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2004 Paper 2, Q9 Optimisation differentiation

Kara

Posted 04 May 2006 - 03:27 PM

Hi im doing an optimisation question (Q9 2004 calculator peper) I've manages part A which gives the volume of the cuboid equal to: V(x)= 2/3x(6 - x^2)

Part B is to find the exact value of x for which this volume is a maximum.

I understand i have to differentiate, set equal to zero to find stationary points, draw a max/min graph to find where x is at its maximum but im struggling with the differentiation, im getting funny numbers.

I would be really grateful for any help.

Nathan

Posted 04 May 2006 - 04:03 PM

when i differentiate, i get 4-2x[^2]

factorised to 2(x[^2] - 4) , therefore getting roots of plus or minus sqrt.gif 2

for my nature table, i had m max tp at -sqrt.gif2 where y= 0.77 and max tp at sqrt.gif2 where y= 4.47

i dont have my answers handy though, so i'm not sure if that's right :S

Kara

Posted 04 May 2006 - 04:14 PM

the answer is the sqaure root of 2 but im still not sure how to get that.

ooooh i get it now i think, was just the differentiation part that i was confussed with. thanks.

Pringles

Posted 04 May 2006 - 05:51 PM

This how I done it


V(x) = \frac{2}{3}x(6 - {x^2}) \\         
        V(x) = 4x - \frac{2}{3}{x^3} \\
 \dfrac{dV}{dx} =  4 - 2{x^2} = 0 \textrm{ for stationary points}\\ 
             4 = {2x^2} \\
             x = \pm \sqrt{2}

Nathan

Posted 04 May 2006 - 09:11 PM

yup, that's right, and then you need to do a nature table to see which is the maximum...making the answer sqrt.gif2

Pringles

Posted 04 May 2006 - 09:25 PM

There is another way of finding the nature that you will find out in if you take Advanced Higher Maths. You differentiate \frac {dV}{dx} then put in the stationary values. If the value you find is >0 then the point is a minimum point if the value you find is <0 then the point is a maxminum

Nathan

Posted 04 May 2006 - 09:56 PM

so you differentiate the derivative?

The Wedge Effect

Posted 04 May 2006 - 10:01 PM

Yeah, second order differentiation, f''(x)\,. or 

\begin{align*} \frac {d^2 y}{dx^2} \end{align*}

which you really don't need to know for Highers.

ad absurdum

Posted 04 May 2006 - 11:42 PM

But it's okay to use that to show the nature of the stationary point in a higher exam, right?

/me really hates drawing nature tables

Dave

Posted 04 May 2006 - 11:50 PM

erm i dont know if it says show or prove with a nature table then obviously doing anything else will get no marks but i guess it wont be a problem since it is factually accurate. However it should be noted that sometimes you can do 2nd derivative and the answer will be zero in which case you have to use a nature table anyway

Pringles

Posted 05 May 2006 - 04:04 PM

QUOTE
However it should be noted that sometimes you can do 2nd derivative and the answer will be zero in which case you have to use a nature table anyway


Yeah should have mentioned that. Im not sure if using this method would get you no marks but to be sure I would use a nature table

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