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J 47 [a dx Problcm Z (12 1 dx points) points) JI itis If it Determine Determine H V evaluate the 1 the the given thc given xenvalue _ e value of the of the Go ivee...

Question

J 47 [a dx Problcm Z (12 1 dx points) points) JI itis If it Determine Determine H V evaluate the 1 the the given thc given xenvalue _ e value of the of the Go iveegeal IX divergent or divergent

J 47 [a dx Problcm Z (12 1 dx points) points) JI itis If it Determine Determine H V evaluate the 1 the the given thc given xenvalue _ e value of the of the Go iveegeal IX divergent or divergent



Answers

Compute the Jacobian (at the point, if indicated).
$$G(u, v)=\left(v \ln u, u^{2} v^{-1}\right), \quad(u, v)=(1,2)$$

For this problem. I'm gonna take a look at the improper and a girl. Um and this is improper. It's a little bit tricky, but it's in between the interval year at zero. And so what we need to do is we need to take a look at this in a girl from negative 10 and then again from 0 to 1. And so we'll split it up, analyze each part, determine if it's convergence or divergence. So the first part, we have the limit as our goes to zero from the left side of the integral. Uh, think of 10 And I'll make this extent of the second little easier to anti differentiate when we get the limit as our goes to zero from left side, um, are anti derivative is going to be native one over X promenade, Evelyn zero. And, uh, actually, I'm sorry. I should have put this in already. So let's make this our not zero based on our limiting value. Okay, so we have limit as our goes to zero from the left of negative one over R plus one over negative one. And so when we look at this first piece the negative one over R for action as our goes to zero from the left. Um, that value that's put in for our is a very small number. Ah, and it is negative because coming from the left zero and so you'll have negative one over a very, very tiny negative number which approaches infinity. And so this particular piece to the interval or to the integral is gonna be infinite. It's gonna diverge. And so it doesn't really matter what happens to that second part. This particular improper interval is divergent.

Using the comparison theory test we have the insignia From 0 to Infinity. Eggs divided by X. Cube Bloods one. The eggs to be less than or equal to the integral from 0 to 1. You have eggs divided by S. q. Plus one the X Plus the bigger from one to infinity. You have X Divided by Esky Blood. zero the S. So we have there is infinity X divided by X. Cube Last one. The eggs to be less than the integer from 0 to 1. You have X divided by X. Key platz one the X plus this parts would give us an integer from one to infinity. You have one divided by X squared the X. So this is going to be from 0 to infinity. Eggs divided by x cubed plus one the X. The X less than or equal to. You have been Ciga From 0 to 1 X. Divided by S. Q. Plus one the X. So this interior will give us plus. So I see this would give us minus one divided by X. The interval from one to infinity. So then We have the instagram from 0 to infinity. You have eggs Divided by S. q. Plus one. The eggs less than or equal to this integra Which is 01, eggs Divided by S. Cube, last one the X. Glass. This would give us one divided by -1 divided by Infinity Plus one divided by one for this part. So then you have this eggs divided by S. Key plus one. The eggs TV less than or equal to the integer from 0-1, eggs divided by S. Q. Plus one, The X. Plus one. So you notice that the right hand side so you realize that this this is fine nights Finite on the Interval 01. And here is the lifts. The left is finally because it is laser, so this is fine nights because it is less done this. Okay, so then this implies that so we can conclude and say that they in figure converges.

Hello, everyone. Today they're going to solve problem number 15 since they're always left than it apart. Minus six. For our ex belongs to real numbers. We have to less than a record two plus. It is the power minus X for all X belongs to cause then began right leg to buy eggs less than two. Plus it'll four minus x. They weren't by X for all eggs belongs to close interval one toe Infinity things one to infinity To buy X the X it could tow limit defense to infinity 123 By X dx It could tow limit. They tends to infinity toe Ln x Like took living he tends to infinity toe I learned more less off key minus two a limp models off one which is equal to limit t tends to infinity toe Ln models off t with physical toe Infinity So the given individual eighth they were then. Thank you

Problem. 18 Jack. Uh huh. G is equal to the determinant off partial X portion you, which is E power bi. Um, partial X portion V, which is you? E power bi abortion. Why fortune you? Which is he? Power you and partial. Why, unfortunately which is equal to zero. So they determine is this item. I'm not buying this coupon Interest is the Roman that this item, the mind these items is negative. You be for you cross the


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