5

'8 = (0)f pue 81 + ITI + gTV ~at (c) / 'waiqoud anie^ IEIIUI 341 JNOS (9)Lg [SYJew €] 0 < I 0 = I Ie snonuiuo) SI I ~T8T 0 > I 281? v) (2)f u...

Question

'8 = (0)f pue 81 + ITI + gTV ~at (c) / 'waiqoud anie^ IEIIUI 341 JNOS (9)Lg [SYJew €] 0 < I 0 = I Ie snonuiuo) SI I ~T8T 0 > I 281? v) (2)f uoipuny 341 Jay1aym euiuJ3120 %(e)L9 [SqJew €]

'8 = (0)f pue 81 + ITI + gTV ~at (c) / 'waiqoud anie^ IEIIUI 341 JNOS (9)Lg [SYJew €] 0 < I 0 = I Ie snonuiuo) SI I ~T8T 0 > I 281? v) (2)f uoipuny 341 Jay1aym euiuJ3120 %(e)L9 [SqJew €]



Answers

In Problems 79-82, find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$ and $\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]$ $$ \mathbf{u}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}-5 \mathbf{k} \quad \text { and } \quad \mathbf{v}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\mathbf{k} $$

Here is asking us to rewrite this in form A plus the eye and they provide some answer choices we're gonna do here is we're going to just multiply this out. So eight minus five I square it is just multiplying. Eight minus five I times itself. So first, we have eight times eight, which is 64. Outer is an eight times a minus five. I gives me a minus 40. I, the owner is going to be the same. And then a negative five I times a negative five. I gives me a positive 25. I squared. Well, we know that I squared is negative one. So this becomes a minus 25. And I'll just put this over here by the 64 because I can combine these whole numbers. When I subtract here, I get 93 so I get a 39 and then these two can combine a minus 40 and another minus 40 is a minus 80 I and this is answer choice G

Okay, we've been asked to find the derivative of the dot product between U. And V. So let's remind ourselves of the equation that we can use. Now notice we can first take the dot product, then take the derivative. Or we can use the formula on the right. So I am going to show you the formula on the right. So what I've done is I just took the derivative of you. So my T square became a two T. My negative T became a negative J. I'm also going to go ahead and take the derivative of my vector V. And I get I plus two T. In the J. Direction. Okay, so to use this, I am now doing the dot product between my um derivative of you and with the V. So I'm taking my two tee times T. I get the two T squared. And then I have my negative J times T squared, so minus T squared. And then I'm going to add it to the dot product between my you and my V. Prime. So t squared times one and then a negative one times two T. So what you'll find is that when you do consider all your light terms zeros out. So that derivative is zero. Now what if we worked on the right side and we first did the dot product. So we're going to go back to our original you and be and do the dot product between those. We'll get a T to the third minus a. T to the third, which is already zero. So when we go to take its derivative or zero, so you can see you can use either side of this equation. Um In a lot of cases are using the left side will be easier. But you're gonna find um some situations when the problems get harder where the right side of the equation is actually the nicer side.

We're being asked to factor the given expression. Remember, Always look for a greatest common factor first. Well, I see that each of our term has this quantity of e plus e, so we can factor that up. So we factor out this V plus eight from each term. We're going to be left with eight You squared minus 38 U minus 33. So now we can never factor by grouping or by trial ever. So I'm gonna use trial ever. So I'm gonna bring down my V plus eight, and we're gonna set apart to buy no meals. So remember, the first term in each binomial needs to multiply to our first term. Well are factors of eight are one and eight and two and four. We also know that our last term's most multiply to our last term negative 33. So are factors of negative. 33 will be one and 33 3 and 11. So we don't have too many choices here. Now we have to keep in mind that our last term is negative. And that means when we put in our factors, get our middle term, bring end up subtracted We also need to remember our middle term is negative, so we're gonna want the highest value to be negative. So now it's really just a matter of guess and check your So let's say we wanted to start with Oh, I don't know. Let's say two and four. I just have a hunch. Sure. So it's going to you and for you. Well, if we really use one and 33 we're gonna end up with numbers that are way too big. So maybe we'll try three and 11. So I'm gonna go 11 1st followed by three. And again because our middle terms negatives are larger. Term has been negative. So we're gonna go negative and then the positive. So let's see if this works well to you. Time ST six. You and negative. 11 times for you is negative. 44. You And when we combine these, we do get negative. 38. You perfect. We found their factors. So our final answer is V plus eight times two U minus 11 times for you plus three

Killer been Rudy. There's why there were prime minus to wipe round that secret. Yeah, to t Now, as usual, started with writing out of the office later equation and that solving it so this is equal to one that is equal to zero. And we can easily see that the solutions are are one hers are to this war. So the number one is a double would off the officer equation. They're for our and therefore our auxiliary solutions are One is e to t r two is tee times e to t. Why shouldn't I shouldn't use the literal ours there, So make me cook. We fixed it. Let's call it. It's called him. Why? Why wanted y two. So why want us here? Right here. And so how do we get for our our y p? So in the IV, guests would have been simply something times, exponential times, special tea, Or as we can see that this is a This is actually one of one type of the auxiliary solution here, So we need to give the goods to more powers off t. So this is not this is not a good guess. And the actual the cracked guesses what their technical, um way is to look a tte the solution to our office area questions. So that solution is one, and it's a double word. So when it's a double route, the way to go is we need to give to Power's off t two this no Hodgins part and there's gonna be some sort of costume that we need to determine here. And this would be our guests. And let's see how his workout. So now we just simply compute to do it. You're so why P Prime is I know that when you're taking product wolves with exponential, the exponential will be prisons in every term. So it's easier to take exponential out. And after we take a one dirty, we will have to t times t squared. And when you take secondary to our and you enter if that this is, that is a again exponential out, and then here will have so the do it there if they were gonna have to t plus two using the part rule and then to t cost T square. So I'm told this will become eight times e x special t squared plus 14 for us two. Now all they have to do is plug this off this back into our original question and throughout the number. Eh? So it's compute that here, this is equal to it. Let me continue on the next speech. This is a cool too Well, let's see why. Double Prime P. That is right there. So I'm gonna take again A and exponential from everything out to simplify writing. And then I'll have t Square plus 40 plus to the minus four times. Why Single prime, which is here. No, they come. That's actually not for its two times. They will become too four times T minus two t squared. And finally we have plus y p. And there is just this. Therefore, we're gonna have, um, t squared here. So we will have some nice cancellations here. So my std squared thistle castor with constantly squared for table casserole. Nice. 40. And india And this will turn out to be to a e t. Are this should vehicle to age times exponential suits eight times exponential. And finally we see that ourselves. Number eight is equal to four. And therefore our solution is why P is four times exponential time off. T


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