5

Ipf Vi-T dy 2. ^" [ - 51 dx 76^ f % 12 2at! [ + dx 5. Tp J7+ dr J #+6sy; t dt 7. Vi-7IVo_a dxdt 01+25 dr IV1 - (In x)10....

Question

Ipf Vi-T dy 2. ^" [ - 51 dx 76^ f % 12 2at! [ + dx 5. Tp J7+ dr J #+6sy; t dt 7. Vi-7IVo_a dxdt 01+25 dr IV1 - (In x)10.

Ip f Vi-T dy 2. ^" [ - 51 dx 76^ f % 12 2at! [ + dx 5. Tp J7+ dr J #+6sy; t dt 7. Vi-7 IVo_a dx dt 01+25 dr IV1 - (In x) 10.



Answers

If $ x^2 + y^2 + z^2 = 9, dx/dt = 5, $ and $ dy/dt = 4, $ find $ dz/dt $ when $ (x, y, z) = (2, 2, 1). $

To use Part one of the fundamental theorem of calculus, we need the upper limit of integration to be a variable. In this case the upper limit of integration is a value in the lower limit is a variable. We can rewrite being a girl by reversing the limits of integration and put it in a negative in front. This will allow us to use Part one of the fundamental theorem of calculus correctly, given us a result of the opposite of X squared minus five that quantity to the 10th power. Yeah, yeah.

You know, first we have the question Why equals integral from CO sign of X. That sign of X, uh, one plus V squared to the 10th Power. Do you V Okay, so we have this and were asked to find the derivative of this function. So as we know by the fundamental theorem of calculus part one, when you take the derivative of an integral it just becomes the function itself times the derivative of the bounds in each situation. So, for example, here's how we're gonna do this. So you have one plus co sign. X Squared to the 10th is gonna be the first half of this integral one plus co sign Squared X to the 10. However, this has to be multiplied by CO sign X or the derivative of Coastline X, which is negative sine x. So we're gonna leave it like that and then, as we know because of fundamental term of calculus Part two, you have to subtract by the second part of the bone. So it's gonna be the second car that bound will be one plus sine squared. Okay, let me rewrite this one plus sine squared of X to the 10th plus, let's think What is it? A rood of sine x? Oh, wait, that's pretty easy. It's co sign of X So rewriting in its final form, the answer is negative sine x negative sine x times one plus oh sine squared of x A coat sine squared of acts to the 10th Power minus one plus sine squared of X all to 10 power times saw er, timecode sine x And the reason, as we know, is because of the fundamental form of calculus Step one. So if we want in written form that she's gonna be negative sine x times one plus co sine squared of X to the 10 So as we know that's what you're going to see here, right? That that is the function itself Times the derivative of the upper bound, minus the function itself. When you plug in sign So one plus I sine squared of X right times co sign of X, which is the derivative of sign of X. So that is a final answer. I hope that helps Remember when you're doing this type of ah, differentiation with into girls. What you're gonna do is you're gonna do the function with the bound clogged in time, the derivative, the bound minus D function with the bound plugged in, uh, times that area of the lower bound. And so that's generally how you combine fundamental theorem of calculus with one and two in order to get the right answer.

In the revolution we have to integrate the functions within the limits of want to eat and the companies yes learn it divided by X. The Mhm. So weird integrate this culture but well integrated letter substitute here Ln X. Right. And that's really what it is. one by its day executes 2 30. Okay now after some tradition we also have to check that new limits. That word should be there new limits since always stands for thanks. So all limits are and and we have to find the new levels that stands for the from here the Ortiz Lennox so here L. And one and here and then he value of alan one and zero value of LNG is one. So finally I am going to substitute too. This observation then my new limits are potato tuber and the place of the S. And its nuclear data. So your duty and at the place of L. N. X. T. So you have testy. Okay we know that intimidation is interrogative. Okay so here you have the derivative of F. T. That is tested and you are integrating that. So it is anti graduating. So it should provide you the function there to us. Okay. And now you have to put the limits that are their approval. So if I put one then I would have but minus have 54 year then I get up here and this table provides me The Weddings of Health one. It shows it is seven and the value of that series. So I put here those who will lose okay have 27 and then zero is fine. So I came to who is my final answer here in this question? Okay thank you.

In the given question we have to integrative function within the limits of similar to what when the function is it with the politics have dash people. That's our that's Bs. Okay so we're able to integrate this function. Finally to integrate it. I am starting by a substitution that is into the politics. It was to be this study with the with it will devour X. Dx because to date now after such conclusion we are both guilty limited to find the new limits for daily living. Okay so I am finding the limits here because the old one stands for That are zero and 1 and we need to find the squad. Mhm. Premier you can see that T. V. To the power X. So here into the power zero that is one and here into the part one that is Okay now I make all the substitution in the integration and many limits are want to eat and anti place of your IgE to devote at seven days you have to go dating. Okay, dating and here you are to put day. So finally that after steve. Okay? No, we have to integrate a test T so we know that it's derivative and you have you are going to integrating the derivative. So finally you should get the function as it is. Okay now we are going to put here their limits. So if you put here the limits then S. E -F. one. Now value of F is you can see here it is trend. So this value is 10 and value of the 27. Okay, here seven. So finally I say that my answer for this question is three years. Okay, thank you.


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