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Problem 27 Find the tangent line to the curve I = et +t2, y = e2t | 3t at the point corresponding to t =...

Question

Problem 27 Find the tangent line to the curve I = et +t2, y = e2t | 3t at the point corresponding to t =

Problem 27 Find the tangent line to the curve I = et +t2, y = e2t | 3t at the point corresponding to t =



Answers

In Exercises $27-30$ , find the value(s) of $t$ so that the tangent line to the
given curve contains the given point.
$$
\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}-t^{2} \mathbf{k} ; \quad(0,-4,4)
$$

Okay, we are going to find the tangent vector of our given vector valued function. So to do that we can just take the derivative of each of our components. So we will have one in the I. Direction negative, one in the J. Direction and three in the K direction. Notice there are no more teas. Everything was linear. So now we just have Constance in each direction, so when we put the zero in, we still get that I minus J plus three K.

So intersex itself at X equals zero like, well, six So t iwas plus or minus the square root of six. Uh, now we take X prime. We have three t squared, minus six. And why is equal to two tea your wife from his duty. So you i d x That's plus Squared six is gonna be cheers. Six over 12 says room six over six and then when it's negative, is gonna be negative. Tiered. Six over 12. So negative. Read six over six. Uh, since x zero, both these are changes. He lines and be for six over six x a six. Is he with Why and why is equal to negative six over six x six.

Schools were given a curve and were given a point and rest. Find the value of teas at the tangent line to this curve contains this pent occurred is R. T. Equals negative T. I plus T squared J. Plus natural log of T. K. J. And the point is to -5 -3. We we have been blessed with Anthony Mr Kanye has allowed us to come in his studio and discuss Jim Jarmusch's name, norman Wilkerson is going to message me and call me so many names for how bad this is this podcast. Right. Even the good ones norman I think that uh I was well we have the derivative our primary Because well this is -12 T one over T. And he destroyed new Orleans like the war was over. And he was like well there's still a war going on now. The point of tangent C. Will be at T. Zero. So the point R. T zero which is in vector form negative T zero T zero squared natural log of T. Zero. So it must be true that are given point to negative five negative theory is equal to negative. It looks pretty badass. But we're laughing the whole time because there's just graffiti everywhere and one of the most prevalent parts of your feet it's just guys negative T. Not T. Not squared natural law that T. Not plus a constant K. Times our prime learning how to do a swastika. On the day he went to six flags. That's a memory he has now Is the day he went to six flags and learn how to draw us watching. Which is a -12 T. Not one over Tina. Therefore it follows that two equals negative T. Not minus K. Negative five equals T. Not squared plus two K. T. Not and -3 because natural law of peanuts plus K. Over t nuts. Um It's very bad. We were laughing at Michelle Pfeiffer remember because she was like she's just the biggest will simplify character. First equation I get that all fucking. Yeah she's taking some snooty bitch. K. Is equal to you guys. A negative T zero minus two. This cat is a fucking nightmare dude. According this. Into the last equation, we get negative three equals natural laws of T. Zero minus T. Zero plus two Hoover T. zero. Well that's one of kurt it's not really the best approach. A better approach to be published in the second equation negative five equals T zero squared plus two times. Well this is uh Negative easier -2 into doing um times T. Zero, which simplifies to t not squared minus two, not squared minus four to not therefore do not swear plus forward to night minus five equals zero. And so the fact that this is T not minus one times 2, not plus five equals zero. So we get that tina is equal to positive one for negative five. Of course you see that Tina, it must be greater than zero. So the only real option it's not because once.

Given to parametric equations, we are asked to find the equation of the tangent line at T equals four. So our first step here is we want to write the generalized formula for the equation of a line and this can be given by point slope form. And for our why not an ex not values that is simply going to be our function. Do you find At the .2 equals 4? So before we do that we're going two right F of T equals X and right G f t equal Y. So now we can again four F&G of four. And for this we just plug into our parametric equations here. FF four is simply going to be the square root of four, And G 04 will be four squared minus two times four, which the square root of four Offly equal to two in four squared minus two times four is 16 minus eight, Which is equal to eight. So now we can plug that back into our equation of a line and we find that why minus eight is equal to mm Times X -2. Now our next step is we need to find this value M. And we know that M. With point slope form is the slope of a line. And because we were asked for the tangent that clues us in to the slope of the tangent line at that point, which is our derivative. So with parametric equations, we know that you slip of line is equal to the derivative of Y with respect to X, which is equal to the derivative of why? With respect to T divided by the derivative of X. With respect to. So when we evaluate that out we find that this is equal to be true to why with respect T Goes out to the to T -2. If we just apply the power rule and for X if we apply our chain rule, We find that it equals one divided by two times the square root keep and simplifying this out. We can factor out to here two times The quantity T -1 times one over R. X, derivative which is 1/1 over to rooty. That just goes out to be T. And we find that our derivative of Y with respect to X is equal to four routine times the quantity t minus one. And we are interested in the slope of this at the Point T equals four. So we can simply evaluate this that T equal four. And we find that we are left with four times the square root of four times four months, one, four times the square root of four. Leave us with eight and four minus one is equal to three. So eight times three we find that come all the way over here that mm is equal to eat times three, Which is equal to 24. So our last step here will be plugging in this value up here. And once we're done with that we find that y minus eight is equal to 24 times x minus two. And we can leave it like this, but we'll take it one step further and simplify and find that y is equal to 24 X minus 48 plus eight. And we can combine these like terms here and find that Y is equal to 24 X- for you. And that is our final answer.


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