VCE Specialist 3/4 Question Thread!

[soccerboi]

If it helps, then answer is [((20root10)pi)/3]-(2pi)/3

Thanks heaps

[b^3]

Should be fixed now.

[soccerboi]

z=2cis(pi/3)
Find the smallest possible integer n for which zⁿ is a real number

I let sin[(n X pi)/3]=0 and solved to get n=0 , but the answer should be n=3.
What have i done wrong?

[kamil9876]

The actual correct answer is that no such smallest integer exists. I'm pretty sure the question should've been "smallest POSITIVE integer", then n=3 is the correct answer, if it was "smallest non-negative integer" then you are right

[soccerboi]

My bad, its suppose to be "smallest positive integer" not "smallest possible integer." Could you show me why it's n=3?

[rife168]

Could you show me why it's n=3?

You had the right idea with
from that, we get

   for



to get the answer of   we just used

to find the smallest positive integer solution, we just use   which gives

[rife168]

For differentiation, we have a definition from first principles, that is



Is there some sort of definition/expression like this, but for integration?
I have come across the definition regarding the definite integral as the limit of an infinite sum, but unlike the first principle definition for differentiation, this didn't seem to actually help in directly calculating an integral... Like, it makes intuitive sense and I understand it, I just don't really see how you can plug values in and arrive at the indefinite integral in the way that you can with the above expression.

What is the connection between
and the actual method of computation of an integral - that is, doing the opposite of what we do for differentiation.




I hope that's clear...

[brightsky]

The fundamental theorem of calculus is what links antidifferentiation to area calculation. Basically, the theorem has two parts, the first of which shows that A'(x) = f(x), where A(x) is the area between a and x, where a<x<b. The second part of the theorem proves that A(b) = F(b) - F(a), where F(x) is the antiderivative of f(x). This is the 'formula' we use nowadays to calculate areas under curves.

It is important to understand that the Riemann sum definition does not relate area calculation to calculus at all. It is due to the fundamental theorem of calculus that the formula Methods students often take for granted holds.

[kamil9876]

What is the connection between \int^b_a{f(x)}dx=\lim_{n \to \infty}\sum_{i=1}^{n} f(x_{i}^{*})\Delta x
and the actual method of computation of an integral - that is, doing the opposite of what we do for differentiation.

You're right, the definition of a (Riemann) Integral is (sort of) that what you wrote down (i.e a limit, but there are in fact some more technical conditions). Computationally it's easier to use Fundamental Theorem of Calculus in many occasions. It's just like with differentiation, you don't always differentiate from first principles but mostly with product rule, chain rule etc. (which you do prove from the limit definition, just as you can prove FTC from the Riemann Sum limit definition).

[rife168]

What is the connection between \int^b_a{f(x)}dx=\lim_{n \to \infty}\sum_{i=1}^{n} f(x_{i}^{*})\Delta x
and the actual method of computation of an integral - that is, doing the opposite of what we do for differentiation.

You're right, the definition of a (Riemann) Integral is (sort of) that what you wrote down (i.e a limit, but there are in fact some more technical conditions). Computationally it's easier to use Fundamental Theorem of Calculus in many occasions. It's just like with differentiation, you don't always differentiate from first principles but mostly with product rule, chain rule etc. (which you do prove from the limit definition, just as you can prove FTC from the Riemann Sum limit definition).

Thanks for answering guys, I didn't realise the distinction between the Riemann sum and the FTC.
Do you know where I could find a proof of what I highlighted? I think actually seeing that could definitely clear things up.

Also, do you guys have a preferred texts or online resources that go into more depth about this stuff? I think I have a copy of Stewart's Calculus on my external HDD for what it's worth.

[brightsky]

Wikipedia's always good.

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

[rife168]

Wikipedia's always good.

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

haha thanks.

And so the endless night of Wikipedia link-clicking begins.....

Last time this happened I started on Taylor series and ended up reading about the Soviet Space program with about 140 tabs open.. -.-

Also could you or kamil perhaps have a look at my other few questions here?

[Shiney94]

If anyone could help me with this question would be great!

√3z² + √2z - i/2 = 0
Find the solutions to this equation

[rife168]

If anyone could help me with this question would be great!

√3z² + √2z - i/2 = 0
Find the solutions to this equation

It could get a bit messy because the LaTex server seems to be down, so I grabbed this from WolframAlpha:
link







Feel free to ask any further questions.

[pi]

Or just chuck it all into the quadratic formula