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Your 053,9)} Une slop - 1 Question 6 1 answer VU 1 IL In(6,) + cfthe tangent arcanekin (6v)} totne ne tOtn? cur neto tne 1(") Jno pianeACme Doim 1 1 (3,6) poln...

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Your 053,9)} Une slop - 1 Question 6 1 answer VU 1 IL In(6,) + cfthe tangent arcanekin (6v)} totne ne tOtn? cur neto tne 1(") Jno pianeACme Doim 1 1 (3,6) polnt (3,6) doint (55 81in2(36) + In(36) 19 36+

Your 053,9)} Une slop - 1 Question 6 1 answer VU 1 IL In(6,) + cfthe tangent arcanekin (6v)} totne ne tOtn? cur neto tne 1 (") Jno piane ACme Doim 1 1 (3,6) polnt (3,6) doint (55 81in2(36) + In(36) 19 36+



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Equations of tangent lines by definition (1) a. Use definition ( 1 ) ( $p .$ I28 ) to find the slope of the line tangent to the graph of $f$ at $P$. b. Determine an equation of the tangent line at $P$. c. Plot the graph of $f$ and the tangent line at $P$. $$f(x)=-3 x^{2}-5 x+1 ; P(1,-7)$$

Hello, everyone. Welcome back to Newman Rate. All right, so now we're Instructor three. Section to number 52 of Catholic captains three. Textbook says number 52. You go given them our vector Art of TV anchor position Becker, which is three T Cube. I, uh, plus two boxes. That's an I was to t squared J. Okay. Plus one over T k. All right, I'm gonna write it t to the negative one, because I'm just gonna use our rule here, but one of a t. K. What they want you to do is calculate the tension, defector. Find the change in vector that tickles one. Okay, so find the tangent vectors. So, um, what is the tangent factor? At T equals one. That's a lot. So are is given. And that's the question. Right? Okay, So how do you do that? Well, the change in vector, uh, will be, um Well, it's been this way. Uh, the position vectors is, say's anchored at 000 That's the tip. The head is whatever you get when you look into right at that same point along your path along the trajectory that ours traversing or describing his tea Barry's. You have your velocity vector, which is 10 gentle to the path. So the tension vectors of Aussie electors. So let's get a velocity. You go. Um, that's it. So what specific Our prime or V of tea? It's gonna be all right. What's a driven off? The X component? Three t cubed the rift. Different inspector team. It's gonna be Bring other three t scratch of 90 square. Okay. Um hi. The ICA phone plus the driver Tutsi Square is gonna be 14. The one J and the derivative Latina. Negative one is negative. T to the negative too, if you like making it one of a T square. Okay. Okay. Now, what is the velocity? Well, the velocity vector at That's where it tickles once a second. I don't know. Are Could be in meters and he could be immediate. Second t could be in seconds, whatever the units are there. But we played in one. Our prime of Wonder V of one is, um list of this component ways. It's gonna be 9 41 You did see square four times. One is four and negative. One over one squared is negative. One. There you go. Now that's the velocity vector. But if they want the tangent, the unit tangent vector, you have to divide by the length of the vector, right? So what's the magnitude of the magnitude of the one? I mean, we could have gotten the magnitude in terms of team playing. One for that, too, also swells. But all right now, when I do this in class, it's it's called a fag arising. So it's like Pythagoras. A squared plus B squared equals C squared breakfast. Where would it so square? It's where added scored it. But we have three components, so it's square. It's where it's square. It added at it. Scored it. So I'm gonna square it. Square it, square it. Adit, Adit and Spirit. So what is that? Well, 81 16. It's 97 plus one. It's not me. It's You have rad 98 which you could simplify. Guess isn't 98 2 times 49. So seven red to like underweight. Separate seven. What's let's see where one tablet tablet switching. Okay, seven. Read to Diego or read 92. So what is tthe e? I think what they're asking for is a unit tension So I'm gonna rent that. Wouldn't you say T hat at one second interview are gonna be, um, one over seven Red, too. Times 94 maybe one of us. You want to write in the night J. K for me, But that's it. And remember how you multiply Affected by scaler. You just smoke play, you distribute. So if you want, you could ride it. And as nine over its seven. Right two for the horizontal component. Um, let's say in terms of I j k so nine over seven Red too. I plus four over 72 J minus one over 72. Okay. And okay, it's right. We could write it that way. Night while in a congressional is. But it's a nine over seven. Red to which could rationalize I plus four over seven Red too, Jay. Yeah, that's us. Tip. It leaves much to be fired this week on tablets. Okay, so seven red, too in the J component and then minus one over seven. Red, too. And if you need to rest in the K direction again, ice the unit vector 100 Jay Z and effective 0101 And that's it. I hope that helped. All right. Thanks for visiting them. Arrayed and good luck with homework. That was helpful.

Thiss problems. We are doing the increases depreciation. So it would take DDX on both side off the equation. We could get some term about divide the eggs and some ex terms some wide Herms, and the rest inside is a constant. So it will goes to Gerard. So let's hunt Duda with the eggs on post side off the equation for this little gives us Mmm. Sure. He two times our 10 geant off three. Why times that you're ever tip about attention? Which one? Over one plus three y squared times three. Why crime? Plus the second term is 18 times arc tangent off to eggs times one over one plus four eggs Square times two equals two So wouldn't simplify it a little bit about this equation And we can get ade times Arc tangent off three. Why divided by one plus nine Why square times why crime? He close negative three times Arc tangent A few eggs he wanted by one plus four X square And now we want to club in the value Our ex ecos square with three over too. Why equals 1/3 So this equation now gives us paid times our attention off one which is pi over four. You added by one plus one is two times why prime? Because negative three times our attention's off square with three is pi older three divided by one plus three. So finally we will get why prime is equal. Chu tight over nature of pi over four you ate the butt party showed the final anniversaries negative one force.

In this case we want to figure out when we have horizontal and vertical tangents. Actually we found both of those points from previous problems. But let's go through how we do it. Attack. Do we find the tangent? Okay. As a function of T. Is mm to t comma to t minus eight. The slope we want to first find when the slope is zero. So that means that this thing needs to be zero. And we get that. That's teak was four. And so we were already figured that out down here. And when that's it, C equals At 7 -16. So when when we had to get a vertical line, we need a slope of infinity. So we need this thing to be zero and that's just zero when T. Is zero. As we've already found that point before. So we get that that is mm. See C at zero is -90. So here we have a vertical tangent and here we have a horizontal

And this problem was asked to figure out where the tangent of the slope of this curve has a value of 1/2. So again we can figure out the tangent like we did in the previous problem. It's two t minus two t comma to T minus eight. So the slope is the This is kind of like the s R D Y. This is the X of the Y t X. And so we get to T minus eight, provided by two tea, and that is simply one minus four over tea. And we want to set that equal to half figure out where the slope is 1/2 and we simply get that that that occurs when t equals eight. And if we plug that into our curve, we see that that occurs at point 55 0 So, actually, if we extend the curve that's shown in the figure you get first Pope is 1/2 right when it comes back into the first quadrant,


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