4U Maths Question Thread

[RuiAce]

Why is the domain of cis(theta) restricted to between -pi and pi?

It's not?

The value of \(\theta\) that satisfies \( -\pi < \theta \leq \pi\) is just the principal argument of a complex number

[forrea099]

Hi,
If this was the graph of derivative function f'(x), what would the graph of f(x) look like?

[RuiAce]

Hi,
If this was the graph of derivative function f'(x), what would the graph of f(x) look like?

It will look like:
- Between 0 and 1: A straight line of gradient 3
- Between 1 and 3: A parabolic shape that eventually flattens out as x approaches 3
- Between 3 and 5: The same parabolic shape, except now going downwards
- Between 5 and 7: A parabolic shape, with the gradient gradually diminishing from -3 to -2
- Between 7 and 8: A parabolic shape, flattening out a bit more quickly at 8
- Between 8 and 9: The same parabolic shape from 7 to 8, except now going upwards
- Between 9 and 10: If you think about that one, it will essentially look like what happens between 8 and 9, but inverted. This is because you want your parabolic curve to flatten out yet again at 10.
and it can start wherever you want it to because you haven't specified anything about a starting point.

Note that the actual shape of the graph is very hard to describe for such a graph because it's literally being composed by a LOT of straight lines. You would need to do a lot of work to determine the exact shape. So if you want a more descriptive answer, you should post your attempt at sketching the shape itself.

[radnan11]

are allowed to use euler's law to prove de moivre's theorem instead of induction?

[RuiAce]

are allowed to use euler's law to prove de moivre's theorem instead of induction?

No. Euler's formula is not a part of the course and therefore must never be used for a final answer.

In fact, attempting to use Euler's law to prove De Moivre's is actually assuming what you're trying to prove. Because you're essentially assuming that \(( e^{i\theta})^n = e^{ni\theta} \).
The index law \( (a^m)^n = a^{mn} \) is NOT a known fact over the complex numbers; De Moivre's law helps to prove that it's true.

[justwannawish]

Hey, I just have a polynomial question that I'm a bit confused about :/
 I've done part (i) but can't figure out the second part
 
It can be proven that cos6x =32cos6^(theta) - 48cos^4(theta) + 19cos^2(Theta) - 1
(i) Find the roots of P(x) = 32x^6 -48x^4 + 18x^2 - 1

Spoiler

My roots were x = cos(pi/12), cos(5pi/12), cos(7pi/12), cos (11pi/12), and positive and negative 1/root 2

(ii) Show that cos^2(pi/12)+cos^2(5pi/12)=1

[RuiAce]

Hey, I just have a polynomial question that I'm a bit confused about :/
 I've done part (i) but can't figure out the second part
 
It can be proven that cos6x =32cos6^(theta) - 48cos^4(theta) + 19cos^2(Theta) - 1
(i) Find the roots of P(x) = 32x^6 -48x^4 + 18x^2 - 1

Spoiler

My roots were x = cos(pi/12), cos(5pi/12), cos(7pi/12), cos (11pi/12), and positive and negative 1/root 2

(ii) Show that cos^2(pi/12)+cos^2(5pi/12)=1

[mxrylyn]

Heyy

Im stuck on integrating x / the root of 1- x^2, with limits at 1/2 and - 1/2

 

[RuiAce]

Heyy

Im stuck on integrating x / the root of 1- x^2, with limits at 1/2 and - 1/2

 

Hint: This is just a 3U integral, but of course in 4U you're not given the substitution. Try \( u = 1-x^2\).

[mxrylyn]

Hint: This is just a 3U integral, but of course in 4U you're not given the substitution. Try \( u = 1-x^2\).

Thank you!

[vikasarkalgud]

Hey Rui,
what would be the most efficient way of integrating I = (integral) sqrt((4+x)/(3-x)) dx ?

[RuiAce]

Hey Rui,
what would be the most efficient way of integrating I = (integral) sqrt((4+x)/(3-x)) dx ?

\[ \int \sqrt{\frac{a+x}{b-x}}dx \text{ and similar forms are usually a little nasty.}\]

Of course, we all know that these integrals are still nasty. We have no choice but to then expand the bottom, complete the square and then split the top into two separate integrals (one which integrates to \( \ln \) and another which integrates to \( \sin^{-1} \)).

If you have some free time, you can try experimenting with this. This may or may not work, but you can perform the substitution \( u = x - \frac12\) to reduce it into \( \int \sqrt{\frac{\frac72 + u}{\frac72 - u}}du \) and then perform another substitution \( u = \frac72\cos2\theta\). You'll need double angles if you walk down this path.

Edit: But the idea is that it ultimately boils down to something like this: \( -14\int \cos^2\theta\,d\theta \). This might've taken longer than the other method tbh, but it's quite interesting to investigate.

[Jeeffffffffffffff]

Hi
I’ve got an integration with trig substitution question and I’ve got an answer for it but I’m not entirely confident in it and was wondering if you could check it for me please?
The question wanted the integral of the square root of 6x-x²-5 and I got
2sin^-1((x-3)/2)+(((x-3)•sqrt(4-(x-3)²))/2) +C
Sorry for all the messy typing I tried uploading a photo but the photo was too big :///

[RuiAce]

Hi
I’ve got an integration with trig substitution question and I’ve got an answer for it but I’m not entirely confident in it and was wondering if you could check it for me please?
The question wanted the integral of the square root of 6x-x²-5 and I got
2sin^-1((x-3)/2)+(((x-3)•sqrt(4-(x-3)²))/2) +C
Sorry for all the messy typing I tried uploading a photo but the photo was too big :///

Looks legit.

(Note that \( \sin^{-1} (-x) = -\sin^{-1}x\), so there's no discrepancy.)

[mxrylyn]

Heyy

Which methods do I use to integrate (×+2) / (the root of: ×^2 - 1)

And also the root of: 9 -  x^2