Mathematics Question Thread

[Sukakadonkadonk]

Hey,

Does anyone know if its possible to integrate ln(x) function?
Or if only way is to convert to exponential and have it restricted with y-axis and do it from there?

Thanks.

[jamonwindeyer]

Hey,

Does anyone know if its possible to integrate ln(x) function?
Or if only way is to convert to exponential and have it restricted with y-axis and do it from there?

Thanks.

Interesting question! You can do it, but for real numbers the integral is only defined for \(x>0\) as \(x\ln{x}-x\). You can prove that because differentiating this yields:



Once you try and do it for negative \(x\) values, you need to start using imaginary numbers. It is still doable though, and indeed, the form above works! But you need to do it in the complex plane, not the real plane

Warning: Non assessable content for this course

[Sukakadonkadonk]

Interesting question! You can do it, but for real numbers the integral is only defined for \(x>0\) as \(x\ln{x}-x\). You can prove that because differentiating this yields:



Once you try and do it for negative \(x\) values, you need to start using imaginary numbers. It is still doable though, and indeed, the form above works! But you need to do it in the complex plane, not the real plane

Warning: Non assessable content for this course

Ok thank you!
This was probably the wrong thread to post in 

[Sukakadonkadonk]

Hey guys,

Another question I have,

Write the sum 5 + 9 + 13 + ... 4n-3
in sigma notation.

Could someone please explain? There was none in the solutions.

Thanks.

[cxmplete]

how would i do question 4 parts c and d?

[jakesilove]

Hey guys,

Another question I have,

Write the sum 5 + 9 + 13 + ... 4n-3
in sigma notation.

Could someone please explain? There was none in the solutions.

Thanks.

Hey! All this is asking for is to write each term of the series in terms of an integer, n. So, we can write the series



as



In sigma notation, this looks like



Does that make sense?


Edit by Rui: can't use n as the summation index. Switched with k

[RuiAce]

Hey,

Does anyone know if its possible to integrate ln(x) function?
Or if only way is to convert to exponential and have it restricted with y-axis and do it from there?

Thanks.

There are actually many functions out there that are integrable. The only thing is that you aren't taught integration techniques in the HSC unless you do 4U, hence what Jamon means by not in this course.

Hey guys,

Another question I have,

Write the sum 5 + 9 + 13 + ... 4n-3
in sigma notation.

Could someone please explain? There was none in the solutions.

Thanks.

[jakesilove]

how would i do question 4 parts c and d?

For c), we just sketch the exponential graph. You can do this by plotting points, or by shifting the regular exponential graph. We know the graph has been shifted UP 60, will be a negative exponential and goes through 100 when t=0. So,



To calculate the distance, we need to integrate the function. We know that



So,



We can say that, at time t=0. x=0 (as the car is just passing A). So,


 

Now, subbing in 3 (three hours)

[cxmplete]

so im doing question 6, and im really confused on how to do it

[jamonwindeyer]

so im doing question 6, and im really confused on how to do it

Sure! Just a note, this question type is super common, so make sure you understand it! Probably worth 5 marks in your HSC this stuff

So we have this equation, \(N=Ae^{kt}\). Now we know that when \(t=0\), there were 124 million users. So we can substitute:



So now our equation is \(N=124000000e^{kt}\). Remember, the letters besides \(t\) are constants, so once we've found them, that is their value for the rest of the question. We also know that at the start of 2009, when \(t=8\), there were 220 million. So:



That's to four decimal places. So now we know all our constants!



To find the internet users at the start of 2012, that's \(t=11\), so:



That is to the nearest whole then finally, just solve for \(N=300\text{ million}\):



So we'd expect it to exceed 300 million somewhere in the 13th year, so somewhere in 2013!

Again, super important question type, so be sure to let me know if this doesn't make sense!

[Kekemato_BAP]

Hi! Can someone help me on Question(ii) in the attached pic? How do I find $M based on what is given? Thanks!!

[jamonwindeyer]

Hi! Can someone help me on Question(ii) in the attached pic? How do I find $M based on what is given? Thanks!!

Hey Kekemato! What was your expression for \(A_{60}\)? You should be able to generalise it to the life of the loan and put it equal to zero. That is, instead of creating an expression for \(A_{60}\), craft a similar one of \(A_{180}\), and solve \(A_{180}=0\)

You should have something like:



You can use the formula for a sum of a geometric series to tidy that series up is this looking familiar based on your work in (i)?

[Sukakadonkadonk]

Hey! All this is asking for is to write each term of the series in terms of an integer, n. So, we can write the series



as



In sigma notation, this looks like



Does that make sense?


Edit by Rui: can't use n as the summation index. Switched with k

Hey,

So are you guys saying that we need to our own intuition to figure what 'n' is in that series?

[RuiAce]

Hey,

So are you guys saying that we need to our own intuition to figure what 'n' is in that series?

Deleted my post because I realised you were asking about n

Write our the formula first. In Jake's example it was 4k-3

Then, it's not hard to just look at the first and last terms to deduce the lower and upper boundaries.

Lower: 5 = 4k-3 so k = 2
Upper: 4n-3 = 4k+3 so k = n

[Kekemato_BAP]

Hey,

Does anyone know if its possible to integrate ln(x) function?
Or if only way is to convert to exponential and have it restricted with y-axis and do it from there?

Thanks.

It is possible although not in the 2-U syllabus. Intergration by Parts.
(I don't know how to use the fancy symbols, someone teach me please)
lnx = 1*lnx
Use the acronym LIATE(logs, inverse trig, algrebraic, trig, exponential) to find u and v'. The first order will become u.
u would equal lnx since lnx is a log.
v' would be 1 since it is algebraic
u' is 1/x
v is x
(integration)(uv')=uv-(integration)(vu')
= lnx*x - (integration)(x*1/x)
= xlnx- (integration)(1)
= xlnx - x
(Don't forget +C for every step too)
Anyways, that's a 3-4U topic and 2U don't need to know it. I never did 3-4U but was taught by this 3U kid who dropped into my 2U class.