Mathematics Question Thread

[fun_jirachi]

Sorry if i post this in the wrong place this is my first time on the site. Can someone please help me with this question. 
f(x) = x^3 - 3x^2 - 9x -2
f'(x) = 3x^2 - 6x - 9
= 3(x-3)(x+1)

By sketching y = f'(x) show that f(x) is increasing when x > 3 or x < -1

You can see that f'(x) has zeroes at 3 and -1, and is a parabola that has a minimum. So you draw a parabola cutting at those points, like in the picture (bad picture, i know). And you can see it's above the x-axis when x>3 and x<-1, so since the gradient is >0 at those points, by definition f(x) is increasing when x>3 and x<-1.

Hope this helps

EDIT:

Hey guys!
I'm having a bit of trouble with part 2 of this question, I can find the time in terms of t = d/s, but not sure how to then find theta.

Thanks!   

Just saw this, gonna edit something in

Ah yes, the dastardly farmhouse question This has already been addressed in the HARD 2U QUESTIONS COMPILATION (soz for caps, im pretty sure the topic is in caps so go and check it out there !

[Mate2425]

I guys i was wondering if you could please  help me with understanding Q16bi from the 2012 HSC paper, i am finding it hard to understand their worked solutions.

Thank you!

[RuiAce]

I guys i was wondering if you could please  help me with understanding Q16bi from the 2012 HSC paper, i am finding it hard to understand their worked solutions.

Thank you!

Already addressed in the compilation

[Mate2425]

Hey, RuiAce could you please tell me a really easy way to understand when you either + probabilities or x.
Also for probability questions could you please tell me what process i follow when they say 'exactly' e.g HSC 2017 Q12e(iv) exactly one of three spins
Thank you, 

[RuiAce]

Hey, RuiAce could you please tell me a really easy way to understand when you either + probabilities or x.
Also for probability questions could you please tell me what process i follow when they say 'exactly' e.g HSC 2017 Q12e(iv) exactly one of three spins
Thank you, 

Simple:
- Times means 'and'
- Plus means 'or' (under the assumption they're mutually exclusive)

The crucial thing about that part is that there's an extra step, in contrast to the part above it. Note that the previous part specifically asked for an even number on the first spin and an odd number on the second and third spins. Whereas this part doesn't cater for that, so you need to consider the possibility where the even number occurs on the second spin instead, or on the third spin instead.

This is why your answer will be similar to the one above it, but you have to multiply by 3 to cater for how there's multiple possibilities.

(Alternatively, you could treat it like an OR scenario, and do \( \frac{18}{125} + \frac{18}{125} + \frac{18}{125} \) instead.

[Opengangs]

Hey, RuiAce could you please tell me a really easy way to understand when you either + probabilities or x.
Also for probability questions could you please tell me what process i follow when they say 'exactly' e.g HSC 2017 Q12e(iv) exactly one of three spins
Thank you, 

Hey, I'm not Rui but hopefully I can provide the same insights as Rui

With probability, you want to see whether your two events are simultaneous or not. If you want two events together, then you use the \(\times\) rule. For example, if you tossed a coin AND rolled a die, then you want these events together. So you'd use the \(\times\) rule. If the question was if you tossed a coin OR rolled a die, then you could have option a or option b. So we use the \(+\) rule.

Exactly one means you could have the following events:

1) Even-odd-odd,
2) Odd-even-odd,
3) Odd-odd-even.

Either one of these have exactly one of the three spins being an even number. So you'd have to use both, addition and multiplication rule.

As an example, to complete the first event, you would want even AND odd AND odd. So you would have \(P(\text{even}) \times P(\text{odd}) \times P(\text{odd})\), since you want these events happening together.

If you have any more questions, feel free to send a reply!

oh, he beat me to it but I'll happily have my explanation up in case you're still confused

[saige_abigail]

Hey - can I please have some help if you don't mind?
[This is from Eddie Woo's YouTube video on Superannuation (2 of 3) from his playlist HSC Series and Sequences]
Johnny regularly invests $750 at an interest rate of 8% p.a. and would like to end up with $50, 000. How long will it take him to reach this amount?
The working out starts off like this...
$50 000 = $750 x (1.08 [1.08^n -1]/0.08)
but it gets to this...
50 000 x (0.08 / 750 x 1.08) = 1.08^n -1
...which I don't get. How did he get the 1.08^n-1 on the right hand side please?
Cheers

[RuiAce]

Hey - can I please have some help if you don't mind?
[This is from Eddie Woo's YouTube video on Superannuation (2 of 3) from his playlist HSC Series and Sequences]
Johnny regularly invests $750 at an interest rate of 8% p.a. and would like to end up with $50, 000. How long will it take him to reach this amount?
The working out starts off like this...
$50 000 = $750 x (1.08 [1.08^n -1]/0.08)
but it gets to this...
50 000 x (0.08 / 750 x 1.08) = 1.08^n -1
...which I don't get. How did he get the 1.08^n-1 on the right hand side please?
Cheers

\begin{align*} 50000 &= 750 \times \left( \frac{1.08(1.08^n - 1)}{0.08} \right)\\ \frac{50000}{750} &= \frac{1.08(1.08^n - 1)}{0.08}\\ \frac{50000}{750} \times 0.08 &= 1.08(1.08^n - 1)\\ \frac{50000}{750} \times 0.08 \times \frac{1}{1.08} &= 1.08^n - 1\end{align*}
Here, I do it step by step so the order in which things got moved over has been shuffled. But it is clear that the 750 and the 1.08 both land on the bottom, whilst the 0.08 lands on the top.

[emily_p]

Hi! Quick question-

How many strips/function values are used in one, two and three applications of the simpsons and trapezoidal rule? I’m not too familiar with the reference sheet methods and prefer to use the short cut.

Thanks 

[kauac]

Hi! Quick question-

How many strips/function values are used in one, two and three applications of the simpsons and trapezoidal rule? I’m not too familiar with the reference sheet methods and prefer to use the short cut.

Thanks 

Hi...

For both Simpson's and trapezoidal's, the rule for the number of function values = number sub-intervals + 1.

E.g. the formulas on the formula sheet use 2 function values for 1 sub-interval.

[fun_jirachi]

Pretty sure Simpson's Rule uses three function values, while trapezoidal uses two. Reason being from the reference sheet for simpsons rule you have f(a), f(b) and f((a+b)/2). THis is for one application.

In general, trapezoidal rule uses n function values for n-1 applications while Simpson's rule does the same for n-2 applications.

Hope this helps

[emily_p]

Pretty sure Simpson's Rule uses three function values, while trapezoidal uses two. Reason being from the reference sheet for simpsons rule you have f(a), f(b) and f((a+b)/2). THis is for one application.

In general, trapezoidal rule uses n function values for n-1 applications while Simpson's rule does the same for n-2 applications.

Hope this helps

Ah yep that definitely helps. So just clarifying if one application of Simpson’s is 2 strips/3 f values, does that mean two applications is 4 strips/5 f values, three applications is 6 strips etc? I’m not sure if the parabolic arcs overlap.

[RuiAce]

Ah yep that definitely helps. So just clarifying if one application of Simpson’s is 2 strips/3 f values, does that mean two applications is 4 strips/5 f values, three applications is 6 strips etc? I’m not sure if the parabolic arcs overlap.

For Simpson's rule, yes you always increment by 2 at a time. So for two applications of Simpson's rule, as you correctly stated you'd require 5 function values.

[emily_p]

For Simpson's rule, yes you always increment by 2 at a time. So for two applications of Simpson's rule, as you correctly stated you'd require 5 function values.

Ohhh right got it, thanks Rui!! 

[fun_jirachi]

For Simpson's rule, yes you always increment by 2 at a time. So for two applications of Simpson's rule, as you correctly stated you'd require 5 function values.

Oops, my mistake. Was meant to write for n applications you need 2n+1 function values, but oh well.