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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: polky on September 25, 2008, 03:42:04 pm

Title: Something about eulers?
Post by: polky on September 25, 2008, 03:42:04 pm: Hey guys, I need help with this question. Found on the Insight 2006 exam 2 multichoice!

Given $\frac{dy}{dx} = \sqrt{sin(2x)}$ and $y=\sqrt{2}$ when $x=\frac{\pi}{12}$ .

The value of y when $x=\frac{\pi}{3}$ is:

A) 0.2500
B) 0.7298
C) 0.9306
D) 1.4369
E) 2.1440

I tried using my Euler's program on this but I ended up with E) as the answer. D) is the correct answer.

Thanks for your help!
Title: Re: Something about eulers?
Post by: Mao on September 25, 2008, 03:56:11 pm: this question didn't specify euler's (i.e. didn't give step-size). I'd try using the fnInt approach:

$y = \int_{\pi / 12}^{\pi / 3} \frac{dy}{dx}\; dx + \sqrt{2} \approx 2.144$ WHAT?

EDIT: found the problem, it was a typo. they meant $y=\frac{1}{\sqrt{2}}$
Title: Re: Something about eulers?
Post by: shinny on September 25, 2008, 03:57:11 pm: Rather than approximating, find the exact value using the fundamental theorem of calculus (I think that's what this one was called at least).

$\frac{dy}{dx}=\sqrt{sin(2x)}$
$y(\frac{\pi}{3})-y(\frac{\pi}{12})=\int^\frac{\pi}{3}_\frac{\pi}{12} \sqrt{sin(2x)}dx$
$y(\frac{\pi}{3})=\int^\frac{\pi}{3}_\frac{\pi}{12} \sqrt{sin(2x)}dx+y(\frac{\pi}{12})$
$y(\frac{\pi}{3})=\int^\frac{\pi}{3}_\frac{\pi}{12} \sqrt{sin(2x)}dx+\sqrt{2}$
Then solve that integral on your calculator and you're done.

EDIT: Ok Mao already did it...I didn't try evaluating it before and just did and its E o_O What the. Guess Insight is wrong?
Title: Re: Something about eulers?
Post by: Mao on September 25, 2008, 04:00:05 pm: Writing by request from polky

When you solve a differential equation $\frac{dy}{dx}=f(x)$

$\implies y=F(x)+c$ , where c is an arbitrary constant.

if you are given the initial conditions $y=y_0,\; x=x_0$

$\implies y_0 = F(x_0) + c$

$\implies c=y_0 - F(x_0)$

$\implies y=F(x)+y_0 - F(x_0)$

$\implies y-y_0 = F(x)-F(x_0)$

remembering the Fundamental Theorem of Calculus (II) that $\int_a^b f(x)\; dx = F(b)-F(a)$

$\implies y-y_0 = \int_{x_0}^x f(t)\; dt$ (I used "t" here to avoid confusion with x)

this is where shinjusuzx got his second line from.

so in this case, you can numerically integrate the derivative function from the initial state (given) and a specific end point.

$y_1=\int_{x_0}^{x_1}\frac{dy}{dx}\; dx + y_0$
Title: Re: Something about eulers?
Post by: shinny on September 25, 2008, 04:09:37 pm: Yeh sorry for not explaining all of that...was trying to avoid it if i didn't have to since I'm bloody slow at latex =P This is assumed knowledge yes? Because for some reason, ALOT of people have never done this method of obtaining numerical answers before. I only learned of this through the project SAC where I had a question involving deriving that from scratch as Mao just did, but yeh, I've never seen it in any textbook =\ (Not that I really read textbooks anyhow though).
Title: Re: Something about eulers?
Post by: Mao on September 25, 2008, 04:12:41 pm: Quote from: shinjitsuzx on September 25, 2008, 04:09:37 pm
Yeh sorry for not explaining all of that...was trying to avoid it if i didn't have to since I'm bloody slow at latex =P This is assumed knowledge yes? Because for some reason, ALOT of people have never done this method of obtaining numerical answers before. I only learned of this through the project SAC where I had a question involving deriving that from scratch as Mao just did, but yeh, I've never seen it in any textbook =\ (Not that I really read textbooks anyhow though).

it is covered in the essentials text.

basically, you have told the calculator to use Euler's method (fnInt uses Euler's method) for many many intervals to give practical accurate answers.
Title: Re: Something about eulers?
Post by: polky on September 25, 2008, 04:22:31 pm: Quote from: shinjitsuzx on September 25, 2008, 04:09:37 pm
Yeh sorry for not explaining all of that...was trying to avoid it if i didn't have to since I'm bloody slow at latex =P This is assumed knowledge yes? Because for some reason, ALOT of people have never done this method of obtaining numerical answers before. I only learned of this through the project SAC where I had a question involving deriving that from scratch as Mao just did, but yeh, I've never seen it in any textbook =\ (Not that I really read textbooks anyhow though).

I had no idea of this Fundamental Theorem of Calculus (II) till just now. Mao has actually tried to explain this to me before, but I didn't get it then and I still don't get it. I'm pretty hopeless :P

I do have the Essentials text, just that I never bothered to look through the explanations and stuff.

Thanks shinjitsuzx and Mao for your help! :D
Title: Re: Something about eulers?
Post by: Mao on September 25, 2008, 04:45:57 pm: YOU DIDNT DO IT IN METHODS?!!!

for reference purposes and the fact that polky's msn does not accept inking, http://vcenotes.com/forum/index.php?topic=5668