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5) Determiue a function f(&) %3, where p and 9 are polycornial functicns, guch that f (v) is undefized at 2 =3ad€ = ~Ibut hes a vertical esymplote cmly &a...

Question

5) Determiue a function f(&) %3, where p and 9 are polycornial functicns, guch that f (v) is undefized at 2 =3ad€ = ~Ibut hes a vertical esymplote cmly &t < =3ln)f(c) =Y.x

5) Determiue a function f(&) %3, where p and 9 are polycornial functicns, guch that f (v) is undefized at 2 =3ad€ = ~Ibut hes a vertical esymplote cmly &t < =3 ln) f(c) = Y.x



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). $

Again this question of function is given that is F X is equal stool three X raised to the power five by six Plus seven X raised to the power to buy three. Okay. And were able to find out the differential of this function So no problem. First of all we will differentiate both are the equation with respect to X and it will be F dash X equals two D over dx off three X raised to the power five by six plus seven. Access to the power to buy three. Okay. And we know the differentiation with respect to X. Of access to the power and it is an access to the power and minus one. So F dash X is here. Okay it is three is already constant and now access to the power five by six. So differentiation will be five by six. Access to the power five by six minus one that is minus one by six. Okay. And plus seven is already there. Okay. And the frustration of access to the Power to buy three it will be two by three. And access to the Power to buy three minus one that is minus one by three. Okay and now FDS X can be written as it is 15 by six Okay. Or we can say five by 25 by two. Okay. And access to the power minus one by 16 numerator. So it will be denominator that is access to the power plus five Plus one x 6 Or we can say 6th wrote off X. Okay and less. It will be 14 x three. Okay and it will be Q wrote off X and the spillover FDS X. Okay. And now to find out the differential we will use the equation that is diva is equals two F dash X. Dx Ok so do I will be we will put the value F dash X from here. Okay. And it is five divided by two and sixth wrote off X plus 14 divided by three and cube root of X. And this is our sx and Dx. So this will be the final answer of discussion. Thank you.

Why did you say we're told that four X squared plus nine y squared equals 36 on the show? So this is the equation of an ellipse where X and y are both functions of t Get this. Pardon me. We're told that dy DT equals one third and ratifying the x tt when X equals two. Yes. Cigarette. Yeah. And why equals two thirds of route five? Yes, disappear. So in order to find the x c t, I'm going to use implicit differentiation on our lips equation. So differentiating both sides of respect the teeth. So we have two times four is eight x times The expertise plus two times nine is 18 times y times dy DT and the right hand side is simply zero solving for the execution, the expert C is equal to negative 18 y times dy DT over eight x which you could simplify to negative nine wide dy DT over guys, we're we're at X plugging in our values. You get negative nine times two thirds route five times one third over four times two it's ms is equal to negative. See the 9/9 minutes to five or four times two, which is Route 5/4. All right, please. This is DX DT Negative. Route 5/4, actually, Sharp party. We're told that the X 15 equals three and rectifying the white tea when X equals negative two. And why it was two thirds or five. Said was plan now, just like in part a use implicit differentiation. And we get the same equation and I want to solve for dy DT instead of the expertise. So we have D y t t is equal to negative eight x the x 30 over 18 y couldn't you just lie up? This simplifies to negative four x over nine y the x 30 substituting our values. This is native four times negative. Two over nine times two thirds Route five Nice three is equal to the negatives. Cancel out 8%. Now this is eight over 19, 0 to 5, which is 4/5. And if you rationalize the denominator, this is equal to four. Route 5/5. Oh,

For three exit, um, and then get with a why cling h e he read to you each of five inch, 307. Keep time calling. He coined six. It's a five time three Yukun eyes four. If X fix to comma Going six. And why to be Klink? Um, minus eight now. Softer partials fee Arsenal Team Eagle Teoh. Thanks. Purcell's See? Oh, sure. Uh uh. Partial X com. Why? Partial x p plus partial X. Why are sure I Why? Partial t now partial sea partial t at keep, uh, keep going. Keep going. Three, Which is equal, Teoh six times five plus minus eight. My multiple remain forever. Just equal, you know, 62.

Hello and welcome back to another differential equation problem. This one is dy dx is equal to three X squared minus E. To the X. All over to I minus five. The initial condition of Y of zero is equal to one. Since this is a separable differential equation will multiply both sides by two Y minus five and D X. To get why is on the left side and X is on the right side. I'll do that. Now we will have to I minus five. Do I. Is equal to this three X squared minus E D X times dx. These are fairly simple integral to do since there are just exponentials and poblano mills. So I'll do that right now. The integral of to I -5 is just quite squared minus five Y. This is equal to on the right side will be X cubed minus E to the X. Don't forget that. Plus see that constant of integration is very important. Two get why all by itself. Now we're going to have to complete the square on the left side. And what I'm going to do is a little trick these problems where I can just add any constant I want to one side and it won't really matter because I still have this plus C. Here in any constant I add to that side will be absorbed into that. Plus C. I'll do that now To complete the square will have to divide this middle number by two and square it. So I'll add 25/4 to both sides. So why squared When is five Y Plus 25 or 4? Mm. And this is going to turn into why minus five halves squared already nat next I'm going to set this equivalent to that. And so for a while so I will have why minus that has is equal to the square plus or minus the square root of X cubed minus E to the X plus C. The last step here is to add five houses to both sides. We'll do that now. Five halves like that. All right now that we have this uh difference equation here we can substitute an X and Y. For of our substitute and the X and Y of our initial condition and get this see value. So if X equals zero and life was one, write this down as one is equal to plus or minus squared of zero -1 Plus C plus five halves. So that will be negative. 3/2ves is equal to plus or minus The Squared of C -1. Looking at this, we realized that it can't be a plus square root because there is a a negative value on the left side and the result of a square can never be a negative number. So we'll have to just take the negative square root, we'll be crossing that out. And then to solve we will get 9/4 Is equal to C -1. To see Ends up being 13 over four. Great. And we'll substitute that value back up to our equation appear. Let's do this one. So if we know see is 13/4, that's right that in here and we also know that this is the negative. But Alright, great. That's our answer to part A to part B. We will have to just use a graphing calculator or some other computer aided software because this involves both exponentials and um paul animals. So this cannot be easily uh graft there will be twists and turns. Um But having done this problem on a graphing calculator, I can show you the graph of this. It will not be very intuitive, but I will do my best. Um This graph will look something like this here where this endpoints and that one point are on the X axis. I'm just right at this graph. Will this end point on the right will be about 4.63. This endpoint on the left to be negative 1.44. Um To solve for these values requires math, that is not taught in this class, whereas knowledge of the uh product log function and non elementary solutions to uh equations. So I thought this was beyond the scope of this problem and we will not be covering it. Lastly to know the range within which the values of X are valid. Looking at the graph, we know that it will be in between -1.44 And 4.63. Again, this problem will be very difficult for the last two parts if you do not have a graphing calculator, so I recommend getting one and plotting this equation right here. I regret this is in the problem.


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