Complex plane and some uni maths (1 Viewer)

[jb_nc]

Ok, I didn't know where to post this but there is one question to do with Ex 2 I've forgotten how to do (for mah test on Monday). There isn't really a uni maths help subforum that gets posted in more than once a month. So uni people any help would be appreciated because I have nfi. I mean I even typed them up in LaTex.

1) Sketch the region of the complex plane. abs(x) means absolute value/modulus just fyi. lol @ not being able to do it, i know. i never paid attention in inequalities.

[This is pretty hard AFAIK, last question in a test] 2) The equation defines a curve in the (x,y)-plane through the point (1,1). Find the tangent to this curve at (1,1).

3) i) [Can do this part just runs onto ii)] Using L'Hôpital's rule, show that
ii) Hence find .

4) An ant walks on the surface z = 9x² + 4y² + 6xy in such a way that its position at time t is determined by the parametric equations:

x(t) = 2cos(t) and y(t) = 3sin(t) for t≥0.

i) [If someone could prove this using the partial derivative chain rule that would be great] Use the Chain Rule to show dz/dt = 36cost(2t)
ii) Hence, find the first time t > 0 that the ant is at its maximum heigh above the xy-plane as it walks on the surface z = 9x² + 4y² + 6xy.

Cheers.

 

[Captin gay]

I just did this question recently, but without the first part.

From the definition of the sinh functionn

divide everything by e^x

"Sub in" infinity.

The other questions meh ill try and do them ater now I gotta eat then go back to shitty algebra.


P.S What software can I use to type latex in LINUX

 

[Captin gay]

[This is pretty hard AFAIK, last question in a test] 2) The equation defines a curve in the (x,y)-plane through the point (1,1). Find the tangent to this curve at (1,1).

Do you mean the equation of the tangent? If so, I got y=-x

lol

EDIT, I mean y = -x +2 which is correct.

EDIT2: To solve this question just differentiate implicitly to get dy/dx, remembering the RHS of the equation becomes zero. Subbing in a point 1,1 you can get m (gradient) = -1. then just use y-y1 = m(x-x1) from HSC 2u to get the eqn of the tangent.

 

[jb_nc]

Ummm, that's the question word for work afaik but yeah it would mean equation of the tangent I guess.

Check ur PMs too btw.

P.S What software can I use to type latex in LINUX

No idea, I used msword equation editor. There's heaps you can use but with LaTeX you still have to type it up the way you would on wikiedpai/wherever. i.e. LaTeX is pretty gay.

 

[jb_nc]

imo every q here can be done by a capable 4u student, they are not really that bad. First q is run of the mill 4u locus, it is simply sketching an inequality involving 2 circles on an argand diagram. Second q is run of the mill implicit differentiation. Third q is just l'hospital. Fourth q im pretty sure u just set dz/dt=0, giving t=pi/4. Check that its a max by checking d2z/dt2=0.

err i never did 4 unit but the methods are different. You don't learn the implicit differentiation chain rule do you? Or defns for sinh and cosh?

But yeah, I wouldn't put it beyond ext 2

 

[pLuvia]

First one you draw the line y=3, and y=-1, and the line in the centre in your locus. The greater sign just means the region is above that y=1

Second one, implicitly differentiate you'll get dy/dx=-(e^y-2ye^x)/(2e^x+xe^y) sub in (1,1) do get dy/dx then just point-gradient formula and you get the tangent