the matrix on at the identity matrix on the left for the 4/3 minus. Good. This one zero minus 1/2. It's okay. Zero minus minus 1/2. It's 1/2 on minus. 1/3 minus suit is minus 36 Which is my heart. Okay, so this is our inverse off the function called a Now it's fine. In burst off. I actually called bay. So be waas. Eat the tea. 00 zero. It's the minus t zero. Is there? Uh, zero. He said to take the inverse of this. This is really easy. Um, because when you got a non zero elements in the leading diagonal on and it's just the reciprocal off those beating darknet values on the rest is all zero. So eat the minus t 000 e to the T they were zero zero. Eat some honesty. Sorry. He's the mind to t expended in verse off X, which is inverse off. Maybe. Which is? They invest a inverse, which is, if the modesty 00 zero e to the T 000 into my studio tea. That's our invested. Be invested a waas one, huh? Minus off that, But it's hot. Six minds of the zero Third. Then when we we'll find them together, it's question, but we got E to the minus. See, huh? Modesty minus ah, the money's team. Bird eats the tea. Mine's 1/2. It's the mind. Yeah, it's the team. Six. It's the team, but Murray get minus. 1/3 eats the minus Tootie zero on the third eats the mind stated, and that's I invest


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