Monash Maths thread

[EspoirTron]

So UoM has a Maths thread so by consequence we need one too! Feel free to ask any Maths related questions here relating to any units across Mathematical sciences at Monash and hopefully someone can help.

Enjoy the Maths!

[kinslayer]

aww yeah, following!

[m.Chemia]

Been waiting for this thread for so long

[LaLaLouise]

Yeahhh maths!

Was wondering if anyone is doing MTH2010 and is attempting to do assignment 2? What the hell is question 2 even about hahaha. Anyone want to give me some hints on 2b and c? I've got a and d

[kinslayer]

Not doing MTH2010 but if you post the question(s) I might be able to help

[keltingmeith]

This is absolutely beautiful, I'm definitely coming back here a lot.

[LaLaLouise]

Thanks kinslayer any help is much appreciated

I've attached the pdf of the assignment. I've done all of the questions except for 2b and c.
 
For 2b I understand how to get a critical point, take the derivative of the function and equate it to zero. From 2a we get the equation and I take the derivative the function with respect to a and b so we have 2 equations and then attempt to solve for a and b but the answer just doesn't come out right. My friend also found this pdf, http://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/54/handouts/MethodLeastSquares.pdf , and got to 3.12 , I can get these equations but MTH2010 doesn't teach those steps afterwards. So confusing haha

For 2c I'm not really sure. With the condition the denominator will be negative for a and b. I understand what a local minimum is, I think it may have to do the second derivative test??

Thanks again and sorry for so much writing

[kinslayer]

The partial derivative calculation might be confusing you- once you have the correct form the rest of it is not hard (but a lot of writing...)

I've done the first part of 2b), see attached.

Showing that it is a local minimum is just a matter of applying the second derivative test for functions of two variables. It's in Stewart somewhere I think or you can see here:

http://mathworld.wolfram.com/SecondDerivativeTest.html

It works out to be very simple and you just have to use the fact that given on the sheet.

[LaLaLouise]

Ohhh I see where I went wrong with the derivative now which makes the question so much easier haha

Yeah I thought we has to use the second derivative test for 2c and I figured out the answer right after I posted the question haha

Thanks so much for you help especially since it's time out of your holidays!

[EspoirTron]

Hey guys, just had a few problems if anyone could please help me out.

Q1. A right circular cylinder is inscribed in a cone with height h and base r. Find the largest possible volume of such a cylinder.

Q2. The base of a solid is a triangular region with vertices (0,0), (1,0) and (0,1). Cross-sections perpendicular to the y-axis are equilateral triangles. Find the volume of such a solid.

Q3. Set up an integral for the volume of a solid torus with radii r and R.

Q4. Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere. (This answer is in terms of r).

If anyone could help I would really appreciate that!

[mrb3n]

Hi guys,

I've got two questions regarding limits. I'll post a picture because trying type them would be very hard.

Any guidance you could give me would be appreciated.

[kinslayer]

Hi guys,

I've got two questions regarding limits. I'll post a picture because trying type them would be very hard.

Any guidance you could give me would be appreciated.

1.a) i)  This is an indeterminate form of type so you can use l'Hôpital's rule. Limit becomes .

1.a) ii) Expand the denominator, then divide numerator and denominator by . Use the fact that for integers n>0. The limit is 1/15.

1.b) f(x) = -3 for all x < 0 and f(x) = -1 for all x > 0. So the left-hand limit is -3 and the right-hand limit is -1.

Not sure what the limit laws are that you are supposed to use but I assume it's just things like;

• the limit of a sum/difference is the sum/difference of limits
• the limit of a ratio is the ratio of the limits (for nonzero denominator)
• the limit of a constant times a function is constant times the limit of a function
• etc...

Just remember to write down which law you are using when you use it.

[mrb3n]

Thanks so much. I could crack all the other problems, but these limit ones just do my head in. I think I should be able to explain the different processes using limit laws, it shouldn't be too hard

[DisaFear]

I can do b), given a) - It will just be the inverse

I can't seem to grasp the process to figure out a) though. I've looked at my lecture notes, and I think he does it sort of intuitively. I'm not sure about whether which bit should be a column or a row. (Yea...basic math flaws )

If anyone could outline the steps I'd need to take for a), would be thankful

[kinslayer]

I have not done 2021 so I'm not sure how it's laid out in the lectures. The basis B' looks like the "standard" basis for so finding the change of basis matrix isn't as hard as it could be .

I used the procedure here: http://www.math.hmc.edu/calculus/tutorials/changebasis/changebasis.pdf

For part a) you should end up with this matrix for the change of basis (I think):



Verifying: if you have a vector with respect to B then AKA .