pi's Specialist Maths Questions

[ariawuu]

 

...

you mean how to work out what tan-1 (3/4) is? use the triangle.. not sure if it answers the question

Which triangle?

nvm i was thinking of something else..

[pi]

Oh god that double derivative question... Spent a few minutes doing that by hand
I saw that in the Math Quest specialist book. I don't really think there is an easy way to do it since most of the time the book would give you the simplest possible answer.

I sped things up a little in the end.

1. Quotient rule
2. Rearrange to put denominator on other side
3. Implicit diff and then rearrange for d^2y/dx^2

Still tedious though 

[pi]

A few questions and course queries (not many are on the course, but you never know ):

Q1
How would I do this:
Tried integration by parts, couldn't manage to get rid of the trig in the integral 


Q2
Is there a general way to solve this (differential equations):

I know its not on the course, but I'm just interested 


Q3
Is trigonometric substitution on the course? Or is it a purely CAS thing?


Q4
Any way to approach integrating trig powers when the power is a fraction? eg. ?
Another 'just interested' question


Q5
This is n00b and trivial, but when sketching , should we sketch the part that is negative on the x-axis too? My CAS doesn't



THANKS!

[sajib_mostofa]

A few questions and course queries (probably none are on the course, but you never know ):

Q1
How would I do this:
Tried integration by parts, couldn't manage to get rid of the trig in the integral 

Q2
Is there a general way to solve this (differential equations):

I know its not on the course, but I'm just interested 

Q3
Is trigonometric substitution on the course? Or is it a purely CAS thing?



THANKS!

haha you're an eager one.

1) If you can't get rid of the integral, it might mean that you may have to swap what expressions you assigned to u and dvdx initially i.e trial and error. Or if that doesn't work either, you may need to repeat by parts integration for the second time. *N.B this isnt on the course.

2) I think differential equations might actually be on the course if I remember correctly i.e Newton's rate of cooling . Sometimes, you might have to flip your expression to be able to integrate. i.e say you have dvdt = v^2, you may have to flip and integrate dt/dv = 1/v^2

3) I'm pretty sure trigonometric substitution is on the course. You may be required to do it by hand if its in the first exam.

* Edit: Sorry for the general answers but you latex doesn't seem to appear to me.

[pi]

#1 My problem with q1, is that when I integrate it by parts (I tried the swapping v and u too), I get a trig (either sin or cos) in the integral again, and this keeps repeating. I'll research some methods

#2 DE's are on the course, but the one mentioned isn't in Mathsquest and Essentials I think the tip you gave (a good one!) was for first degree ones. I can't think of a way for 2nd degree as they don't obey fraction laws apparently (could be wrong?)

#3 Thanks!



Posed two more qs

A few questions and course queries (not many are on the course, but you never know ):

Q1
How would I do this:
Tried integration by parts, couldn't manage to get rid of the trig in the integral  


Q2
Is there a general way to solve this (differential equations):

I know its not on the course, but I'm just interested  


Q3
Is trigonometric substitution on the course? Or is it a purely CAS thing?


Q4
Any way to approach integrating trig powers when the power is a fraction? eg. ?
Another 'just interested' question


Q5
This is n00b and trivial, but when sketching , should we sketch the part that is negative on the x-axis too? My CAS doesn't



THANKS!

[sajib_mostofa]

Q4) Let me think about that Q 

Q5) You should sketch both positive and negative parts. Maybe try adjusting the scale in your CAS?

[TrueTears]

eh doesn't have an elementary integral lol

[Mao]

Q1.









Q2.
For this one, you are venturing into second order differential equations. If f(y)=ky, then you have an linear ODE (ordinary differential equation), which is solvable but the technique is beyond specialist. If f(y) is not a linear function of y, then you are venturing into nonlinear ODE, many of which don't have elementary solutions (or have elliptical functions as solutions, which you simply don't want to deal with. I've finished my maths major and I still haven't had a need to deal with these functions yet).

Q3.
IIRC it is part of VCE.

Q4.
There is a solution to that, expressed in terms of an elliptic function. (you can see what the solution is on wolfram integrator.

Q5.
Sketch the negative branch.

[HenryP]

Your CAS might not be showing it because it's in Imaginary numbers mode. Change the CAS to real mode and it should sketch the negative branch

[pi]

Thanks sajib_mostofa, TT, HenryP and Mao!

Re: Mao's post, thanks for Q1, seems like the rest were out of the course

Re: HenryP's post, THANKS

[pi]

Is something like this allowed?



Especially during Weierstrass substitution where the original function had a limit of 0 to pi. Is that even allowed, especially as only if -pi<x<pi...


Sorry if question is a bit ambiguous, but I don't want to post the question as it is assessment (got the right answer, just want to know if what I have done is mathematically legit)


EDIT: When evaluating the above integral on CAS, appears, but not by hand... Can I ignore it?




Thanks!

[moekamo]

normally you'd take the 'icky' point and replace it with or something, then you'd evaluate the integral in terms of . Then at the last stage you'd take the limit as where is the value that made the original integral go pear shaped.

This is basically how you integrate improper integrals(have infinities or asymptotes in the terminals/between the terminals) and determine weather they converge to some number or diverge to infinity.

This way, you don't have infinities or undefined's on your terminals. You also avoid potential situations as well.

So in this case(if i understand your problem correctly) you would replace the with , carry out the integration in terms of that, then take the limit as .

But yea, without the question its hard to know if what you did is rigorous enough, probably at VCE level it is though...but then again, im not your teacher

[pi]

THANKS!

I'll try that, report back tmrw

[QuantumJG]

normally you'd take the 'icky' point and replace it with or something, then you'd evaluate the integral in terms of . Then at the last stage you'd take the limit as where is the value that made the original integral go pear shaped.

This is basically how you integrate improper integrals(have infinities or asymptotes in the terminals/between the terminals) and determine weather they converge to some number or diverge to infinity.

This way, you don't have infinities or undefined's on your terminals. You also avoid potential situations as well.

So in this case(if i understand your problem correctly) you would replace the with , carry out the integration in terms of that, then take the limit as .

But yea, without the question its hard to know if what you did is rigorous enough, probably at VCE level it is though...but then again, im not your teacher

With integration you can essentially be as pedantic as you want.

For example:



Actually doesn't converge, whereas:



So in general you say that the above integral is the same as the first integral and thus apply this rule to any odd function that's the integrand (especially if it's a minor step in a huge problem).

With highschool maths you don't really have to worry about the behaviour of the integrand at the limits of integration, simply because you don't get the technology until analysis. Obviously if you're interested in looking at the maths in more detail then definitely go further.

[/0]

If you were doing integration by parts you would have to do it twice, bit of a pain